A Functional (Monadic) Second-Order Theory of Infinite Trees

TL;DR

This paper provides a complete axiomatization of Monadic Second-Order Logic over infinite trees, enabling formal proof of Rabin's Tree Theorem through a polynomial-time recognizable deduction system.

Contribution

It introduces Functional (Monadic) Second-Order Logic (FSO), extending MSO with higher abstraction to facilitate axiomatization and proof of key decidability results.

Findings

Complete axiomatization of MSO over infinite trees

Formal proof of Rabin's Tree Theorem

Extension of MSO to FSO for higher abstraction

Abstract

This paper presents a complete axiomatization of Monadic Second-Order Logic (MSO) over infinite trees. MSO on infinite trees is a rich system, and its decidability ("Rabin's Tree Theorem") is one of the most powerful known results concerning the decidability of logics. By a complete axiomatization we mean a complete deduction system with a polynomial-time recognizable set of axioms. By naive enumeration of formal derivations, this formally gives a proof of Rabin's Tree Theorem. The deduction system consists of the usual rules for second-order logic seen as two-sorted first-order logic, together with the natural adaptation In addition, it contains an axiom scheme expressing the (positional) determinacy of certain parity games. The main difficulty resides in the limited expressive power of the language of MSO. We actually devise an extension of MSO, called Functional (Monadic) Second-Order Logic (FSO), which allows us to uniformly manipulate (hereditarily) finite sets and corresponding labeled trees, and whose language allows for higher abstraction than that of MSO.

Tables

Table 1. Table 1. Operations on Automata.

Table 2. Table 2. Mutual Interpretability of 𝖥𝖲𝖮 𝖥𝖲𝖮 \mathsf{FSO} and 𝖬𝖲𝖮 𝖬𝖲𝖮 \mathsf{MSO} (§ 3.6 ).

Equations

$$t::=x|\dot{\epsilon }|{\mathsf{S}}_d(t)$$

$$\phi ,\psi \in {\mathbf{\Lambda }}_{\mathcal{D}}::=X(t)|t\doteq u|t\dot{<}u|(\phi \vee \psi )|\neg \phi |(\exists x)\phi |(\exists X)\phi$$

$$\begin{array}[]{r !{~~\deq~~} l !{\qquad} r !{~~\deq~~} l}(\forall x)\varphi\lx@intercol\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrel{:=}\leavevmode\nobreak\ \leavevmode\nobreak\ &\lnot(\exists x)(\lnot\varphi)&\varphi\land\psi\lx@intercol\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrel{:=}\leavevmode\nobreak\ \leavevmode\nobreak\ &\lnot(\lnot\varphi\lor\lnot\psi)\\ (\forall X)\varphi\lx@intercol\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrel{:=}\leavevmode\nobreak\ \leavevmode\nobreak\ &\lnot(\exists X)(\lnot\varphi)&\varphi\Rightarrow\psi\lx@intercol\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrel{:=}\leavevmode\nobreak\ \leavevmode\nobreak\ &\lnot\varphi\lor\psi\\ \top\lx@intercol\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrel{:=}\leavevmode\nobreak\ \leavevmode\nobreak\ &(\forall x)(x\mathrel{\doteq}x)&\bot\lx@intercol\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrel{:=}\leavevmode\nobreak\ \leavevmode\nobreak\ &\lnot\top\\ (t\mathrel{\dot{\leq}}u)\lx@intercol\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrel{:=}\leavevmode\nobreak\ \leavevmode\nobreak\ &(t\mathrel{\dot{<}}u)\lor(t\mathrel{\doteq}u)\end{array}$$

$$(\forall x)(x\mathrel{\doteq}x)\qquad\text{and}\qquad(\forall x)(\forall y)\big{(}x\mathrel{\doteq}y\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \varphi[x/z]\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \varphi[y/z]\big{)}$$

$$(\exists X)(\forall y)\big{[}X(y)\leavevmode\nobreak\ \leavevmode\nobreak\ \Longleftrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \varphi\big{]}$$

$$(\forall X)\left(X(\dot{\varepsilon})\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \mathord{\mathop{\bigwedge}}_{d\in\mathscr{D}}(\forall y)\big{[}X(y)\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ X(\mathsf{S}_{d}(y))\big{]}\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ (\forall y)X(y)\right)$$

$$\begin{array}[]{c !{\qquad} c !{\qquad} c}\dfrac{}{\Phi\vdash\varphi\lor\lnot\varphi}&\dfrac{}{\Phi,\varphi\vdash\varphi}&\dfrac{\Phi\vdash\varphi\qquad\Phi\vdash\lnot\varphi}{\Phi\vdash\psi}\\ \\ \dfrac{\Phi\vdash\varphi}{\Phi\vdash\varphi\lor\psi}&\dfrac{\Phi\vdash\psi}{\Phi\vdash\varphi\lor\psi}&\dfrac{\Phi\vdash\varphi\lor\psi\qquad\Phi,\varphi\vdash\vartheta\qquad\Phi,\psi\vdash\vartheta}{\Phi\vdash\vartheta}\\[10.00002pt] \end{array}$$

$$\begin{array}[]{c}\dfrac{\Phi\vdash\varphi[t/x]}{\Phi\vdash(\exists x)\varphi}\qquad\qquad\dfrac{\Phi\vdash(\exists x)\varphi\qquad\Phi,\varphi\vdash\psi}{\Phi\vdash\psi}\leavevmode\nobreak\ \text{($x$ not free in $\Phi,\psi$)}\\ \\ \dfrac{\Phi\vdash\varphi[Y/X]}{\Phi\vdash(\exists X)\varphi}\qquad\qquad\dfrac{\Phi\vdash(\exists X)\varphi\qquad\Phi,\varphi\vdash\psi}{\Phi\vdash\psi}\leavevmode\nobreak\ \text{($X$ not free in $\Phi,\psi$)}\\[10.00002pt] \end{array}$$

$$\Phi ,\phi ⊢\psi \text{iff}\Phi ⊢\phi \Rightarrow \psi$$

$$\frac{\Phi ⊢\psi \Rightarrow \phi \Phi ⊢\psi }{\Phi ⊢\phi }$$

$$\begin{array}[]{c !{\qquad\quad} c}\lnot(\exists x)\mathop{\bigvee}_{d\neq d^{\prime}}\big{(}\mathsf{S}_{d}(x)\mathrel{\doteq}\mathsf{S}_{d^{\prime}}(x)\big{)}&(\forall x)(\forall y)\mathop{\bigwedge}_{d\in\mathscr{D}}\big{(}\mathsf{S}_{d}(x)\mathrel{\doteq}\mathsf{S}_{d}(y)\leavevmode\nobreak\ \leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ x\mathrel{\doteq}y\big{)}\\ \\ \lnot(\exists x)\big{(}x\mathrel{\dot{<}}x\big{)}&(\forall x)(\forall y)(\forall z)\big{(}x\mathrel{\dot{<}}y\leavevmode\nobreak\ \leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ y\mathrel{\dot{<}}z\leavevmode\nobreak\ \leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ x\mathrel{\dot{<}}z\big{)}\\ \\ (\forall x)\big{(}\dot{\varepsilon}\mathrel{\dot{\leq}}x\big{)}&(\forall x)(\forall y)\Big{(}\big{(}\mathop{\bigvee}_{d\in\mathscr{D}}x\mathrel{\dot{<}}\mathsf{S}_{d}(y)\big{)}\leavevmode\nobreak\ \leavevmode\nobreak\ \Leftrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ x\mathrel{\dot{\leq}}y\Big{)}\end{array}$$

$$T:{\mathcal{D}}^{\ast }⟶\Sigma$$

$$x\in X_i\text{iff}T(x)={\mathtt{a}}_i$$

$$F:{\mathcal{D}}^{\ast }⟶\Sigma$$

$$V_{\omega }:=\bigcup _{n\in \mathbb{N}}{V}_n$$

$$F(t)\doteq L$$

$$(\forall x)(\exists k\in K)\phi (x,k)\text{leavevmode}\text{leavevmode}⟹\text{leavevmode}\text{leavevmode}(\exists F:{\mathcal{D}}^{\ast }\to K)(\forall x)\phi (x,F(x))$$

$$K,L::=k|\dot{\kappa }|{\dot{\mathsf{g}}}_{n,m}(L_1,\dots ,L_n)$$

$$\begin{array}[]{r r l}\varphi,\psi&\mathrel{::=}&t\mathrel{\doteq}u\quad|\quad t\mathrel{\dot{<}}u\\ &\quad|&K\mathrel{\doteq}L\quad|\quad K\mathrel{\dot{\in}}L\quad|\quad K\mathrel{\dot{\subseteq}}L\quad|\quad F(t)\mathrel{\doteq}K\\ &\quad|&\varphi\lor\psi\quad|\quad\lnot\varphi\\ &\quad|&(\exists x)\varphi\quad|\quad(\exists F:K)\varphi\quad|\quad(\exists k\mathrel{\dot{\in}}L)\varphi\quad|\quad(\exists k\mathrel{\dot{\subseteq}}L)\varphi\end{array}$$

$$\begin{array}[]{r !{~~\deq~~} l !{\qquad} r !{~~\deq~~} l}(\forall x)\varphi\lx@intercol\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrel{:=}\leavevmode\nobreak\ \leavevmode\nobreak\ &\lnot(\exists x)(\lnot\varphi)&\varphi\land\psi\lx@intercol\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrel{:=}\leavevmode\nobreak\ \leavevmode\nobreak\ &\lnot(\lnot\varphi\lor\lnot\psi)\\ (\forall F:L)\varphi\lx@intercol\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrel{:=}\leavevmode\nobreak\ \leavevmode\nobreak\ &\lnot(\exists F:L)(\lnot\varphi)&\varphi\Rightarrow\psi\lx@intercol\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrel{:=}\leavevmode\nobreak\ \leavevmode\nobreak\ &\lnot\varphi\lor\psi\\ (\forall k\mathrel{\star}L)\varphi\lx@intercol\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrel{:=}\leavevmode\nobreak\ \leavevmode\nobreak\ &\lnot(\exists k\mathrel{\star}L)(\lnot\varphi)&(t\mathrel{\dot{\leq}}u)\lx@intercol\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrel{:=}\leavevmode\nobreak\ \leavevmode\nobreak\ &(t\mathrel{\dot{<}}u)\lor(t\mathrel{\doteq}u)\\ \top\lx@intercol\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrel{:=}\leavevmode\nobreak\ \leavevmode\nobreak\ &(\forall x)(x\mathrel{\doteq}x)&\bot\lx@intercol\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrel{:=}\leavevmode\nobreak\ \leavevmode\nobreak\ &\lnot\top\end{array}$$

$$(F:K):=(\forall x)(\exists k\dot{\in }K)(F(x)\doteq k)$$

$$\mathbf{K}\star \mathbf{L}=(K_1,\dots ,K_n)\star (L_1,\dots ,L_n):={\bigwedge }_{1\le i\le n}{K}_i\star L_i$$

$$\begin{array}[]{c}\dfrac{\Phi\vdash\varphi[t/x]}{\Phi\vdash(\exists x)\varphi}\qquad\qquad\dfrac{\Phi\vdash(\exists x)\varphi\qquad\Phi,\varphi\vdash\psi}{\Phi\vdash\psi}\leavevmode\nobreak\ \leavevmode\nobreak\ \text{($x$ not free in $\Phi,\psi$)}\\ \\ \dfrac{\Phi\vdash\varphi[G/F]\qquad\Phi\vdash(G:K)}{\Phi\vdash(\exists F:K)\varphi}\quad\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \,\dfrac{\Phi\vdash(\exists F:K)\varphi\qquad\Phi,\,(F:K),\,\varphi\vdash\psi}{\Phi\vdash\psi}\leavevmode\nobreak\ \text{($F$ not free in $\Phi,\psi$)}\\ \\ \dfrac{\Phi\vdash\varphi[K/k]\qquad\Phi\vdash K\mathrel{\star}L}{\Phi\vdash(\exists k\mathrel{\star}L)\varphi}\leavevmode\nobreak\ \leavevmode\nobreak\ \text{(for $\mathrel{\star}$ either $\mathrel{\dot{\in}}$ or $\mathrel{\dot{\subseteq}}$)}\\ \\ \dfrac{\Phi\vdash(\exists k\mathrel{\star}K)\varphi\qquad\Phi,\,k\mathrel{\star}K,\,\varphi\vdash\psi}{\Phi\vdash\psi}\leavevmode\nobreak\ \leavevmode\nobreak\ \text{(for $\mathrel{\star}$ either $\mathrel{\dot{\in}}$ or $\mathrel{\dot{\subseteq}}$, and $k$ not free in $\Phi,\psi$)}\\ \\ \dfrac{\Phi\vdash\varphi}{\Phi[F(t)/k]\vdash\varphi[F(t)/k]}\leavevmode\nobreak\ \leavevmode\nobreak\ \text{($\Phi[F(t)/k],\varphi[F(t)/k]$ $\mathsf{FSO}$-formulae)}\\[10.00002pt] \end{array}$$

$$\begin{array}[]{l l r c}\hline\cr\hline\cr\lx@intercol\textbf{Language}\hfil\lx@intercol\\ \hline\cr\text{Individual Terms}&t\quad\mathrel{::=}\quad x\quad|\quad\dot{\varepsilon}\quad|\quad\mathsf{S}_{d}(t)&\text{($d\in\mathscr{D}$)}&\text{(\S\ref{sec:mso:lgge})}\\ \text{Functions}&F,G,H,\text{etc}&\text{(only variables)}&\text{(\S\ref{sec:fso:fso})}\\ \text{$\mathrm{HF}$-Terms}&K,L\quad\mathrel{::=}\quad k&\text{($k$ $\mathrm{HF}$-variable)}&\text{(\S\ref{sec:fso:fso})}\\ &\quad|\quad\dot{\mathsf{\kappa}}&\text{($\kappa\in V_{\omega}$)}\\ &\quad|\quad\dot{\mathsf{g}}_{n,m}(L_{1},\dots,L_{n})&\text{($\dot{\mathsf{g}}_{n,m}$ $\mathrm{HF}$-Function)}\\ \\ \text{Formulae}&\varphi,\psi\quad\mathrel{::=}&&\text{(\S\ref{sec:fso:fso})}\\ &\lx@intercol\quad|\quad t\mathrel{\doteq}u\quad|\quad t\mathrel{\dot{<}}u\quad|\quad F(t)\mathrel{\doteq}K\hfil\lx@intercol\\ &\lx@intercol\quad|\quad K\mathrel{\doteq}L\quad|\quad K\mathrel{\dot{\in}}L\quad|\quad K\mathrel{\dot{\subseteq}}L\hfil\lx@intercol\\ &\lx@intercol\quad|\quad\varphi\lor\psi\quad|\quad\lnot\varphi\hfil\lx@intercol\\ &\lx@intercol\quad|\quad(\exists x)\varphi\quad|\quad(\exists F:K)\varphi\quad|\quad(\exists k\mathrel{\dot{\in}}L)\varphi\quad|\quad(\exists k\mathrel{\dot{\subseteq}}L)\varphi\hfil\lx@intercol\par\par\\ \\ \hline\cr\hline\cr\lx@intercol\textbf{Axioms}\hfil\lx@intercol\\ \hline\cr\lx@intercol\text{\emph{Equality:}}\hfil\lx@intercol&&\text{(\S\ref{sec:ax:eq})}\\ \lx@intercol\hfil\begin{array}[]{c !{\qquad} c}(\forall x)(x\mathrel{\doteq}x)&(\forall x)(\forall y)\big{(}x\mathrel{\doteq}y\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \varphi[x/z]\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \varphi[y/z]\big{)}\\ \\ K\mathrel{\doteq}K&\left(K\mathrel{\doteq}L\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \varphi[K/m]\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \varphi[L/m]\right)\end{array}\hfil\lx@intercol\\ \\ \lx@intercol\text{\emph{Induction:}}\hfil\lx@intercol&&\text{(\S\ref{sec:ax:ind})}\\ \lx@intercol\hfil\varphi(\dot{\varepsilon})\quad\Longrightarrow\quad\mathord{\mathop{\bigwedge}}_{d\in\mathscr{D}}(\forall x)\big{[}\varphi(x)\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \varphi(\mathsf{S}_{d}(x))\big{]}\quad\Longrightarrow\quad(\forall x)\varphi(x)\hfil\lx@intercol\\ \\ \lx@intercol\text{\emph{Tree Axioms:}}\hfil\lx@intercol&&\text{(\S\ref{sec:ax:tree})}\\ \lx@intercol\hfil\begin{array}[]{c !{\quad} c}\lnot(\exists x)\mathop{\bigvee}_{d\neq d^{\prime}}\big{(}\mathsf{S}_{d}(x)\mathrel{\doteq}\mathsf{S}_{d^{\prime}}(x)\big{)}&(\forall x)(\forall y)\mathop{\bigwedge}_{d\in\mathscr{D}}\big{(}\mathsf{S}_{d}(x)\mathrel{\doteq}\mathsf{S}_{d}(y)\leavevmode\nobreak\ \leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ x\mathrel{\doteq}y\big{)}\\ \\ \lnot(\exists x)\big{(}x\mathrel{\dot{<}}x\big{)}&(\forall x)(\forall y)(\forall z)\big{(}x\mathrel{\dot{<}}y\leavevmode\nobreak\ \leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ y\mathrel{\dot{<}}z\leavevmode\nobreak\ \leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ x\mathrel{\dot{<}}z\big{)}\\ \\ (\forall x)\big{(}\dot{\varepsilon}\mathrel{\dot{\leq}}x\big{)}&(\forall x)(\forall y)\Big{(}\big{(}\mathop{\bigvee}_{d\in\mathscr{D}}x\mathrel{\dot{<}}\mathsf{S}_{d}(y)\big{)}\leavevmode\nobreak\ \leavevmode\nobreak\ \Leftrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ x\mathrel{\dot{\leq}}y\Big{)}\end{array}\hfil\lx@intercol\\ \\ \lx@intercol\text{\emph{Axioms on $\mathrm{HF}$-Sets:}}\hfil\lx@intercol&&\text{(\S\ref{sec:ax:hf})}\\ &\lx@intercol\varphi_{n,m}[\bm{K}/\bm{k}][\dot{\mathsf{g}}_{n,m}(\bm{K})/\ell]\hfil\lx@intercol\\ &\lx@intercol\hfil\text{(provided\quad$\mathbf{Sk}(\mathsf{ZFC}^{-})\vdash(\forall k_{1},\dots,k_{n})(\exists!\ell)\varphi_{n,m}$)}\lx@intercol\\ \\ \lx@intercol\text{\emph{Functional Choice Axioms:}}\hfil\lx@intercol&&\text{(\S\ref{sec:ax:choice})}\\ \lx@intercol\hfil\begin{array}[]{r !{\quad\longlimp\quad} l}(\forall k\mathrel{\dot{\in}}K)(\exists\ell\mathrel{\dot{\in}}L)\varphi(k,\ell)\lx@intercol\quad\Longrightarrow&\big{(}\exists f\mathrel{\dot{\in}}L^{K}\big{)}(\forall k\mathrel{\dot{\in}}K)\varphi(k,f(k))\\[10.00002pt] (\forall x)(\exists k\mathrel{\dot{\in}}K)\varphi(x,k)\lx@intercol\quad\Longrightarrow&(\exists F:K)(\forall x)(\exists k\mathrel{\dot{\in}}K)\left(F(x)\mathrel{\doteq}k\leavevmode\nobreak\ \land\leavevmode\nobreak\ \varphi(x,k)\right)\\[10.00002pt] (\forall k\mathrel{\dot{\in}}K)(\exists F:L)\varphi(k,F)\lx@intercol\quad\Longrightarrow&\big{(}\exists G:L^{K}\big{)}(\forall k\mathrel{\dot{\in}}K)\varphi(k,F)[G(k)\sslash F]\end{array}\hfil\lx@intercol\par\\ \\ \hline\cr\hline\cr\end{array}$$

$$(\forall x)(x\mathrel{\doteq}x)\qquad\text{and}\qquad(\forall x)(\forall y)\big{(}x\mathrel{\doteq}y\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \varphi[x/z]\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \varphi[y/z]\big{)}$$

$$K\doteq K\text{and}\left(K\doteq L\text{leavevmode}\text{leavevmode}⟹\text{leavevmode}\text{leavevmode}\phi [K/m]\text{leavevmode}\text{leavevmode}⟹\text{leavevmode}\text{leavevmode}\phi [L/m]\right)$$

$$\frac{\Phi ⊢\phi }{\Phi [F(t)/k]⊢\phi [F(t)/k]}$$

$$(F(t)\doteq K)\text{leavevmode}\text{leavevmode}⟹\text{leavevmode}\text{leavevmode}\phi (K)\text{leavevmode}\text{leavevmode}⟹\text{leavevmode}\text{leavevmode}\phi (F(t))$$

$$\frac{\Phi ⊢\phi (F(t))\Phi ⊢(F:K)}{\Phi ⊢(\exists k\dot{\in }K)\phi (k)}$$

$$(k\dot{\in }K)\text{leavevmode}\text{leavevmode}⟹\text{leavevmode}\text{leavevmode}\phi (k)\text{leavevmode}\text{leavevmode}⟹\text{leavevmode}\text{leavevmode}(\exists k\dot{\in }K)\phi (k)$$

Full text

By a complete axiomatization we mean a complete deduction system with a polynomial-time recognizable set of axioms. By naive enumeration of formal derivations, this formally gives a proof of Rabin’s Tree Theorem. The deduction system consists of the usual rules for second-order logic seen as two-sorted first-order logic, together with the natural adaptation to infinite trees of the axioms of $\mathsf{MSO}$ on $\omega$-words. In addition, it contains an axiom scheme expressing the (positional) determinacy of certain parity games.

The main difficulty resides in the limited expressive power of the language of $\mathsf{MSO}$. We actually devise an extension of $\mathsf{MSO}$, called Functional (Monadic) Second-Order Logic ($\mathsf{FSO}$), which allows us to uniformly manipulate (hereditarily) finite sets and corresponding labeled trees, and whose language allows for higher abstraction than that of $\mathsf{MSO}$.

1. Introduction

This paper presents a complete axiomatization of Monadic Second-Order Logic ($\mathsf{MSO}$) over infinite trees. $\mathsf{MSO}$ on infinite trees is a rich system which contains non trivial mathematical theories (see e.g. [Rab69, BGG97]) and subsumes many logics, including modal logics (see e.g. [BdRV02]) and logics for verification (see e.g. [VW08]). Rabin’s Tree Theorem [Rab69], the decidability of $\mathsf{MSO}$ on infinite trees, is one of the most powerful known results concerning the decidability of logics (see e.g. [BGG97]).

The original decidability proof of [Rab69] relied on an effective translation of formulae to finite state automata running on infinite trees. Since then, there has been considerable work on Rabin’s Tree Theorem, culminating in streamlined decidability proofs, as presented e.g. in  [Tho97, GTW02, PP04]. Most current approaches to $\mathsf{MSO}$ on infinite trees (with the notable exception of [Blu13]) are based on translations of formulae to automata.

By a ‘complete axiomatization’ we mean a complete deduction system with a polynomial-time recognizable set of axioms and rules. This condition on axiom/rule recognizability is typical in proof theory, where it is known as the Cook-Reckhow criterion [CR79]. The point is that proofs should be ‘easily checkable’, which rules out axiomatizations based on enumerations of all true formulae. In this way, a complete axiomatization not only constitutes an alternative demonstration of Rabin’s Tree theorem itself, by naive enumeration of formal derivations, but also yields a meaningful notion of ‘proof certificate’ for theorems.

Our deduction system consists of the usual rules for second-order logic seen as two-sorted first-order logic (see e.g. [Rib12]), together with the natural adaptation to infinite trees of the axioms of $\mathsf{MSO}$ on $\omega$-words [Sie70]. In addition, it contains an axiom scheme expressing the (positional) determinacy of certain parity games.

We continue a line of work begun by Büchi and Siefkes, who gave axiomatizations of $\mathsf{MSO}$ on various classes of linear orders (see e.g. [Sie70, BS73]), as well as an axiomatization of Weak $\mathsf{MSO}$ ($\mathsf{WMSO}$) over infinite trees [Sie78] ($\mathsf{WMSO}$ is $\mathsf{MSO}$ with set quantifications restricted to finite sets). These works essentially rely on formalizations of automata in the logic. A major result in the axiomatic treatment of logics over infinite structures is Walukiewicz’s proof of completeness of Kozen’s axiomatization of the modal $\mu$-calculus [Wal00] (see also [AL17] for an alternative recent proof of this result). Another trend relies on model-theoretic techniques. For instance [tCF10, GtC12] give complete axiomatizations of $\mathsf{MSO}$ and the modal $\mu$-calculus over finite trees; a reworking of the completeness of $\mathsf{MSO}$ on $\omega$-words [Sie70] is proposed in [Rib12]; and [SV10] gives a model-theoretic completeness proof for a fragment of the modal $\mu$-calculus. An attractive feature of model-theoretic completeness proofs for the aforementioned logics is that they allow elegant reformulations of algebraic approaches to these logics. Unfortunately, in the case of $\mathsf{MSO}$ over infinite trees, the only known algebraic approach [Blu13] seems too complex to be easily formalized. We therefore directly formalize a translation of formulae to automata in the axiomatic theory.

Mirroring usual automata based decidability proofs (see e.g. [Tho97, GTW02, PP04]), our method for proving completeness proceeds in two steps. We first formalize a translation of $\mathsf{MSO}$-formulae to tree automata (using the positional determinacy of parity games to prove the complementation lemma), so that each closed formula is provably equivalent to an automaton over the singleton alphabet. The second (and much shorter) step is a variant of the Büchi-Landweber Theorem [BL69] which states that $\mathsf{MSO}$ decides winning for (definable) games of finite graphs, and which is obtained thanks to the completeness of $\mathsf{MSO}$ over $\omega$-words.

The main expositional difficulty resides in the limited expressive power of the language of $\mathsf{MSO}$. To ameliorate this we actually devise an extension of $\mathsf{MSO}$, called Functional (Monadic) Second-Order Logic ($\mathsf{FSO}$), allowing uniform manipulation of (hereditarily) finite sets and corresponding labeled infinite trees. We intuitively see $\mathsf{FSO}$ as providing a language for higher abstraction than that of $\mathsf{MSO}$, allowing a uniform formalization of automata and games which would have been difficult to write down in $\mathsf{MSO}$. However, since $\mathsf{FSO}$ is interpretable in $\mathsf{MSO}$ (as we show), its language has the same intrinsic limitations as the language of $\mathsf{MSO}$. In particular it suffers from the inexpressibility of choice over tree positions [GS83, CL07], and so predicates such as length comparison of tree positions are not expressible in $\mathsf{FSO}$. This implies that only positional strategies (w.r.t. our specific notion of acceptance games), are expressible in $\mathsf{FSO}$ and moreover that usually unproblematic reasoning on infinite plays can become cumbersome in this setting.

There are several ways to translate $\mathsf{MSO}$ to tree automata. We choose to translate formulae to alternating parity automata, following [Wal02]. The two non-trivial steps in the translation are negation and (existential) quantification. Negation requires the complementation of automata, relying on the determinacy of acceptance games, while existential quantifiers require us to simulate an alternating automaton by an equivalent non-deterministic one (this is the Simulation Theorem [EJ91, MS95]), thence obtaining an automaton computing the appropriate projection.

As usual with translations of $\mathsf{MSO}$ to tree automata, we rely on McNaughton’s Theorem [McN66] (see also e.g. [Tho90, PP04]), stating that non-deterministic Büchi automata on $\omega$-words are effectively equivalent to deterministic parity (or Muller, Rabin, Streett) automata on $\omega$-words. In translations of $\mathsf{MSO}$ to alternating tree automata, McNaughton’s Theorem is usually invoked for the Simulation Theorem.111The approach of [MS95] to the Simulation Theorem actually contains a proof of McNaughton Theorem, but we do not see how to easily formalize it in our context. In our context, the relevant instances of McNaughton’s Theorem are imported into $\mathsf{FSO}$ via the completeness of $\mathsf{MSO}$ on $\omega$-words [Sie70].

It is well-known that the $\mathsf{MSO}$ theory of $k$-ary trees can be embedded in that of the binary tree [Rab69]. However, it does not seem that such an embedding yields an axiomatization of $k$-ary trees from an axiomatization of the binary tree. Therefore, in this work, we axiomatize the $\mathsf{MSO}$ theory of the full infinite $\mathscr{D}$-ary tree for an arbitrary non-empty finite set $\mathscr{D}$.

This paper is a corrected version of [DR15], which contains a flaw in the positional determinacy argument (Thm. VI.15). In the present paper, we augment the systems $\mathsf{FSO}$ and $\mathsf{MSO}$ with an axiom expressing the positional determinacy of parity games, thereby obtaining complete axiomatizations. We do not know yet whether the theory $\mathsf{MSO}$ of [DR15] is complete, but let us mention that the axiomatization of $\mathsf{WMSO}$ over infinite trees given in [Sie78] augments the natural analogue for trees of Peano’s arithmetic with an axiom of induction over finite trees.

Outline.

The paper is organized as follows. We present the basic formal theory for $\mathsf{MSO}$ in §2 Our theory $\mathsf{FSO}$ is then presented in §3 and we sketch its mutual interpretability with $\mathsf{MSO}$. §4 and §5 discuss a formalization of two-players infinite games in $\mathsf{FSO}$, and, in particular, we give a formulation of the axiom $(\mathit{PosDet})$ of positional determinacy of parity games. This provides us with the required tools to formalize in §6 (alternating) tree automata, acceptance games and basic operations on them (including complementation in $\mathsf{FSO}+(\mathit{PosDet})$). §7 is an interlude discussing a complete theory of $\mathsf{MSO}$ over $\omega$-words within the infinite paths of $\mathsf{FSO}$. Building on §6 and §7, we then give our completeness argument for $\mathsf{FSO}+(\mathit{PosDet})$ and $\mathsf{MSO}+(\mathit{PosDet})$ in §8. Finally, §9 contains a proof of the Simulation Theorem in $\mathsf{FSO}$. The full version [DR20] gives details of proofs omitted here.

2. Preliminaries: $\mathsf{MSO}$ on Infinite Trees as a Second-Order Logic

We present here a basic formal theory of Monadic Second-Order Logic ($\mathsf{MSO}$) over infinite trees. This theory can be seen as an analogue for trees of Peano’s axioms for second order arithmetic. In order to obtain a complete theory, $\mathsf{MSO}$ will be augmented with an axiom of positional determinacy of parity games (see §3.6, §5.6 and §8).

We are going to define the theory ${\mathsf{MSO}_{\mathscr{D}}}$ of the infinite full $\mathscr{D}$-ary tree $\mathscr{D}^{*}$, for $\mathscr{D}$ a finite non-empty set. Both the language and the axioms of ${\mathsf{MSO}_{\mathscr{D}}}$ will depend on $\mathscr{D}$. The language of ${\mathsf{MSO}_{\mathscr{D}}}$ is the usual language of two-sorted first-order logic, with one sort for Individuals and one sort for (Monadic) Predicates. The axioms of ${\mathsf{MSO}_{\mathscr{D}}}$ are the expected axioms on the relational structure of the full $\mathscr{D}$-ary tree, together with induction and comprehension. The theory ${\mathsf{MSO}_{\mathscr{D}}}$ is essentially that of [Sie78], but with second-order quantifications intended to range over arbitrary subsets of $\mathscr{D}^{*}$ (instead of just finite ones), and without the axiom of induction over finite trees.

We fix for the rest of this Section a finite non-empty set $\mathscr{D}$ of tree directions.

2.1. The Language of ${\mathsf{MSO}_{\mathscr{D}}}$

The language of ${\mathsf{MSO}_{\mathscr{D}}}$ has two sorts:

• •

The sort of Individuals, intended to range over tree positions $p\in\mathscr{D}^{*}$. We have infinitely many Individual variables $x,y,z$ etc. We also have one constant symbol $\dot{\varepsilon}$ (for the root of $\mathscr{D}^{*}$), and one unary function symbol $\mathsf{S}_{d}$ for each $d\in\mathscr{D}$ (for the successor function $p\mapsto p.d$). Individual terms, written $t,u,\text{etc}.$ are given by:

[TABLE]

• •

The sort of (Monadic) Predicates, with variables $X,Y,Z,\text{etc}$, intended to range over sets of tree positions $A\in\mathcal{P}(\mathscr{D}^{*})$. There are no other term formers for this sort.

Formulae of ${\mathsf{MSO}_{\mathscr{D}}}$ are given by the following grammar:

[TABLE]

where $t$ and $u$ are Individual terms. We use the usual derived formulae:

[TABLE]

We employ usual writing conventions for formulae, for instance omitting internal and external brackets when appropriate.

2.2. The Deduction System of ${\mathsf{MSO}_{\mathscr{D}}}$

Deduction for ${\mathsf{MSO}_{\mathscr{D}}}$ is defined by the system presented in Figure 1 and Figure 2 (where $\Phi$ stands for a multiset of formulae), together with the following axioms.

• •

Equality on Individuals:

[TABLE]

• •

The Tree Axioms of Figure 3.

• •

Comprehension Scheme:

[TABLE]

• •

Induction Axiom:

[TABLE]

Remark 1**.**

As usual, one can derive $\vdash(\varphi\Rightarrow\psi\Rightarrow\vartheta)\leavevmode\nobreak\ \Leftrightarrow\leavevmode\nobreak\ ((\varphi\land\psi)\Rightarrow\vartheta)$ and we have the Deduction Theorem:

[TABLE]

Indeed, if $\Phi,\varphi\vdash\psi$, then one gets $\Phi\vdash\lnot\varphi\lor\psi$ by $\lor$-Elimination on the Excluded Middle $\Phi\vdash\varphi\lor\lnot\varphi$. Conversely, if $\Phi\vdash\lnot\varphi\lor\psi$, then one gets $\Phi,\varphi\vdash\psi$ by $\lor$-Elimination. One similarly obtains the Modus Ponens as a derived rule

[TABLE]

{nota}

Henceforth, we write $\mathsf{MSO}$ instead of ${\mathsf{MSO}_{\mathscr{D}}}$ when the set of directions $\mathscr{D}$ is clear from the context.

3. A Functional Extension of $\mathsf{MSO}$ on Infinite Trees

In this Section, we present (bounded) Functional (Monadic) Second-Order Logic over the full $\mathscr{D}$-ary tree (${\mathsf{FSO}_{\mathscr{D}}}$), an extension of ${\mathsf{MSO}_{\mathscr{D}}}$ with (hereditarily) finite sets and bounded quantification over them. As with ${\mathsf{MSO}_{\mathscr{D}}}$ in §2, we will simply write $\mathsf{FSO}$ for ${\mathsf{FSO}_{\mathscr{D}}}$ when $\mathscr{D}$ is irrelevant or clear from the context.

${\mathsf{FSO}_{\mathscr{D}}}$ is equipped with a basic axiomatization which will allow us, in §4-§6, to formalize a basic theory of games and automata, and in particular to state an axiom scheme $(\mathit{PosDet})$ expressing the positional determinacy of (suitably represented) parity games (§5.6). We will then show in §8 that ${\mathsf{FSO}_{\mathscr{D}}}+(\mathit{PosDet})$ is complete.

3.1. Motivations and Overview.

Let us first discuss the motivations and guiding principles in the design of ${\mathsf{FSO}_{\mathscr{D}}}$. As usual, within the language of ${\mathsf{MSO}_{\mathscr{D}}}$ presented in §2, we can simulate a labeling of $\mathscr{D}^{*}$ over a finite non-empty set $\Sigma$

[TABLE]

There are different ways to achieve this. A possibility is, for say $\Sigma=\{\mathtt{a}_{1},\dots,\mathtt{a}_{n}\}$, to code $T:\mathscr{D}^{*}\to\Sigma$ using a tuple of Monadic variables $X_{1},\dots,X_{n}$ such that

[TABLE]

A more succinct coding could be obtained using $\lceil\log n\rceil$ monadic variables to encode the letter index $i$ of $\mathtt{a}_{i}$ in binary. However, directly working with such codings would make it cumbersome to formalize games and automata as presented in this paper. We will therefore rather work in the system ${\mathsf{FSO}_{\mathscr{D}}}$, which is built around the following principles:

1. (1)

${\mathsf{FSO}_{\mathscr{D}}}$ has no primitive notion of Monadic variables. Instead, ${\mathsf{FSO}_{\mathscr{D}}}$ has a primitive notion of Function variables, of the form

[TABLE] 2. (2)

In addition, ${\mathsf{FSO}_{\mathscr{D}}}$ allows us to work uniformly with arbitrary finite sets. In particular, we have an explicit sort for them, including terms, variables and quantifications. 3. (3)

${\mathsf{FSO}_{\mathscr{D}}}$ is faithfully interpretable in ${\mathsf{MSO}_{\mathscr{D}}}$. To this end, all quantifications over finite sets in ${\mathsf{FSO}_{\mathscr{D}}}$-formulae are required to be bounded.

In particular, there is a syntactic translation $\langle-\rangle$ of ${\mathsf{FSO}_{\mathscr{D}}}$-formulae to ${\mathsf{MSO}_{\mathscr{D}}}$-formulae. The basic idea of this translation is to interpret finite sets using propositional logic, and to interpret Functions $F:\mathscr{D}^{*}\to\{\mathtt{a}_{1},\dots,\mathtt{a}_{n}\}$ as partitions $X_{1},\dots,X_{n}$ of $\mathscr{D}^{*}$. But while ${\mathsf{FSO}_{\mathscr{D}}}$ handles free variables over finite sets in a uniform way, the translation $\langle-\rangle$ only applies to ${\mathsf{FSO}_{\mathscr{D}}}$-formulae without free variables over finite sets. This means that for an ${\mathsf{FSO}_{\mathscr{D}}}$-formula $\varphi(k)$ with $k$ a variable over finite sets, for each finite set $\kappa$ we will have a specific ${\mathsf{MSO}_{\mathscr{D}}}$-formula $\langle\varphi(\kappa)\rangle$.

Technically, the finite sets of ${\mathsf{FSO}_{\mathscr{D}}}$ will be the usual hereditarily finite sets.

{defi}

Let $V_{0}\mathrel{:=}\varnothing$, and $V_{n+1}\mathrel{:=}\mathcal{P}(V_{n})$ for each $n\in\mathbb{N}$. The set $V_{\omega}$ of hereditarily finite sets (HF-sets) is defined as

[TABLE]

Remark 2**.**

In the context of this paper, it is useful to note that, as is well-known (see e.g. [Jec06, Exercise 12.9]), $V_{\omega}$ is a model of $\mathsf{ZFC}^{-}$ (i.e. of $\mathsf{ZFC}$ without the infinity axiom).

{conv}

We will always assume the finite non-empty set $\mathscr{D}$ of tree directions to be an $\mathrm{HF}$-set.

The language of ${\mathsf{FSO}_{\mathscr{D}}}$ will have the same sort of Individuals as ${\mathsf{MSO}_{\mathscr{D}}}$ and a sort for $\mathrm{HF}$-sets, and its Function variables will be of the form $F:\mathscr{D}^{*}\to K$ for $K$ a term over $\mathrm{HF}$-sets ($\mathrm{HF}$-term). The design of ${\mathsf{FSO}_{\mathscr{D}}}$ is obtained as a compromise between the following two conflicting desiderata:

1. (1)

To be as flexible as possible to allow an easy formalization of games and automata. 2. (2)

To be as simple as possible to allow an easy translation to ${\mathsf{MSO}_{\mathscr{D}}}$.

This leads us to two peculiar design choices.

1. (1)

We have, in addition to the above mentioned sorts, a distinct sort of Functions over $\mathrm{HF}$-sets. This sort contains only constants (so these functions cannot be quantified over), whose purpose is to provide Skolem functions for those $\forall\exists$ (bounded) statements over $\mathrm{HF}$-sets which are provable in $\mathsf{ZFC}^{-}$. 2. (2)

In order to facilitate the translation of ${\mathsf{FSO}_{\mathscr{D}}}$ to ${\mathsf{MSO}_{\mathscr{D}}}$, Function variables, written $(F:K)$ (“$F$ has codomain $K$”), cannot occur in $\mathrm{HF}$-terms. Formally, Functions $(F:K)$ are only allowed in atomic formulae of the form

[TABLE]

The axioms of ${\mathsf{FSO}_{\mathscr{D}}}$ will contain the obvious adaptation of the Tree Axioms and the Induction Axiom of ${\mathsf{MSO}_{\mathscr{D}}}$. We also have axioms defining the aforementioned Skolem functions. In addition, the Comprehension Scheme of ${\mathsf{MSO}_{\mathscr{D}}}$ will be replaced by Functional Choice Axioms allowing us to define Functions $F:\mathscr{D}^{*}\to K$ from $\forall\exists$-statements:

[TABLE]

Remark 3**.**

Functional Choice Axioms* as above actually amount to Comprehension in $\mathsf{MSO}$ (§2.2). Such axioms do not create choice predicates for Individuals, which are known to be undefinable in $\mathsf{MSO}$ [GS83, CL07], and moreover to break decidability when added to the language of $\mathsf{MSO}$ [BG00, CL07].*

The rest of this Section is organized as follows. The system ${\mathsf{FSO}_{\mathscr{D}}}$ is defined in §3.2-3.4, and its (expected) interpretation in the standard model of $\mathscr{D}$-ary trees is given in §3.5. Then in §3.6 we discuss the interpretation of ${\mathsf{FSO}_{\mathscr{D}}}$ in ${\mathsf{MSO}_{\mathscr{D}}}$ and a straightforward embedding of ${\mathsf{MSO}_{\mathscr{D}}}$ in ${\mathsf{FSO}_{\mathscr{D}}}$. Finally, §3.7 presents notation whose purpose is to allow some flexibility in the manipulation of functions. The language and axioms of ${\mathsf{FSO}_{\mathscr{D}}}$ are summarized in Figure 5, with references to the relevant parts of the text.

3.2. The Language of ${\mathsf{FSO}_{\mathscr{D}}}$

We now formally define the language of ${\mathsf{FSO}_{\mathscr{D}}}$, for $\mathscr{D}$ an $\mathrm{HF}$-set. It consists of the the following sorts:

• •

The sort of Hereditarily finite ($\mathrm{HF}$) sets, with infinitely many $\mathrm{HF}$-variables $k,\ell$ etc., and with one constant symbol $\dot{\mathsf{\kappa}}$ for each $\kappa\in V_{\omega}$ (we often simply write $\kappa$ for $\dot{\mathsf{\kappa}}$ in formulae, omitting the overset dot).

• •

The same sort of Individuals as ${\mathsf{MSO}_{\mathscr{D}}}$ (see §2.1).

• •

The sort of Functions, with infinitely many variables $F,G,H,\text{etc}$.

• •

The sort of $\mathrm{HF}$-Functions, with no variable. For each pair $(n,m)\in\mathbb{N}\times\mathbb{N}$, we assume given a constant symbol $\dot{\mathsf{g}}_{n,m}$ of arity $n$. The interpretation of these constant symbols is discussed in §3.4.4.

The language of ${\mathsf{FSO}_{\mathscr{D}}}$ has two kinds of terms. The Individual terms are the same as those of ${\mathsf{MSO}_{\mathscr{D}}}$. In addition, ${\mathsf{FSO}_{\mathscr{D}}}$ also has $\mathrm{HF}$-terms, which are given by

[TABLE]

The formulae of ${\mathsf{FSO}_{\mathscr{D}}}$ are built as follows:

[TABLE]

An ${\mathsf{FSO}_{\mathscr{D}}}$-formula $\varphi$ is $\mathrm{HF}$-closed if it contains no free $\mathrm{HF}$-variable.

{nota}

1. (1)

Usual derived formulae are defined similarly as with $\mathsf{MSO}$ (where $\mathrel{\star}$ is either $\mathrel{\dot{\in}}$ or $\mathrel{\dot{\subseteq}}$):

[TABLE] 2. (2)

In addition to bounded quantification $(\exists F:K)$, we use the notation $(F:K)$ within formulae as the defined formula:

[TABLE] 3. (3)

For variables $\bm{K}=K_{1},\dots,K_{n}$ and $\bm{L}=L_{1},\dots,L_{n}$, and $\mathrel{\star}$ either $\mathrel{\doteq}$, $\mathrel{\dot{\in}}$ or $\mathrel{\dot{\subseteq}}$, we let

[TABLE]

Remark 4**.**

The (hereditarily) finite set $\mathscr{D}$ of tree directions is considered both as a parameter in the definition of ${\mathsf{FSO}_{\mathscr{D}}}$, via the successor term constructors $\mathsf{S}_{d}$ (for $d\in\mathscr{D}$) and the corresponding axioms (see §3.4), and as a (constant) $\mathrm{HF}$ set, which can occur as such in ${\mathsf{FSO}_{\mathscr{D}}}$ formulae. Strictly speaking, we should write $\dot{\mathsf{\mathscr{D}}}$ rather than $\mathscr{D}$ in the latter case, but we usually simply omit the overset dot, as with other HF-sets.

3.3. The Deduction System of ${\mathsf{FSO}_{\mathscr{D}}}$

Deduction for ${\mathsf{FSO}_{\mathscr{D}}}$ is defined by the system presented on Figure 1 (with ${\mathsf{FSO}_{\mathscr{D}}}$ formulae instead of ${\mathsf{MSO}_{\mathscr{D}}}$ formulae) and Figure 4, together with all the axioms of §3.4. The language and axioms of ${\mathsf{FSO}_{\mathscr{D}}}$ are summarized in Figure 5.

\opt

draftnotes

3.4. Basic Axiomatization

We now present the axioms of ${\mathsf{FSO}_{\mathscr{D}}}$. The first group (Equality, Tree Axioms and Induction, §3.4.1-§3.4.2) corresponds to its counterpart in ${\mathsf{MSO}_{\mathscr{D}}}$. We then present our specific axioms for $\mathrm{HF}$-Sets in §3.4.4 and our Functional Choice Axioms in §3.4.5.

3.4.1. Equality

The theory ${\mathsf{FSO}_{\mathscr{D}}}$ has usual equality axioms for individuals and $\mathrm{HF}$-Sets.

• •

Equality on Individuals.

[TABLE]

• •

Equality on $\mathrm{HF}$-Sets (for all formula $\varphi$, all $\mathrm{HF}$-terms $K,L$ and all $\mathrm{HF}$-variable $m$):

[TABLE]

Remark 5**.**

Note that ${\mathsf{FSO}_{\mathscr{D}}}$ is equipped with an explicit Substitution Rule

[TABLE]

Substitution entails the following (where $\varphi(F(t))$ is an $\mathsf{FSO}$-formula):

[TABLE]

as well as the derived rule

[TABLE]

The former is a direct consequence of the Substitution rule together with elimination of equality on $\mathrm{HF}$-Sets. For the latter, first note that Remark 1 also holds for ${\mathsf{FSO}_{\mathscr{D}}}$. In particular, one can derive

[TABLE]

On the other hand, we have

[TABLE]

We therefore get

[TABLE]

and the Substitution rule gives

[TABLE]

3.4.2. Induction

We have the following Induction Scheme:

[TABLE]

3.4.3. Tree Axioms

For the tree structure of $\mathscr{D}^{*}$, we have the same Tree Axioms as ${\mathsf{MSO}_{\mathscr{D}}}$, displayed in Figure 3 (recall that ${\mathsf{FSO}_{\mathscr{D}}}$ has the same Individuals as ${\mathsf{MSO}_{\mathscr{D}}}$).

We now state expected results on the axioms so far introduced. To this end, let ${\mathsf{FSO}_{\mathscr{D}}^{0}}$ be the system consisting of the deduction rules of Figures 1 and 4, together with the Equality Axioms (§3.4.1) the Induction Scheme (§3.4.2) and the Tree Axioms (Figure 3).

Proposition 6**.**

${\mathsf{FSO}_{\mathscr{D}}^{0}}$* proves the following.*

1. (1)

$(\forall x)(\forall y)(x\mathrel{\dot{\leq}}y\mathrel{\dot{\leq}}x\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ x\mathrel{\doteq}y)$** 2. (2)

$(\forall x)(x\mathrel{\dot{\leq}}\dot{\varepsilon}\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ x\mathrel{\doteq}\dot{\varepsilon})$** 3. (3)

$\lnot(\exists x)(x\mathrel{\dot{<}}\dot{\varepsilon})$** 4. (4)

$\lnot(\exists x)(\mathsf{S}_{d}(x)\mathrel{\doteq}\dot{\varepsilon})$** 5. (5)

$(\forall x)(x\mathrel{\doteq}\dot{\varepsilon}\leavevmode\nobreak\ \leavevmode\nobreak\ \lor\leavevmode\nobreak\ \leavevmode\nobreak\ (\exists y)\lor_{d\in\mathscr{D}}x\mathrel{\doteq}\mathsf{S}_{d}(y))$** 6. (6)

$(\forall x)(\forall y)(x\mathrel{\dot{<}}y\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \lor_{d\in\mathscr{D}}\mathsf{S}_{d}(x)\mathrel{\dot{\leq}}y)$**

A consequence of Proposition 6 is that the Induction Scheme of ${\mathsf{FSO}_{\mathscr{D}}}$ (§3.4.2) implies the usual scheme of Well-Founded Induction w.r.t. the strict prefix order $\mathrel{\dot{<}}$.

Theorem 7** (Well-Founded Induction).**

${\mathsf{FSO}_{\mathscr{D}}^{0}}$* proves the following form of well-founded induction:*

[TABLE]

Remark 8**.**

Both Proposition 6 and Theorem 7 also hold for ${\mathsf{MSO}_{\mathscr{D}}}$.

3.4.4. $\mathrm{HF}$-Sets

We now present our axioms on $\mathrm{HF}$-Sets. Their purpose is to ease formalization in ${\mathsf{FSO}_{\mathscr{D}}}$. Recall that $\mathrm{HF}$-sets range over $V_{\omega}$ (Definition 3.1). The idea of these axioms is to incorporate in ${\mathsf{FSO}_{\mathscr{D}}}$ as much of the theory of $V_{\omega}$ as possible, while keeping ${\mathsf{FSO}_{\mathscr{D}}}$ interpretable in ${\mathsf{MSO}_{\mathscr{D}}}$ and with a semi-recursive notion of provability. The interpretation of ${\mathsf{FSO}_{\mathscr{D}}}$ in ${\mathsf{MSO}_{\mathscr{D}}}$ relies on the fact that in ${\mathsf{FSO}_{\mathscr{D}}}$-formulae, all quantifications over $\mathrm{HF}$-Sets are bounded (either by $\mathrel{\dot{\in}}$ or $\mathrel{\dot{\subseteq}}$), so that in a closed ${\mathsf{FSO}_{\mathscr{D}}}$-formula, quantifications over $\mathrm{HF}$-Sets can be interpreted using usual propositional logic.

We will have, as particular cases of our axioms on $\mathrm{HF}$-Sets, all bounded formulae over $\mathrm{HF}$-Sets which are true in $V_{\omega}$. Moreover, w.r.t. the interpretation of ${\mathsf{FSO}_{\mathscr{D}}}$ in ${\mathsf{MSO}_{\mathscr{D}}}$ (§3.6) and in particular w.r.t. its application to ${\mathsf{MSO}_{\mathscr{D}}}$ over $\omega$-words (§7, §8 and §9), it is important to have sufficiently many functions over $V_{\omega}$ available within closed $\mathrm{HF}$-terms. This is the main purpose of our axioms on $\mathrm{HF}$-Sets. They state that the $\mathrm{HF}$-Functions $\dot{\mathsf{g}}_{n,m}$ are Skolem functions for $\forall\exists!$-statements over $\mathrm{HF}$-Sets. These axioms are further commented in §8.5.

{defi}

[$\mathrm{HF}$-Formula]

An $\mathrm{HF}$-formula is an ${\mathsf{FSO}_{\mathscr{D}}}$-formula with atoms of the form $K\mathrel{\doteq}L$, $K\mathrel{\dot{\in}}L$ or $K\mathrel{\dot{\subseteq}}L$ where $K$ and $L$ are $\mathrm{HF}$-terms.

Fix a distinguished $\mathrm{HF}$-variable $\ell$, and an enumeration $k_{1},k_{2},\dots$ of distinct $\mathrm{HF}$-variables all different from $\ell$. Furthermore, fix an enumeration $(\varphi_{n,m})_{n,m\in\mathbb{N}}$ of $\mathrm{HF}$-formulae satisfying the following conditions:

1. (1)

Each formula $\varphi_{n,m}$ has free variables among $k_{1},\dots,k_{n},\ell$. 2. (2)

All $\mathrm{HF}$-Functions occurring in $\varphi_{n,m}$ have the form $\dot{\mathsf{g}}_{n^{\prime},m^{\prime}}$ with $m^{\prime}<m$. 3. (3)

Each $\mathrm{HF}$-formula $\varphi$ satisfying (1) and (2) occurs infinitely often in $(\varphi_{n,m})_{n,m\in\mathbb{N}}$, in the following sense. If $\varphi$ has free variables among $k_{1},\dots,k_{n},\ell$, then there are infinitely many $m\in\mathbb{N}$ such that $\varphi$ is $\varphi_{n,m}$.

Recall from Remark 2 that $V_{\omega}$ is a model of $\mathsf{ZFC}^{-}$. The idea of our Axioms on $\mathrm{HF}$-Sets is that

[TABLE]

(where $(k\mathrel{\dot{\subseteq}}k^{\prime})$ is interpreted as $(\forall m\mathrel{\dot{\in}}k)(m\mathrel{\dot{\in}}k^{\prime})$). However, recall that $\varphi_{n,m}$ in (1) may contain $\mathrm{HF}$-Functions $\dot{\mathsf{g}}_{n^{\prime},m^{\prime}}$ with $m^{\prime}<m$. The premise of (1) can thus not be formulated in $\mathsf{ZFC}^{-}$, but requires a suitable extension of it. We let $\mathbf{Sk}(\mathsf{ZFC}^{-})$ consist of $\mathsf{ZFC}^{-}$ augmented with the axioms

[TABLE]

\opt

draftnotes

It is well-known that $\mathbf{Sk}(\mathsf{ZFC}^{-})$ is thus a conservative extension of $\mathsf{ZFC}^{-}$ (see e.g. [vD04, §3.4]). ${\mathsf{FSO}_{\mathscr{D}}}$ has the following axiom scheme for $\mathrm{HF}$-Sets, which simply consists of (1) formulated for $\mathbf{Sk}(\mathsf{ZFC}^{-})$ rather than $\mathsf{ZFC}^{-}$:

• •

Axioms on $\mathrm{HF}$-Sets. For each $n,m\in\mathbb{N}$ such that

[TABLE]

and for all $\mathrm{HF}$-terms $\bm{K}=K_{1},\dots,K_{n}$, we have the axiom

[TABLE]

Remark 9**.**

Note that this axiom scheme makes the axiom set of ${\mathsf{FSO}_{\mathscr{D}}}$ not recursive. But as expected for a proof system, provability in ${\mathsf{FSO}_{\mathscr{D}}}$ remains semi-recursive.

We fix here once and for all an interpretation of the $\mathrm{HF}$-Function symbols $\dot{\mathsf{g}}_{n,m}$ as functions over $V_{\omega}$. The idea is that if (2) holds, then $\dot{\mathsf{g}}_{n,m}$ is interpreted as a computable function $\mathtt{g}_{n,m}:V_{\omega}^{n}\to V_{\omega}$ such that

[TABLE]

But again recall that $\varphi_{n,m}$ may contain $\mathrm{HF}$-Functions $\dot{\mathsf{g}}_{n^{\prime},m^{\prime}}$ with $m^{\prime}<m$. We therefore proceed inductively, as follows.

{conv}

By induction on $m\in\mathbb{N}$, we interpret the $\mathrm{HF}$-Function symbols $\dot{\mathsf{g}}_{n,m}$ as computable functions

[TABLE]

Consider the formula $\varphi_{n,m}$, and assume that all $\mathrm{HF}$-Functions $\dot{\mathsf{g}}_{n^{\prime},m^{\prime}}$ with $m^{\prime}<m$ are already interpreted. If (2) holds then by the Countable Axiom of Choice we interpret $\dot{\mathsf{g}}_{n,m}$ as the unique function $\mathtt{g}_{n,m}:V^{n}_{\omega}\to V_{\omega}$ such that (3) holds. Note that such functions are computable. Otherwise, we interpret $\dot{\mathsf{g}}_{n,m}$ as the function with constant value $\varnothing$.

Remark 10**.**

Note each $\mathrm{HF}$-Function symbol is interpreted by a recursive function in Convention 3.4.4. However, since (2) is undecidable, there is no algorithm taking $(n,m)\in\mathbb{N}^{2}$ to the interpretation of $\dot{\mathsf{g}}_{n,m}$. This point is further discussed in §8.5, where a natural workaround is proposed, as well as some explanations for our present choice of Axioms on $\mathrm{HF}$-Sets.

We now discuss some consequences of these axioms. First note that if $\varphi$ is a closed $\mathrm{HF}$-formula, then it is provable in $\mathbf{Sk}(\mathsf{ZFC}^{-})$ if and only if it holds in $V_{\omega}$. We state this fact as a Remark for the record, and also to reiterate how much deductive power underlies the axioms on $\mathrm{HF}$-sets.

Remark 11**.**

Given a closed $\mathrm{HF}$-formula $\varphi$,

[TABLE]

\opt

draftnotes

Moreover, we have all instances of the following:

1. (1)

Extensionality.

[TABLE] 2. (2)

Finite sets. For each $n\in\mathbb{N}$ we have an $n$-ary $\mathrm{HF}$-Function symbol $\{-,\dots,-\}$ such that

[TABLE]

We have in particular singletons $\{-\}$ and unordered pairs $\{-,-\}$. Using Extensionality, ${\mathsf{FSO}_{\mathscr{D}}}$ proves that

[TABLE]

We use the following shorthand:

[TABLE] 3. (3)

Union. We have an $\mathrm{HF}$-Function symbol $\cup(-)$ such that

[TABLE] 4. (4)

Powerset.

We have an $\mathrm{HF}$-Function symbol $\mathcal{P}(-)$ such that

[TABLE]

The powerset is the reason for our introduction of inclusion ($\mathrel{\dot{\subseteq}}$) as an atomic formula: It is well-known that the powerset cannot be defined by a $\Delta_{0}(\mathrel{\dot{\in}})$-formula. A possible formula defining it is:

[TABLE]

The quantification $\forall\ell$ in the right conjunct is not $\mathrel{\dot{\in}}$-bounded, and cannot be so. In addition, we also have an $\mathrm{HF}$-Function symbol $\mathcal{P}_{*}(-)$ for the non-empty powerset, that is such that

[TABLE] 5. (5)

Comprehension. Given an $\mathrm{HF}$-formula $\varphi$ with free variables among $k_{1},\dots,k_{n},k$, we have an $\mathrm{HF}$-Function $\{k\mathrel{\dot{\in}}(-)\leavevmode\nobreak\ |\leavevmode\nobreak\ \varphi[-,\dots,-,k]\}$ such that

[TABLE] 6. (6)

Products.

We have a binary $\mathrm{HF}$-Function $(-)\times(-)$ such that

[TABLE]

Moreover, we have binary projections given by $\mathrm{HF}$-Functions $\pi_{1}^{-,-}(-)$ and $\pi_{2}^{-,-}(-)$ such that

[TABLE]

and similarly for $\pi_{2}^{-,-}(-)$. Whenever possible, we write $\pi_{i}(-)$ instead of $\pi_{i}^{k,\ell}(-)$. Note that by composing binary projections $\pi_{1}$ and $\pi_{2}$ we obtain projections

[TABLE]

for any $k_{1},\dots,k_{n}$ and $i\in\{1,\dots,n\}$ 7. (7)

Function Spaces and Application.

We have an exponent $\mathrm{HF}$-Function $(-)^{(-)}$ such that

[TABLE]

Moreover, function application is given by the $\mathrm{HF}$-Function $@_{-,-}(-,-)$ with

[TABLE]

(here $f$ and $a$ are $\mathrm{HF}$-variables). Whenever possible, we omit the subscripts $k,\ell$ of $@_{k,\ell}(f,a)$ and write simply $f(a)$ for $@_{k,\ell}(f,a)$. 8. (8)

Disjoint Unions.

We have a binary $\mathrm{HF}$-Function $(-)+(-)$ with

[TABLE]

(see Convention 5.5 for a further discussion on finite ordinals in our context). We moreover have $\mathrm{HF}$-Functions $\mathsf{in}^{k,\ell}_{k}(-)$, $\mathsf{in}^{k,\ell}_{\ell}(-)$, and $[-,-]_{k,\ell}^{\imath}$ such that (dropping subscripts and superscripts)

[TABLE]

for $m\mathrel{\dot{\in}}k$, $\ell\mathrel{\dot{\in}}n$ and $f\mathrel{\dot{\in}}\imath^{k}$, $g\mathrel{\dot{\in}}\imath^{\ell}$.

\opt

draftnotes

{conv}

Regarding function spaces as in 7 above, ${\mathsf{FSO}_{\mathscr{D}}}$ proves

[TABLE]

In the following, we reason modulo that bijection, and simply identify $(k^{\ell})^{m}$ with $k^{\ell\times m}$.

Remark 12**.**

An $\mathrm{HF}$-relation $\mathord{\preceq}\subseteq K\times K$ is a partial order on an $\mathrm{HF}$-term $K$, if the formula $\mathrm{PO}(\preceq,K)$ holds in $V_{\omega}$, where $\mathrm{PO}(\preceq,K)$ is the $\mathrm{HF}$-formula:

[TABLE]

A partial order $\mathord{\preceq}\subseteq K\times K$ is a well-order if every subset of $K$ has a $\preceq$-least element, that is, if the following formula $\mathrm{WO}(\preceq,K)$ holds in $V_{\omega}$:

[TABLE]

Since every $\mathrm{HF}$-set is finite and can be well-ordered, we have

[TABLE]

Since $\varphi(k)$ is an $\mathrm{HF}$-formula $\varphi(k)$, it follows that ${\mathsf{FSO}_{\mathscr{D}}}$ proves $\varphi(k)$, hence in particular that every $\mathrm{HF}$-set is well-ordered. ∎

3.4.5. Functional Choice Axioms

We have the following functional choice axiom schemes.

• •

$\mathrm{HF}$-Bounded Choice for $\mathrm{HF}$-Sets.

[TABLE]

• •

$\mathrm{HF}$-Bounded Choice for Functions.

[TABLE]

• •

Iterated $\mathrm{HF}$-Bounded Choice.

[TABLE]

where the substitution $[G(k)\sslash F]$ is defined as the usual substitution operation but with

[TABLE]

We insist that none of these axioms create choice functions for the individuals of ${\mathsf{FSO}_{\mathscr{D}}}$ (cf Remark 3). Despite their common shape, these three axiom schemes are actually of different nature. First, the axiom of $\mathrm{HF}$-Bounded Choice for Functions

[TABLE]

is a counterpart in ${\mathsf{FSO}_{\mathscr{D}}}$ of the Comprehension Scheme of ${\mathsf{MSO}_{\mathscr{D}}}$. Recalling the informal discussion in §3.1 and anticipating §3.5 and §3.6, let us assume a translation $\langle-\rangle$ from ($\mathrm{HF}$-closed) ${\mathsf{FSO}_{\mathscr{D}}}$-formulae to ${\mathsf{MSO}_{\mathscr{D}}}$-formulae, and let us assume that $K$ is a closed $\mathrm{HF}$-term representing the $\mathrm{HF}$-set $\{\kappa_{1},\dots,\kappa_{n}\}$. Then the premise of (4) can be read as

[TABLE]

The conclusion easily follows from the fact that using Comprehension, one can define in ${\mathsf{MSO}_{\mathscr{D}}}$ a partition $X_{1},\dots,X_{n}$ of $\mathscr{D}^{*}$ such that

[TABLE]

Second, $\mathrm{HF}$-Bounded Choice for $\mathrm{HF}$-Sets

[TABLE]

may look similar to the Axioms on $\mathrm{HF}$-Sets of §3.4.4. The differences are that the formula $\varphi$ here is an arbitrary formula of ${\mathsf{FSO}_{\mathscr{D}}}$, not necessarily an $\mathrm{HF}$-formula in the sense of Definition 3.4.4, and moreover that this axiom only involves ${\mathsf{FSO}_{\mathscr{D}}}$ (i.e. bounded) quantifications, contrary to (2). Note that for $\mathrm{HF}$-formulae $\varphi$, this axiom is indeed an instance of the axioms of §3.4.4. In the general case, this axiom can be seen as following from the fact that quantifications over $\mathrm{HF}$-Sets in ${\mathsf{FSO}_{\mathscr{D}}}$ are ultimately interpreted in propositional logic. Assume that the $\mathrm{HF}$-terms $K$ and $L$ are closed, and correspond to the $\mathrm{HF}$-sets $\kappa$ and $\lambda$ respectively. Then the premise of (5) can be read as

[TABLE]

which is equivalent in propositional logic to the interpretation of the conclusion

[TABLE]

Similarly, Iterated $\mathrm{HF}$-Bounded Choice reduces to an equivalence of the form

[TABLE]

and follows from Comprehension.

The definition of ${\mathsf{FSO}_{\mathscr{D}}}$ is now complete.

{nota}

Similarly as with ${\mathsf{MSO}_{\mathscr{D}}}$, we shall write $\mathsf{FSO}$ for ${\mathsf{FSO}_{\mathscr{D}}}$ when the set of tree directions $\mathscr{D}$ is clear from the context.

3.5. The Standard Model of $\mathsf{FSO}$

The standard model $\mathfrak{T}$ of ${\mathsf{FSO}_{\mathscr{D}}}$ is the full $\mathscr{D}$-ary tree $\mathscr{D}^{*}$ equipped with suitable domains for each sort:

• •

$\mathrm{HF}$-Sets range over $V_{\omega}$, and each constant $\dot{\mathsf{\kappa}}$ is interpreted by the corresponding $\mathrm{HF}$-set $\kappa\in V_{\omega}$.

• •

Individuals range over $\mathscr{D}^{*}$, the constant $\dot{\varepsilon}$ is interpreted by the empty sequence $\varepsilon\in\mathscr{D}^{*}$ and $\mathsf{S}_{d}$ as the map taking $p\in\mathscr{D}^{*}$ to $p.d\in\mathscr{D}^{*}$. Moreover, we write $<$ for the strict prefix order on $\mathscr{D}^{*}$.

• •

Functions range over

[TABLE]

• •

For each $n,m\in\mathbb{N}$, the $\mathrm{HF}$-Function $\dot{\mathsf{g}}_{n,m}$ (of arity $n$) is interpreted as the function

[TABLE]

fixed in Convention 3.4.4.

Remark 13**.**

*Note that $\mathfrak{T}$ has the same individuals as the standard model of ${\mathsf{MSO}_{\mathscr{D}}}$. Moreover we write $\mathfrak{T}$ for both the standard model of ${\mathsf{FSO}_{\mathscr{D}}}$ and that of ${\mathsf{MSO}_{\mathscr{D}}}$, as an abuse of notation. *

We have the usual interpretation $\llbracket t\rrbracket\in\mathscr{D}^{*}$ for each closed individual term $t$ with parameters in $\mathfrak{T}$, and an interpretation $\llbracket K\rrbracket\in V_{\omega}$ for each closed $\mathrm{HF}$-term $K$ with parameters in $\mathfrak{T}$. The relation $\mathfrak{T}\models\varphi$, for a closed $\mathsf{FSO}$-formula $\varphi$ with parameters in $\mathfrak{T}$, is defined by induction on $\varphi$ as follows:

[TABLE]

By a routine induction argument, we can show the soundness of ${\mathsf{FSO}_{\mathscr{D}}}$ w.r.t. $\mathfrak{T}$:

Proposition 14**.**

Given $\mathsf{FSO}$-formulae $\psi_{1},\dots,\psi_{n},\varphi$ with free $\mathrm{HF}$-variables among $\bm{k}$, free Individual variables among $\bm{x}$, and free Function variables among $\bm{F}$, if

[TABLE]

then for all $\mathrm{HF}$-Sets $\bm{\kappa}$, all $\bm{p}\in\mathscr{D}^{*}$ and all $\bm{\mathtt{F}}\in\bigcup_{\kappa\in V_{\omega}}\left(\mathscr{D}^{*}\to\kappa\right)$, we have

[TABLE]

Remark 15**.**

It follows from Remark 10 that the map $\llbracket-\rrbracket$ is not computable on $\mathrm{HF}$-terms. We refer to §8.5 for a discussion and a workaround.

3.6. Mutual Interpretability of $\mathsf{FSO}$ and $\mathsf{MSO}$

While $\mathsf{FSO}$ seems more expressive than $\mathsf{MSO}$ (and, indeed, is easier to work with), the two theories can mutually interpret each other via two formula-level translations:

[TABLE]

As we shall see, both translations preserve and reflect provability:

[TABLE]

The interpretation $(-)^{\circ}$ of $\mathsf{MSO}$ in $\mathsf{FSO}$ simply amounts to simulate the (Monadic) Predicate variables of $\mathsf{MSO}$ by $\mathsf{FSO}$-Function variables $\mathscr{D}^{*}\to 2$. We therefore see $(-)^{\circ}$ as an embedding, and see $\mathsf{FSO}$ as a conservative extension of $\mathsf{MSO}$ which is faithfully interpretable in $\mathsf{MSO}$. This property is not only a sanity check: we actually rely on it in our completeness argument (see Rem. 21). We discuss the translations $\langle-\rangle$ and $(-)^{\circ}$ separately in §3.6.1 and §3.6.2 below. Full proofs are detailed in [DR20, App. A].

3.6.1. From $\mathsf{FSO}$ to $\mathsf{MSO}$

The translation $\langle-\rangle:\mathsf{FSO}\to\mathsf{MSO}$ interprets the $\mathrm{HF}$-part of $\mathsf{FSO}$ using propositional logic. It is essentially straightforward, except for the case of Functions, which require some care. We will work with the following convention:

{conv}

We assume that each $\mathrm{HF}$-set $\kappa$ comes with a fixed enumeration $\bm{\kappa}=\kappa_{1},\dots,\kappa_{n}$ of its elements.

The translation $\langle-\rangle$ will map an $\mathrm{HF}$-closed $\mathsf{FSO}$-formula $\varphi$ without free Function variables to an $\mathsf{MSO}$-formula $\langle\varphi\rangle$. As stated earlier, quantifications over $\mathrm{HF}$-Sets will be interpreted using propositional logic. For instance we have,

[TABLE]

where $\llbracket K\rrbracket\in V_{\omega}$ is the standard interpretation of the closed $\mathrm{HF}$-term $K$ defined in §3.5. As a consequence, the translation $\langle-\rangle$ is non-uniform w.r.t. $\mathrm{HF}$-Sets. In particular, for an $\mathsf{FSO}$-formula $\varphi$ with free $\mathrm{HF}$-variables among $\bm{k}=k_{1},\dots,k_{p}$, each tuple of $\mathrm{HF}$-sets $\bm{\kappa}=\kappa_{1},\dots,\kappa_{p}$ will induce a specific $\mathsf{MSO}$-formula $\langle\varphi[\bm{\kappa}/\bm{k}]\rangle$.

The interpretation of Function variables is more complex. Consider a closed $\mathrm{HF}$-term $K$ and assume $\llbracket K\rrbracket=\{\kappa_{1},\dots,\kappa_{c}\}$. Then a Function $(F:K)$ can be seen as a function

[TABLE]

As indicated in §3.1, we interpret $F$ as a tuple $X_{1},\dots,X_{c}$ of Monadic variables such that

[TABLE]

In other words, $F:\mathscr{D}^{*}\to\{\kappa_{1},\dots,\kappa_{c}\}$ is seen as a partition $X_{1},\dots,X_{c}$ of $\mathscr{D}^{*}$. To handle the interpretation of Functions in the inductive definition of $\langle-\rangle$, it is actually convenient to temporarily work in an extension of $\mathsf{FSO}$ with the following atomic formulae:

• •

$\arrowvert X_{1}\dots X_{n}(t)\mathrel{\doteq}_{\bm{\kappa}}L\arrowvert$ where $\bm{\kappa}=\kappa_{1},\dots,\kappa_{n}$ enumerates an $\mathrm{HF}$-set and $X_{1},\dots,X_{n}$ are monadic variables of $\mathsf{MSO}$.

Extended $\mathsf{FSO}$-formulae are built just like $\mathsf{FSO}$-formulae, but possibly using the atomic formulae above. Extended atomic formulae are useful for dealing with $\mathrm{HF}$-bounded quantifications over Functions, say $(\exists F:K)\varphi$. The point is that $F$ occurs in subformulae of $\varphi$ of the form $F(t)\mathrel{\doteq}L$, where the $\mathrm{HF}$-term $L$ may contain free $\mathrm{HF}$-variables. Hence the value of $L$ is not known when the translation of $(\exists F:K)$ has to be computed. Extended atomic formulae allow us to delay the interpretation of $F(t)\mathrel{\doteq}L$ until $\llbracket L\rrbracket$ is known.

The interpretation of an extended $\mathrm{HF}$-closed $\mathsf{FSO}$-formula $\varphi$ without free Function variables is the $\mathsf{MSO}$-formula $\langle\varphi\rangle$ defined by induction on $\varphi$ as follows:

[TABLE]

where in the last clause, ${\llbracket K\rrbracket}$ is enumerated by $\bm{\kappa}=\kappa_{1},\dots,\kappa_{c}$, and $\Upsilon_{c}(X_{1},\dots,X_{c})$ is the following $\mathsf{MSO}$-formula, expressing that $X_{1},\dots,X_{c}$ form a partition of $\mathscr{D}^{*}$:

[TABLE]

Note that in the definition of $\langle\varphi\rangle$ above, since $\varphi$ is assumed to be $\mathrm{HF}$-closed, the displayed $\mathrm{HF}$-terms $K$ and $L$ are closed, so that their $\mathfrak{T}$-interpretation $\llbracket K\rrbracket,\llbracket L\rrbracket\in V_{\omega}$ is defined (see §3.5).

Remark 16**.**

Since it involves the standard interpretation map $\llbracket-\rrbracket$ on $\mathrm{HF}$-terms, it follows from Remark 15 (§3.5) that the interpretation $\langle-\rangle$ is not recursive. We refer to §5.6.1 and §8.5 for discussions and workarounds.

Theorem 17**.**

For every closed $\mathsf{FSO}$-formula $\varphi$, we have

[TABLE]

The proof of Theorem 17 is deferred to [DR20, App. A]. The logical rules of $\mathsf{FSO}$ are handled routinely. The interpretations of most of the axioms of $\mathsf{FSO}$ are almost trivially provable from the corresponding axioms of $\mathsf{MSO}$. The Functional Choice Axioms are dealt-with essentially as explained in §3.4.5.

3.6.2. From $\mathsf{MSO}$ to $\mathsf{FSO}$

The translation $(-)^{\circ}:\mathsf{MSO}\to\mathsf{FSO}$ is much simpler than $\langle-\rangle$. Assume given a $\mathsf{FSO}$-Function variable $F_{X}$ for each monadic $\mathsf{MSO}$-variable $X$. The map $(-)^{\circ}$ is inductively defined as follows:

[TABLE]

It is easy to see that $(-)^{\circ}$ is truth preserving (and reflecting) w.r.t. the standard model $\mathfrak{T}$, by a direct induction on formulae relying on the bijection $\mathcal{P}(\mathscr{D}^{*})\cong 2^{\mathscr{D}^{*}}$.

Lemma 18**.**

Given a closed $\mathsf{MSO}$-formula $\varphi$, we have

[TABLE]

The main result on $(-)^{\circ}$ is the following. Its proof is deferred to [DR20, App. A].

Theorem 19**.**

Given a closed $\mathsf{MSO}$-formula $\varphi$,

[TABLE]

Theorem 19 can actually be extended to $\mathsf{FSO}$ formulae. This is essentially the content of the following result.

Proposition 20**.**

For a closed $\mathsf{FSO}$-formula $\varphi$, we have the following.

[TABLE]

Remark 21**.**

Theorem 19 and Proposition 20 will be used in our completeness argument (§8) in two different ways:

1. (1)

We first obtain completeness of $\mathsf{FSO}$ (augmented with the Axiom $(\mathit{PosDet})$ of §5.6) for $\mathsf{MSO}$ formulae via a usual translation of formulae to automata. From this result, completeness of $\mathsf{FSO}+(\mathit{PosDet})$ follows by Proposition 20, while completeness of $\mathsf{MSO}$ (augmented with $\langle-\rangle$-translations of suitable instances of $(\mathit{PosDet})$) follows by Theorem 19. 2. (2)

In addition, we will use Proposition 20 in §7 in order to import the $\mathsf{MSO}$-theory of $\mathbb{N}$ for the infinite paths of $\mathfrak{T}$. We rely on this for the version of the Büchi-Landweber Theorem (namely that $\mathsf{FSO}$ decides parity games on finite graphs) used in the completeness argument of §8, as well as for the Simulation Theorem in §9.

3.7. Notations

We now introduce some notation that we will use throughout our formalization of games and automata in $\mathsf{FSO}$.

3.7.1. $\mathsf{FSO}$ with Extended $\mathrm{HF}$-Terms

First, recall that the syntax of $\mathsf{FSO}$ formally disallows Functions in $\mathrm{HF}$-terms. We propose here a notation system that allows them in some circumstances. For instance, assuming $(F:K)$, we can use the notation

[TABLE]

More generally, consider an atomic formula

[TABLE]

with $M$ and $N$ terms on the following grammar:

[TABLE]

Such formulae can be interpreted in $\mathsf{FSO}$, provided one assumes bounds for the Function variables occurring in them. Let $M$ and $N$ be as above, and assume their free Function variables to be among $\bm{F}=F_{1},\dots,F_{n}$. Note that there are (proper) $\mathrm{HF}$-terms $M^{\prime}$ and $N^{\prime}$ such that

[TABLE]

for some $\mathrm{HF}$-variables $\bm{\ell}=\ell_{1},\dots,\ell_{c}$ and where $\bm{F(t)}=F_{i_{1}}(t_{1}),\dots,F_{i_{c}}(t_{c})$. Given proper $\mathrm{HF}$-terms $K_{1},\dots,K_{n}$, assuming $\bm{F}:\bm{K}$, one can let

[TABLE]

where $\bm{L}=L_{1},\dots,L_{c}$ is such that $L_{j}=K_{i}$ iff the $j$th element of $\bm{F(t)}$ is $F_{i}(t_{j})$. Note however that the above defined formula $M\mathrel{\star}N$ actually depends on the choice of $\bm{K}$, so we rather write it as:

[TABLE]

Generalizing further we can, with the above method, interpret in $\mathsf{FSO}$ formulae build with $\mathrm{HF}$-terms in the sense of (9). The interpretation in $\mathsf{FSO}$ of such a formula $\varphi$ with free Function variables among $\bm{F}=F_{1},\dots,F_{n}$ is defined by induction, and depends on a choice of proper $\mathrm{HF}$-terms $\bm{K}=K_{1},\dots,K_{n}$. Using notation as above, we arrive at the following definition:

[TABLE]

Beware that $(\varphi)_{\bm{F},\bm{K}}$ only makes sense under the assumptions $\bm{F}:\bm{K}$. Keeping this in mind we may obtain, for instance, the following formulations of the Functional Choice Axioms of §3.4.5.

• •

$\mathrm{HF}$-Bounded Choice for Functions.

[TABLE]

• •

Iterated $\mathrm{HF}$-Bounded Choice.

[TABLE]

3.7.2. Notations for Products and Functions

We now introduce notation for a form of product type, based on the function spaces and application functions of §3.4.4.7. The main idea is to be able to manipulate a Function variable

[TABLE]

as a function

[TABLE]

Furthermore, it is convenient to allow such $F$ to have a domain defined by an $\mathsf{FSO}$ formula $\psi(-)$, and to write

[TABLE]

We develop here a notation system for such “function” and “product types”. In §3.7.3, we discuss formulations of the Functional Choice Axioms of §3.4.5 induced by this notation. In order not to overload the arrow symbol $\longrightarrow$ (which will be used with games later on), we will write typing declarations as

[TABLE]

{nota}

[Product Types]Product types are given by the following grammar, where $\psi(-)$ is an $\mathsf{FSO}$ formula of an individual variable (with possibly other free variables of any sort), and where $K$ is an $\mathrm{HF}$-term.

[TABLE]

The arity of a product type $\Pi$ is:

• •

$(1,n)$ if $\Pi$ is of the form $\psi(-)\times K_{1}\times\dots\times K_{n}$,

• •

$(0,n)$ if $\Pi$ is of the form $K_{1}\times\dots\times K_{n}$.

Product types are to be used with the following defined formulae.

[TABLE]

Here f stands for a Function variable $F$ if the arity of $\Pi$ is of the form $(1,n)$, and for an $\mathrm{HF}$-variable $f$ if the arity of $\Pi$ is of the form $(0,n)$. Moreover, for $\Pi=\psi(-)\times K_{1}\times\dots\times K_{n}$, we let

[TABLE]

{rems}

1. (1)

Thanks to Rem. 12, using the Axioms of $\mathrm{HF}$-Bounded Choice (§3.4.5), we have

[TABLE] 2. (2)

Using Convention 3.4.4, for each product type $\Pi$ we have

[TABLE]

It follows that for each product type $\Pi$ and each formula $\varphi$ we have

[TABLE]

{nota}

In the following, given a product type $\Pi$, we use the notation $\tilde{t}:\Pi$, where $\tilde{t}$ stands for a tuple of the form $(t,K_{1},\dots,K_{n})$ if $\Pi$ has arity $(1,n)$, or of the form $(K_{1},\dots,K_{n})$ if $\Pi$ has arity $(0,n)$. When $\Pi$ is clear from the context, we write $\tilde{t}$ instead of $\tilde{t}:\Pi$, and furthermore we may omit the overset tilde, writing $t$ instead of $\tilde{t}$.

Write $\tilde{x}$ for tuples of variables of the form $(x,k_{1},\dots,k_{n})$ or of the form $(k_{1},\dots,k_{n})$.

1. (1)

If $\Pi=\psi(-)\times K_{1}\times\dots\times K_{n}$ and $\tilde{t}=(t,L_{1},\dots,L_{n})$, we write $F^{\Pi\to K}(\tilde{t})$ for the $\mathrm{HF}$-term

[TABLE]

If $\Pi=K_{1}\times\dots\times K_{n}$ and $\tilde{t}=(L_{1},\dots,L_{n})$, we write $f^{\Pi\to K}(\tilde{t})$ for the $\mathrm{HF}$-term

[TABLE]

When $\Pi$ and $K$ are clear from the context, in either case above we write $\textsc{{f}}(\tilde{t})$ for $\textsc{{f}}^{\Pi\to K}(\tilde{t})$. 2. (2)

We furthermore write $\tilde{t}\mathrel{\dot{\in}}\textsc{{f}}$ or even $\textsc{{f}}(\tilde{t})$ for the formula

[TABLE] 3. (3)

We extend product types as follows

[TABLE]

where $X$ is a Function variable. We let

[TABLE]

Note that

[TABLE]

3.7.3. Choice and Comprehension

We list here some important straightforward consequences of the Functional Choice Axioms of §3.4.5 pertaining to Product Types.

Theorem 22**.**

${\mathsf{FSO}_{\mathscr{D}}}$* proves the following generalizations of the Functional Choice Axioms of §3.4.5:*

• •

$\mathrm{HF}$-Bounded Choice*.*

[TABLE]

• •

Iterated $\mathrm{HF}$-Bounded Choice*.*

[TABLE]

Theorem 23** (Comprehension for Product Types).**

${\mathsf{FSO}_{\mathscr{D}}}$* proves the following form of Comprehension, where $V$ does not occur free in $\varphi$:*

[TABLE]

Proof 3.5**.**

We require

[TABLE]

By excluded middle, bounded existentials and generalization we have,

[TABLE]

and we conclude by $\mathrm{HF}$-Bounded Choice.

{rems}

1. (1)

In the case of $\Pi=\mathscr{D}^{*}$, Theorem 23 gives Comprehension for characteristic functions:

[TABLE] 2. (2)

We have the following form of Comprehension for $\mathrm{HF}$-Sets:

[TABLE] 3. (3)

Using Comprehension for $\mathrm{HF}$-Sets, the well-orders on $\mathrm{HF}$-Sets given by Remark 12 give the following Induction Scheme for $\mathrm{HF}$-Sets:

[TABLE]

where $\preceq_{K}$ is the well-order on $K$ given by Remark 12.

4. Game Positions

This Section and the next one describe our setting for games. The games we consider ultimately aim at formalizing acceptance games of tree automata (§6), and thus must encompass acceptance games for non-deterministic tree automata. We shall therefore give a setting for infinite games, with players Proponent $\mathsf{P}$ (corresponding to Automaton or $\exists$loïse) and Opponent $\mathsf{O}$ (corresponding to Pathfinder or $\forall$bérlard). In the case of acceptance games, $\mathsf{P}$ plays for acceptance and $\mathsf{O}$ plays for rejection, and in the particular case of non-deterministic automata, $\mathsf{P}$ chooses transitions from the non-deterministic transition relation, while $\mathsf{O}$ chooses tree directions $d\in\mathscr{D}$, with the aim of building an infinite path. This leads to an inherent asymmetry in the very notion of games, where, from a game position with a given tree position $x\in\mathscr{D}^{*}$, $\mathsf{P}$ can only go to game positions with tree position $x$, while $\mathsf{O}$ must go to a game position with tree position a successor of $x$.

Due to the fact that we cannot access the usual primitive recursive codings in the monadic language, we will only consider games that are ‘superposed’ onto the infinite $\mathscr{D}$-tree, with only boundedly many positions associated with each tree node. Such a setting indeed suffices for the case of acceptance games arising from tree automata. Assume that we are given disjoint non-empty $\mathrm{HF}$-Sets $\mathsf{P}_{\mathcal{G}}$ and $\mathsf{O}_{\mathcal{G}}$ of Proponent and Opponent labels respectively. Intuitively, Proponent will play from game positions of the form

[TABLE]

while Opponent will play from positions of the form

[TABLE]

A game will be given by specifying edge relations of the form

[TABLE]

So $\mathsf{P}$ can only move to a game position with the same underlying tree position, while $\mathsf{O}$ is forced to move to a game position with a successor underlying tree position. This induces a dag structure on game positions, whose underlying partial order $\trianglelefteq_{\mathcal{G}}$ is the lexicographic product of the usual tree order with the one setting $\mathsf{P}$-labels smaller than all $\mathsf{O}$-labels. The games we shall consider will all be subrelations of $\trianglelefteq_{\mathcal{G}}$.

This Section is devoted to the definition of this dag structure. We shall also prove some basic results related to induction in §4.2 and to infinite paths in §4.3. These will help proving some similar results for games in §5, for which arguments are more naturally given at the level of $\trianglelefteq_{\mathcal{G}}$.

4.1. A Partial Order of Game Positions

We first introduce the formal notion of labels of game positions.

{defi}

[Labels of Game Positions] Labels of game positions are pairs $(\mathsf{P}_{\mathcal{G}},\mathsf{O}_{\mathcal{G}})$ of $\mathrm{HF}$-terms satisfying the following formula:

[TABLE]

We write $\mathsf{P}\mathsf{O}_{\mathcal{G}}$ for $\mathsf{P}_{\mathcal{G}}\cup\mathsf{O}_{\mathcal{G}}$. When no ambiguity arises, we write $\mathsf{P}$, $\mathsf{O}$ and $\mathsf{P}\mathsf{O}$ for $\mathsf{P}_{\mathcal{G}}$, $\mathsf{O}_{\mathcal{G}}$ and $\mathsf{P}\mathsf{O}_{\mathcal{G}}$ respectively.

Assume $(\mathsf{P},\mathsf{O})$ are labels of game positions. Intuitively, game positions are pairs $(x,k)$ with $x\in\mathscr{D}^{*}$ and $k\in\mathsf{P}\mathsf{O}$, Proponent’s positions are game positions with $k\in\mathsf{P}$ and Opponent’s positions are game positions with $k\in\mathsf{O}$. To summarize, we have the informal correspondence:

[TABLE]

We will throughout the paper use the following notation to manipulate game positions and sets of game positions.

{nota}

[Game Positions]

We introduce the following notation, assuming $\mathrm{Labels}(\mathsf{P}_{\mathcal{G}},\mathsf{O}_{\mathcal{G}})$.

1. (1)

Variables, written $u,v,w,\text{etc}$, range over game positions, that is over pairs $(x,k)$ with $x$ an Individual variable and $k$ an $\mathrm{HF}$-variable ranging over $\mathsf{P}\mathsf{O}_{\mathcal{G}}$. 2. (2)

Sets of game positions, written $U,V,W,\text{etc}$, range over Functions $\mathscr{D}^{*}\times\mathsf{P}\mathsf{O}_{\mathcal{G}}\mathrel{\mathfrak{to}}2$. We will systematically use the following notation:

[TABLE]

We often write $v\in V$ or $V(v)$ for $V(v)\mathrel{\doteq}1$. 3. (3)

For a set of game positions $V$, we write $V_{\mathsf{P}}$ and $V_{\mathsf{O}}$ for the $\mathsf{P}$ and $\mathsf{O}$ subsets of $V$ respectively. This amounts to the following abbreviations:

[TABLE]

Intuitively, $V_{\mathsf{P}}$ represents $V\cap(\mathscr{D}^{*}\times\mathsf{P}_{\mathcal{G}})$ while $V_{\mathsf{O}}$ represents $V\cap(\mathscr{D}^{*}\times\mathsf{O}_{\mathcal{G}})$. 4. (4)

In formulae we interpret quantifiers over (sets of) game positions as follows:

[TABLE]

where, in the $\exists v$ case, we choose $x,\ell$ not free in $\varphi$.

We now introduce the partial order $\trianglelefteq$ on game positions.

{defi}

[Partial Order on Game Positions]

The relations $\vartriangleleft_{(\mathsf{P}_{\mathcal{G}},\mathsf{O}_{\mathcal{G}})}$, $\trianglelefteq_{(\mathsf{P}_{\mathcal{G}},\mathsf{O}_{\mathcal{G}})}$ and $\mathsf{s}^{\vartriangleleft}_{(\mathsf{P}_{\mathcal{G}},\mathsf{O}_{\mathcal{G}})}$, where $\mathsf{P}_{\mathcal{G}}$ and $\mathsf{O}_{\mathcal{G}}$ are $\mathrm{HF}$-variables, are defined as follows:

[TABLE]

When no ambiguity arises, we write $\vartriangleleft_{\mathcal{G}}$, $\trianglelefteq_{\mathcal{G}}$ and $\mathsf{s}^{\vartriangleleft}_{\mathcal{G}}$, or even $\vartriangleleft$, $\trianglelefteq$ and $\mathsf{s}^{\vartriangleleft}$ for $\vartriangleleft_{(\mathsf{P}_{\mathcal{G}},\mathsf{O}_{\mathcal{G}})}$, $\trianglelefteq_{(\mathsf{P}_{\mathcal{G}},\mathsf{O}_{\mathcal{G}})}$ and $\mathsf{s}^{\vartriangleleft}_{(\mathsf{P}_{\mathcal{G}},\mathsf{O}_{\mathcal{G}})}$ respectively.

Note that the formula $\mathsf{s}^{\vartriangleleft}(-,-)$ is actually bounded, since by Notation 4.1.(1), the variable $w$ ranges over game positions, so that $(\exists w)$ stands for $(\exists w\in\mathscr{D}^{*}\times\mathsf{P}\mathsf{O})$.

We note a number of useful properties of $\vartriangleleft$, in particular that it is a discrete partial order.

Proposition 24**.**

${\mathsf{FSO}_{\mathscr{D}}}$* proves following, under the assumption $\mathrm{Labels}(\mathsf{P},\mathsf{O})$.*

1. (1)

$u\vartriangleleft v\vartriangleleft w\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ u\vartriangleleft w$** 2. (2)

$\lnot(u\vartriangleleft u)$** 3. (3)

$u\trianglelefteq v\trianglelefteq u\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ u=v$** 4. (4)

$u\vartriangleleft v\quad\Longleftrightarrow\quad(\exists w\trianglelefteq v)(\mathsf{s}^{\vartriangleleft}(u,w))\quad\Longleftrightarrow\quad(\exists w^{\prime}\trianglerighteq u)(\mathsf{s}^{\vartriangleleft}(w^{\prime},v))$** 5. (5)

(\forall k\in\mathsf{P})\big{(}u\trianglelefteq(\dot{\varepsilon},k)\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ u=(\dot{\varepsilon},k)\big{)}* *

4.2. Induction and Recursion

We now present some basic results on induction and recursion w.r.t. the partial order on game positions.

\opt

draftnotes

We can show that $\vartriangleleft$ satisfies well-founded induction from the induction principle on the underlying tree.

Theorem 25** ($\vartriangleleft$-Induction).**

${\mathsf{FSO}_{\mathscr{D}}}$* proves the following, under the assumption $\mathrm{Labels}(\mathsf{P},\mathsf{O})$.*

[TABLE]

Proof 4.2**.**

Let $V$ be such that, for any game position $v$:

[TABLE]

We show that

[TABLE]

by induction on $x$, whence the theorem will follow.

Suppose that $x=\dot{\varepsilon}$, and so $y=\dot{\varepsilon}$. We first prove the statement for arbitrary $\ell\mathrel{\dot{\in}}\mathsf{P}$; in this case notice that there is no $u$ such that $u\vartriangleleft(\dot{\varepsilon},\ell)$, and so we vacuously satisfy the LHS of (10) above. Therefore we have that $(\dot{\varepsilon},\ell)\in V$. Otherwise $\ell\mathrel{\dot{\in}}\mathsf{O}$ and every $u\vartriangleleft(\dot{\varepsilon},\ell)$ is of the form $(\dot{\varepsilon},k)$ for some $k\mathrel{\dot{\in}}\mathsf{P}$, and we have just shown that such $u$ must be contained in $V$. Therefore we can conclude that $(\dot{\varepsilon},\ell)\in V$, again by (10), as required.

*Now we consider the inductive step, assuming the statement above is already true for $x$ and considering the case of $\mathsf{S}_{d}x$. If $y\mathrel{\dot{\leq}}\mathsf{S}_{d}x$ then either $y\mathrel{\dot{\leq}}x$ or $y\mathrel{\doteq}\mathsf{S}_{d}x$. In the former case we have by the inductive hypothesis that, for any $\ell\mathrel{\dot{\in}}\mathsf{P}\mathsf{O}$, $(y,\ell)\in V$. So assume that $y\mathrel{\doteq}\mathsf{S}_{d}x$. Again we distinguish when $\ell\mathrel{\dot{\in}}\mathsf{P}$ and when $\ell\mathrel{\dot{\in}}\mathsf{O}$ in order to exhibit the LHS of (10) above. In the former case, notice that any $(z,k)\vartriangleleft(y,\ell)$ is such that $z\mathrel{\dot{\leq}}x$, and so we have that $(z,k)\in V$ by the inductive hypothesis; thus $(y,\ell)\in V$ by (10). In the latter case (when $\ell\mathrel{\dot{\in}}\mathsf{O}$) we have for any $(z,k)\vartriangleleft(y,\ell)$ either $z\mathrel{\dot{\leq}}x$ or ($z\mathrel{\doteq}\mathsf{S}_{d}x$ and $k\mathrel{\dot{\in}}\mathsf{P}$). In both cases we have seen that $(z,k)\in V$, and so again we have that $(y,\ell)\in V$ by (10). *

Since $\vartriangleleft$ is a partial order with induction, comprehension (Theorem 23) gives a Recursion Theorem, which allows us to define a set of game positions $V$ by induction on game positions. This requires the value of $V$ at a position $v$ to be determined by its values at positions $u\vartriangleleft v$. Thus, if the value of $V$ at $v$ is given by a formula $\varphi(V,v)$, we assume that the following formula holds

[TABLE]

The Recursion Theorem says that, assuming $\mathrm{Rec}(\varphi)$, the set of game positions $V$ given by

[TABLE]

is the unique set of game positions such that

[TABLE]

Proposition 26** (Recursion Theorem).**

${\mathsf{FSO}_{\mathscr{D}}}$* proves that $\mathrm{Labels}(\mathsf{P},\mathsf{O})\land\mathrm{Rec}(\varphi)$ implies*

[TABLE]

Proof 4.3**.**

Consider a formula $\varphi(V,v)$ and assume $\mathrm{Rec}(\varphi)$ and $\mathrm{Labels}(\mathsf{P},\mathsf{O})$. We begin with the second part of the statement, namely the uniqueness part. Fix $V,U$. By $\vartriangleleft$-induction on $v$, we show that $\mathsf{FSO}$ proves the following formula $\psi(v)=\psi(V,U,v)$:

[TABLE]

Let $v$ and assume both premises of $\psi(v)$, as well as $\psi(w)$ for all $w\vartriangleleft v$. The premises of $\psi(v)$ imply those of $\psi(w)$ for $w\vartriangleleft v$, so that we have $(Vw\Leftrightarrow Uw)$ for all $w\vartriangleleft v$. Hence, given $u\trianglelefteq v$, if $u\vartriangleleft v$ then we are done. It thus remains to show $(Vv\Leftrightarrow Uv)$. Thanks to the premises of $\psi(v)$, this amounts to showing $\varphi(V,v)\Leftrightarrow\varphi(U,v)$, which itself follows from $\mathrm{Rec}(\varphi)$, since $(Vw\Leftrightarrow Uw)$ for all $w\vartriangleleft v$.

We now turn to the first part of the statement. Let $V$ such that

[TABLE]

By $\vartriangleleft$-induction on $v$, we show that $\mathsf{FSO}$ proves the following formula

[TABLE]

So let $v$ and assume $\theta(w)$ for all $w\vartriangleleft v$. Given $u\trianglelefteq v$, if $u\vartriangleleft v$ then $\vartheta(u)$ follows from $\theta(u)$. It thus remains to show $\vartheta(v)$. We consider the two implications separately.

• •

Case of $\varphi(V,v)\Longrightarrow Vv$.* Assume $\varphi(V,v)$. By definition of $V$, we are done if we show*

[TABLE]

Given $U$ such that \big{(}Uu\leavevmode\nobreak\ \Leftrightarrow\leavevmode\nobreak\ \varphi(U,u)\big{)} for all $u\trianglelefteq v$, we obtain $Uv$ from $\varphi(U,v)$, which itself follows $\varphi(V,v)$ and $\mathrm{Rec}(\varphi)$. The premise of $\mathrm{Rec}(\varphi)$ follows from $(\forall w\vartriangleleft v)\psi(V,U,w)$, whose premises are in turn given by resp. $(\forall w\vartriangleleft v)\vartheta(w)$ and the assumption on $U$.

• •

Case of $Vv\Longrightarrow\varphi(V,v)$.* Assume $Vv$. By comprehension (Theorem 23) let $U$ such that*

[TABLE]

We obtain $\varphi(V,v)$ from $Uv$, which in turn by def. of $V$ follows from (\forall u\trianglelefteq v)\big{(}Uu\leavevmode\nobreak\ \Leftrightarrow\leavevmode\nobreak\ \varphi(U,u)\big{)}. In order to show the latter, note that by definition of $U$ we have $(Uu\Leftrightarrow Vu)$ for all $u\vartriangleleft v$. Hence $\mathrm{Rec}(\varphi)$ gives $\varphi(U,v)\Leftrightarrow\varphi(V,v)$ and we get $(Uv\Leftrightarrow\varphi(U,v))$ from the definition of $U$. In the case of $u\vartriangleleft v$, namely $(Uu\Leftrightarrow\varphi(U,u))$, we have $(\forall w\trianglelefteq u)(Uw\Leftrightarrow Vw)$ so that $\mathrm{Rec}(\varphi)$ implies $\varphi(U,u)\Leftrightarrow\varphi(V,u)$ and the result follows form $\vartheta(u)$.

4.3. Infinite Paths

We develop here a notion of infinite paths (i.e. unbounded linearly order sets) for the partial order $\trianglelefteq$ on game positions. This material will be useful in Section 5.2 to handle properties of infinite plays in games which intrinsically rely on the particular structure of the relation $\trianglelefteq$ on game positions. A typical example is the Predecessor Lemma 37.

{defi}

[Game Paths]

Let $\mathsf{P},\mathsf{O}$ be $\mathrm{HF}$-variables. Given a game position $u$ and a set of game positions $U$, we say that $U$ is a path from $u$ if the following formula $\mathrm{Path}(\mathsf{P},\mathsf{O},u,U)$ holds:

[TABLE]

We write $\mathrm{Path}(u,U)$ when $\mathsf{P}$ and $\mathsf{O}$ are clear from the context.

As a preparation to the Predecessor Lemma 37 for Infinite Plays, we prove here the analogous property for infinite paths.

Lemma 27** (Predecessor Lemma for Game Paths).**

${\mathsf{FSO}_{\mathscr{D}}}$* proves the following. Assuming that $\mathrm{Labels}(\mathsf{P},\mathsf{O})$ and that $\mathrm{Path}(\mathsf{P},\mathsf{O},u_{0},U)$ hold for a game position $u_{0}$ and a set of game positions $U$, we have*

[TABLE]

The proof of Lemma 27 relies on the following usual maximality principle for non-empty linearly-ordered bounded sets.

Lemma 28**.**

${\mathsf{FSO}_{\mathscr{D}}}$* proves the following, assuming $\mathrm{Labels}(\mathsf{P},\mathsf{O})$. Given a set of game positions $V$, if $V$ is bounded (i.e. $(\exists u)(\forall v\in V)(v\vartriangleleft u)$), non-empty and linearly ordered, then $V$ has a maximum element: $(\exists u\in V)(\forall v\in V)(v\trianglelefteq u)$.*

Proof 4.4**.**

By $\vartriangleleft$-induction, we prove the following property:

1. (1)

For all $u$, for all $V$, if $V$ is non-empty, linearly ordered by $\vartriangleleft$ and such that $\forall v\in V(v\trianglelefteq u)$, then $V$ has a $\vartriangleleft$-maximal element.

Let $u$ and $V$ satisfy the assumptions from 1 above, and assume 1 for all $c\vartriangleleft u$. First, if $u=v$ for some $v\in V$, then $u$ is indeed the maximal element of $V$. So we can assume $v\vartriangleleft u$ for all $v\in V$.

By Comprehension for Product Types (Thm. 23), let $U$ be the set of all $w$ such that $\mathsf{s}^{\vartriangleleft}(w,u)$ and such that $v\trianglelefteq w$ for some $v\in V$. For each $v\in V$, it follows from Proposition 24.(4) that there is some $w\in U$ such that $v\trianglelefteq w$. In particular, $U$ is non-empty since $V$ is non-empty.

We claim the following:

Claim 29**.**

[TABLE]

Proof 4.5** (Proof of Claim 29).**

Let $w\in U$. By Comprehension for Product Types (Thm. 23), let $W$ be the set of all $v\in V$ such that $v\trianglelefteq w$. Note that $W$ is non-empty by definition of $U$. It is inherits the property of being linearly ordered from $V$, and by construction it is bounded by $w$ with $w\vartriangleleft u$. By induction hypothesis, $W$ has a maximal element, say $\tilde{w}$. We indeed have $\tilde{w}\in V$ and $v\trianglelefteq\tilde{w}$ for all $v\in V$ with $v\trianglelefteq w$. Since $\tilde{w}\trianglelefteq w$, uniqueness follows from the antisymmetry of $\trianglelefteq$.

The remainder of the argument relies on the particular structure of $\vartriangleleft$. Using Comprehension on $\mathrm{HF}$-Sets, it follows from the definition of $\vartriangleleft$ that there is some $x\in\mathscr{D}^{*}$ and some $\mathrm{HF}$-Set $k$ such that $U$ is exactly the set of all $(x,\ell)$ with $\ell\in k$. This observation allows us to show

Claim 30**.**

[TABLE]

Proof 4.6** (Proof of Claim 30).**

Write $\preceq$ for the well-order on $k$ given by Remark 12. By $\preceq$-Induction (Remark 3.7.3.(3)) we show the following:

[TABLE]

Let $\ell\in k$ be such that the property holds for all $\ell^{\prime}\prec\ell$. If $\ell$ is $\preceq$-minimal, the result follows from the existence of a unique $\tilde{w}$ such that $\vartheta((x,\ell),\tilde{w})$. Otherwise, let $\ell^{\prime}$ be the $\preceq$-predecessor of $\ell$, and let $m\in k$ such that $\psi(\ell^{\prime},m)$ be given by induction hypothesis. By Claim 29, let $\tilde{w}_{\ell},\tilde{w}_{m}$ be the unique elements of $V$ such that $\vartheta((x,\ell),\tilde{w}_{\ell})$ and $\vartheta((x,m),\tilde{w}_{m})$. Since $V$ is linearly ordered, we have either that $\tilde{w}_{m}\trianglelefteq\tilde{w}_{\ell}$ or that $\tilde{w}_{m}\trianglelefteq\tilde{w}_{\ell}$. In the former case, we take $\ell$ for the new $m$, and in the latter we keep the same $m$.

Since $U$ is non-empty, there is a $\preceq$-maximal $\ell\in k$. Let $m\in k$ such that $\psi(\ell,m)$, and by Claim 29, let $\tilde{w}_{m}\in V$ such that $\vartheta((x,m),\tilde{w}_{m})$. By definition of $k$, we do have $\tilde{w}\trianglelefteq\tilde{w}_{m}$ for all $\tilde{w}\in V$ with $\vartheta(w,\tilde{w})$ for some $w\in U$. Hence we have that $\varphi(\tilde{w}_{m})$.

Consider now $\tilde{w}_{m}\in V$ such that $\varphi(\tilde{w}_{m})$. As noted above, for all $v\in V$ there is some $w\in U$ such that $v\vartriangleleft w$. But we also have $v\trianglelefteq\tilde{w}$ where $\tilde{w}$ is unique such that $\vartheta(w,\tilde{w})$. It thus follows that $v\trianglelefteq\tilde{w}_{m}$ for all $v\in V$.

This concludes the proof of Lemma 28.

We can now prove Lemma 27.

Proof 4.7** (Proof of Lemma 27).**

Fix $v\in U$ with $u_{0}\vartriangleleft v$. By Comprehension for Product Types (Thm. 23), let $W$ be the set of all $w\in U$ such that $w\vartriangleleft v$. Since $u_{0}\vartriangleleft v$ and $\mathrm{Path}(u_{0},U)$, the set $W$ is non-empty, linearly ordered and bounded by $v$. By Lemma 28, it has a maximal element, say $w$. We have $u_{0}\trianglelefteq w$ and $w\vartriangleleft v$. Moreover, by $\mathrm{Path}(u_{0},U)$ there is some $\tilde{w}\in U$ such that $\mathsf{s}^{\vartriangleleft}(w,\tilde{w})$. Again by $\mathrm{Path}(u_{0},U)$, we have

[TABLE]

But $\tilde{w}\vartriangleleft v$ implies $\tilde{w}\trianglelefteq w$, a contradiction, while $v\vartriangleleft\tilde{w}$ implies $w\vartriangleleft v\vartriangleleft\tilde{w}$, contradicting $\mathsf{s}^{\vartriangleleft}(w,\tilde{w})$. It thus follows that $\tilde{w}=v$ and we are done.

5. Infinite Two-Player Games

This Section is devoted to definitions and basic properties relating to games, building on §4. We will use these games in §6 and §9 to formalize a basic theory of tree automata in $\mathsf{FSO}$.

Our games are played on bipartite dags (with partial order $\trianglelefteq_{\mathcal{G}}$) induced by labels of game positions $(\mathsf{P}_{\mathcal{G}},\mathsf{O}_{\mathcal{G}})$ in the sense of §4. Continuing §4, Proponent will play from positions of the form

[TABLE]

while Opponent will play from positions of the form

[TABLE]

A game will be given by specifying edge relations of the form

[TABLE]

so that, for $\mathsf{J}$ either $\mathsf{P}$ or $\mathsf{O}$,

[TABLE]

(actually even $\mathsf{s}^{\vartriangleleft}(u,v)$). We insist on the fact that $\mathsf{P}$ can only move to a game position with the same underlying tree position, while $\mathsf{O}$ is forced to move to a game position with a successor tree position.

We first give basic definitions and results on games (§5.1) and infinite plays (§5.2). Besides the above mentioned constraints on the shape of games, these notions are standard. Our notion of strategy is presented in §5.3. A crucial point here is that, w.r.t. our games, the monadic language imposes all strategies to be by construction positional in the usual sense (see e.g. [Tho97]). Finally, §5.4 briefly discusses our setting for winning in games, and §5.5 presents in more detail the important particular case of parity conditions. Parity conditions are one of the prominent formulations of winning conditions for $\omega$-regular games. This is in particular due to the fact that they are positionally determined, i.e. the winner of a parity game can always win with a positional winning strategy [EJ91] (see also [Tho97, Wal02, PP04]). This is of crucial importance in our setting as all our strategies are inherently positional, due to the underlying limits on expressiveness in the language of $\mathsf{MSO}$. Finally, the Axiom $(\mathit{PosDet})$ of Positional Determinacy of Parity Games is formulated in §5.6.

5.1. Games

A game $\mathcal{G}$ will be given by labels of game positions $\mathsf{P}_{\mathcal{G}}$ and $\mathsf{O}_{\mathcal{G}}$ together with Functions

[TABLE]

where $\mathcal{P}_{*}(-)$ is the $\mathrm{HF}$-Function of §3.4.4.4. Such Functions $E_{\mathsf{P}},E_{\mathsf{O}}$ induce edge relations $\mathchoice{\mathrel{\overset{}{\underset{\mathsf{P}_{\mathcal{G}}}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\mathsf{P}_{\mathcal{G}}}}}{\longrightarrow_{\mathsf{P}_{\mathcal{G}}}}{\longrightarrow_{\mathsf{P}_{\mathcal{G}}}}$ and $\mathchoice{\mathrel{\overset{}{\underset{\mathsf{O}_{\mathcal{G}}}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\mathsf{O}_{\mathcal{G}}}}}{\longrightarrow_{\mathsf{O}_{\mathcal{G}}}}{\longrightarrow_{\mathsf{O}_{\mathcal{G}}}}$ given by

[TABLE]

We make this formal in the following definition.

{defi}

[Games and Edge Relations]

1. (1)

A game $\mathcal{G}$ is given by $\mathrm{HF}$-terms $\mathsf{P}_{\mathcal{G}},\mathsf{O}_{\mathcal{G}}$ and Functions $E(\mathcal{G})_{\mathsf{P}},E(\mathcal{G})_{\mathsf{O}}$ which satisfy the following formula

[TABLE]

We often write $\mathrm{Game}(\mathcal{G})$ for $\mathrm{Game}(\mathsf{P}_{\mathcal{G}},\mathsf{O}_{\mathcal{G}},E(\mathcal{G})_{\mathsf{P}},E(\mathcal{G})_{\mathsf{O}})$. Moreover, when no ambiguity arises, we abbreviate $\mathcal{G}=(\mathsf{P}_{\mathcal{G}},\mathsf{O}_{\mathcal{G}},E(\mathcal{G})_{\mathsf{P}},E(\mathcal{G})_{\mathsf{O}})$ as $\mathcal{G}=(\mathsf{P}_{\mathcal{G}},\mathsf{O}_{\mathcal{G}},E_{\mathsf{P}},E_{\mathsf{O}})$ or even $\mathcal{G}=(\mathsf{P}_{\mathcal{G}},\mathsf{O}_{\mathcal{G}},E)$ or $\mathcal{G}=(\mathsf{P},\mathsf{O},E)$. 2. (2)

The edge relations induced by $\mathcal{G}=(\mathsf{P},\mathsf{O},E_{\mathsf{P}},E_{\mathsf{O}})$ are defined as follows:

[TABLE]

When no ambiguity arises, we write $\mathchoice{\mathrel{\overset{}{\underset{\mathsf{P}}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\mathsf{P}}}}{\longrightarrow_{\mathsf{P}}}{\longrightarrow_{\mathsf{P}}}$, $\mathchoice{\mathrel{\overset{}{\underset{\mathsf{O}}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\mathsf{O}}}}{\longrightarrow_{\mathsf{O}}}{\longrightarrow_{\mathsf{O}}}$ and $\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}$, for $\mathchoice{\mathrel{\overset{}{\underset{\mathsf{P}_{\mathcal{G}}}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\mathsf{P}_{\mathcal{G}}}}}{\longrightarrow_{\mathsf{P}_{\mathcal{G}}}}{\longrightarrow_{\mathsf{P}_{\mathcal{G}}}}$, $\mathchoice{\mathrel{\overset{}{\underset{\mathsf{O}_{\mathcal{G}}}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\mathsf{O}_{\mathcal{G}}}}}{\longrightarrow_{\mathsf{O}_{\mathcal{G}}}}{\longrightarrow_{\mathsf{O}_{\mathcal{G}}}}$ and $\mathchoice{\mathrel{\overset{}{\underset{\mathcal{G}}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\mathcal{G}}}}{\longrightarrow_{\mathcal{G}}}{\longrightarrow_{\mathcal{G}}}$.

Note that $\mathrm{Game}(\mathcal{G})$ implies that the edge relation $\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}$ has no dead ends, i.e. that from any position, a move can always be made by one of the players. It follows that the edge relation $\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}$ induces an unbounded partial order. (Note that it already follows from the structure of $\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}$ that it induces a partial order.)

Lemma 31**.**

${\mathsf{FSO}_{\mathscr{D}}}$* proves*

[TABLE]

Games are equipped with a natural notion of subgame. In this paper we will use subgames to ease some reasoning on automata (in particular in §9), and also to more easily define certain strategies that are more naturally seen as concepts at the game level (see §5.3). We only need the following weak notion of subgame.

{defi}

[Subgame]

We say that $\mathcal{G}^{\prime}$ is a subgame of $\mathcal{G}$ whenever the following formula holds

[TABLE]

Remark 32**.**

Let $\mathcal{G}=(\mathsf{P}_{\mathcal{G}},\mathsf{O}_{\mathcal{G}},E(\mathcal{G})_{\mathsf{P}},E(\mathcal{G})_{\mathsf{O}})$ with $\mathrm{Game}(\mathcal{G})$. Then we have $\mathrm{Sub}(\mathcal{G},\mathcal{G}(\trianglelefteq))$, where $\mathcal{G}(\trianglelefteq)$ stands for the game

[TABLE]

in which by $\mathrm{HF}$-Bounded Choice we let

[TABLE]

Note that the edge relation of $\mathcal{G}(\trianglelefteq)$ is precisely the relation $\trianglelefteq_{(\mathsf{P}_{\mathcal{G}},\mathsf{O}_{\mathcal{G}})}$ of Definition 4.1, hence the notation.

The edge relation $\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}$ of a game $\mathcal{G}$ only specifies the moves of $\mathcal{G}$. In order to manipulate plays (i.e. sequences of moves) we define the reflexive-transitive closure $\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}$ and the transitive closure $\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}$ of $\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}$. As expected, these are second-order notions.222It is well known (see e.g. [Lib04, Chap. 4]) that transitive closure in graphs is not expressible in first-order logic over the edge relation.

{defi}

Let $\mathcal{G}=(\mathsf{P},\mathsf{O},E_{\mathsf{P}},E_{\mathsf{O}})$ where $\mathsf{P},\mathsf{O}$ are $\mathrm{HF}$-variables and $E_{\mathsf{P}},E_{\mathsf{O}}$ are Function variables. We define the following formulae.

[TABLE]

Whenever possible, we write $\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}$ and $\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}$ for $\mathchoice{\mathrel{\overset{*}{\underset{\mathcal{G}}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}_{\mathcal{G}}}}{\longrightarrow^{*}_{\mathcal{G}}}{\longrightarrow^{*}_{\mathcal{G}}}$ and $\mathchoice{\mathrel{\overset{+}{\underset{\mathcal{G}}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}_{\mathcal{G}}}}{\longrightarrow^{+}_{\mathcal{G}}}{\longrightarrow^{+}_{\mathcal{G}}}$.

The relations $\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}$ and $\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}$ satisfy properties analogous to those of Proposition 24:

Proposition 33** (Properties of Edge Relations).**

${\mathsf{FSO}_{\mathscr{D}}}$* proves the following, under the assumption $\mathrm{Game}(\mathcal{G})$.*

1. (1)

$u\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}v\leavevmode\nobreak\ \leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ \mathsf{s}^{\vartriangleleft}(u,v)$** 2. (2)

$\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}$* is irreflexive and asymmetric.* 3. (3)

$\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}$* is reflexive and transitive.* 4. (4)

$u\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}v\quad\Leftrightarrow\quad u=v\lor(\exists w)\left(u\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}w\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}v\right)\quad\Leftrightarrow\quad u=v\lor(\exists w)\left(u\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}w\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}v\right)$** 5. (5)

$u\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}v\leavevmode\nobreak\ \leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ u\trianglelefteq v$** 6. (6)

$\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}$* is antisymmetric.* 7. (7)

$u\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}v\leavevmode\nobreak\ \leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ u\vartriangleleft v$** 8. (8)

$\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}$* is irreflexive and transitive.* 9. (9)

$(\forall k\in\mathsf{P})\left(u\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}(\dot{\varepsilon},k)\leavevmode\nobreak\ \leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ u=(\dot{\varepsilon},k)\right)$**

Induction for games (i.e. w.r.t. edge relations) is an immediate corollary to Theorem 25 and Proposition 33.

Corollary 34** (Game Induction).**

${\mathsf{FSO}_{\mathscr{D}}}$* proves the following, under the assumption $\mathrm{Game}(\mathcal{G})$.*

[TABLE]

5.2. Infinite Plays

We now define our notion of infinite play. They are sets of game positions which are unbounded and linearly ordered w.r.t. $\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}$. Infinite plays will allow us to define winning in games (§5.4) and thus acceptance for tree automata (§6). Furthermore, we prove a number of basic properties on infinite plays on which we rely for the formalization of usual operations on tree automata.

In the following, given $\mathcal{G}=(\mathsf{P},\mathsf{O},E)$, we write $\mathrm{Path}(\mathcal{G},u,U)$ for $\mathrm{Path}(\mathsf{P},\mathsf{O},u,U)$, where $\mathrm{Path}$ is as in Definition 4.3.

{defi}

[Infinite Plays] Let $\mathcal{G}=(\mathsf{P},\mathsf{O},E_{\mathsf{P}},E_{\mathsf{O}})$ where $\mathsf{P},\mathsf{O}$ are $\mathrm{HF}$-variables and $E_{\mathsf{P}},E_{\mathsf{O}}$ are Function variables. Given a position $u$ and a set of game positions $U$, we say that $U$ is an infinite play in $\mathcal{G}$ from $u$ when the following formula $\mathrm{Play}(\mathcal{G},u,U)$ holds:

[TABLE]

Note that $\mathrm{Play}(\mathcal{G},u,U)$ is literally just the formula $\mathrm{Path}(\mathcal{G},u,U)$ in which $\mathchoice{\mathrel{\overset{*}{\underset{\mathcal{G}}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}_{\mathcal{G}}}}{\longrightarrow^{*}_{\mathcal{G}}}{\longrightarrow^{*}_{\mathcal{G}}}$ replaces $\trianglelefteq$, $\mathchoice{\mathrel{\overset{}{\underset{\mathcal{G}}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\mathcal{G}}}}{\longrightarrow_{\mathcal{G}}}{\longrightarrow_{\mathcal{G}}}$ replaces $\mathsf{s}^{\vartriangleleft}(-,-)$ and $\mathchoice{\mathrel{\overset{+}{\underset{\mathcal{G}}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}_{\mathcal{G}}}}{\longrightarrow^{+}_{\mathcal{G}}}{\longrightarrow^{+}_{\mathcal{G}}}$ replaces $\vartriangleleft$. It follows from Proposition 33 that $\mathrm{Play}(\mathcal{G},u,U)$ implies $\mathrm{Path}(\mathcal{G},u,U)$. In other words, an infinite play in $\mathcal{G}=(\mathsf{P},\mathsf{O},E)$ is simply an infinite path of the underlying partial order $\trianglelefteq_{(\mathsf{P},\mathsf{O})}$ which respects the transitions of $\mathcal{G}$ induced by $E$. Also, if $\mathcal{G}^{\prime}$ is a subgame of $\mathcal{G}$, then $\mathrm{Play}(\mathcal{G}^{\prime},u,U)$ implies $\mathrm{Play}(\mathcal{G},u,U)$.

We now gather some basic properties on infinite plays. The first one will help to show that a set of game positions is linearly ordered.

Proposition 35**.**

${\mathsf{FSO}_{\mathscr{D}}}$* proves the following, assuming $\mathrm{Game}(\mathcal{G})$. Let $V$ and $u_{0}\in V$ be such that*

[TABLE]

Then

[TABLE]

Proof 5.1**.**

First, it follows from Proposition 33.(6) that $u_{0}$ is unique such that $(\forall v\in V)(u_{0}\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}v)$. By induction on the edge relation $\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}$ (cf. Corollary 34) we show

[TABLE]

Let $u\in V$, and assume that $\theta(v)$ holds for all $v\in V$ such that $v\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}u$. If $u=u_{0}$ then we are done since $u_{0}\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}v$ for all $v\in V$. Otherwise, by assumption there is $v\in V$ with $v\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}u$, and moreover such that $u$ is the unique $\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}$-successor of $v$ in $U$.

Note that $v\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}u$ implies $v\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}u$ (Proposition 33, (2) & (4)), so that $\theta(v)$ follows from the induction hypothesis. Given $w\in V$, if $w\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}v$ then we get $w\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}u$ and we are done. Otherwise, since $\theta(v)$ implies $v\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}w$, we may appeal to the following.

Claim 36**.**

[TABLE]

Proof 5.2** (Proof of Claim 36).**

We reason by induction on $\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}$. So let $w\in V$ with $v\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}w$ and such that

[TABLE]

Since $u_{0}\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}v\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}w$ we have $w\neq u_{0}$ by Proposition 33.(6), so that there is $w^{\prime}\in V$ with $w^{\prime}\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}w$. If $v\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}w^{\prime}$ then the induction hypothesis implies $u\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}w^{\prime}$, so that $u\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}w$ and we are done. Otherwise $\theta(v)$ implies $w^{\prime}\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}v$. Assume for contradiction that $w^{\prime}\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}v$. We thus have

[TABLE]

Proposition 33.(7) then gives $w^{\prime}\vartriangleleft v\vartriangleleft w$. But this contradicts $w^{\prime}\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}w$ since the latter implies $\mathsf{s}^{\vartriangleleft}(w^{\prime},w)$ by Proposition 33.(1). Hence $w^{\prime}=v$. But then $v=w^{\prime}\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}w\in V$ and, since $u$ is the unique $\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}$-successor of $v$ in $V$, we have $u=w$, as required.

This concludes the proof of Proposition 35.

Proposition 35 is a useful tool to prove that given sets of game positions are infinite plays. Some constructions on automata (see §6, §9) furthermore require us to build plays in one game from plays in another game. To this end, we note here the following property, which we informally see as a partial converse to Proposition 35.

Lemma 37** (Predecessor Lemma for Infinite Plays).**

${\mathsf{FSO}_{\mathscr{D}}}$* proves the following. Assuming $\mathrm{Game}(\mathcal{G})$ and $\mathrm{Play}(\mathcal{G},u_{0},U)$, we have*

[TABLE]

Proof 5.3**.**

First, it follows from Proposition 33 that $\mathrm{Play}(\mathcal{G},u_{0},U)$ implies $\mathrm{Path}(\mathcal{G},u_{0},U)$. We invoke the Predecessor Lemma 27 for Game Paths. Assuming $u_{0}\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}v$, Proposition 33 implies $u_{0}\vartriangleleft v$, so there is $u\in U$ such that $\mathsf{s}^{\vartriangleleft}(u,v)$. Since $U$ is an infinite play, $u\in U$ has an $\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}$-successor in $U$, i.e. there is some $u^{\prime}\in U$ such that $u\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}u^{\prime}$. Again since $U$ is an infinite play, we have

[TABLE]

But by Proposition 33 again, $v\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}u^{\prime}$ implies $u\vartriangleleft v\vartriangleleft u^{\prime}$, contradicting $\mathsf{s}^{\vartriangleleft}(u,u^{\prime})$, while $u^{\prime}\mathchoice{\mathrel{\overset{+}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{+}}}{\longrightarrow^{+}}{\longrightarrow^{+}}v$ implies $u\vartriangleleft u^{\prime}\vartriangleleft v$, contradicting $\mathsf{s}^{\vartriangleleft}(u,v)$. Hence $u^{\prime}=v$ and we are done.

Next, we show that games have infinite plays from any position, relying on Remark 12.

Lemma 38**.**

${\mathsf{FSO}_{\mathscr{D}}}$* proves that $\mathrm{Game}(\mathcal{G})$ implies*

[TABLE]

Proof 5.4**.**

Let $\mathcal{G}=(\mathsf{P},\mathsf{O},E_{\mathsf{P}},E_{\mathsf{O}})$. Fix $v\in V$. Using Remark 12, let $\preceq$ be a well-order on $\mathsf{P}\cup\mathsf{O}$. We extend the relation $\preceq$ to $\mathscr{D}^{*}\times(\mathsf{P}\cup\mathsf{O})$ by setting:

[TABLE]

Remark 12 implies that every non-empty $W$ such that

[TABLE]

has a $\preceq$-least element. By $\mathrm{HF}$-Bounded Choice (Theorem 22), we define

[TABLE]

by setting, for $\mathsf{J}$ either $\mathsf{P}$ or $\mathsf{O}$,

[TABLE]

Let $\mathcal{G}^{\prime}\mathrel{:=}(\mathsf{P},\mathsf{O},E^{\prime})$. Note that we have $\mathrm{Game}(\mathcal{G}^{\prime})$ and that

[TABLE]

By Comprehension for Product Types (Theorem 23), we then let

[TABLE]

It remains to show

[TABLE]

First, we have $v\in U$ by reflexivity of $\mathchoice{\mathrel{\overset{*}{\underset{\mathcal{G}^{\prime}}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}_{\mathcal{G}^{\prime}}}}{\longrightarrow^{*}_{\mathcal{G}^{\prime}}}{\longrightarrow^{*}_{\mathcal{G}^{\prime}}}$ (Proposition 33.(3)), and $(\forall u\in U)\left(v\mathchoice{\mathrel{\overset{*}{\underset{\mathcal{G}}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}_{\mathcal{G}}}}{\longrightarrow^{*}_{\mathcal{G}}}{\longrightarrow^{*}_{\mathcal{G}}}u\right)$ follows from (11). Moreover, we have

[TABLE]

thanks to (11), since this property already holds for $\mathcal{G}^{\prime}$. It remains to show that $U$ is linearly ordered w.r.t. $\mathchoice{\mathrel{\overset{*}{\underset{\mathcal{G}}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}_{\mathcal{G}}}}{\longrightarrow^{*}_{\mathcal{G}}}{\longrightarrow^{*}_{\mathcal{G}}}$. We invoke Proposition 35: its first premise has already been discussed, its second follows from the definition of $E^{\prime}$, and its last one is Proposition 33.(4).

Finally, in some situations (typically for the Simulation Theorem in §9), it is convenient to build infinite plays from paths (in the sense of Definition 4.3).

Lemma 39** (Infinite Plays From Paths).**

Assume $\mathrm{Game}(\mathcal{G})$ and let $u_{0}$ and $U$ be such that

[TABLE]

Then ${\mathsf{FSO}_{\mathscr{D}}}$ proves $\mathrm{Play}(\mathcal{G},u_{0},U)$.

Proof 5.5**.**

Thanks to Proposition 33, the result directly follows from the fact that

[TABLE]

Fix $u\in U$. By $\vartriangleleft$-induction we show $(\forall v\in U)(u\trianglelefteq v\leavevmode\nobreak\ \leavevmode\nobreak\ \Rightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ u\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}v)$. So let $v\in U$ such that the property holds for all $w\vartriangleleft v$, and assume $u\trianglelefteq v$. If $u=v$ then we are done. Otherwise, by the Predecessor Lemma 27 for Paths, we have $\mathsf{s}^{\vartriangleleft}(w,v)$ for some $w\in U$ with $u\trianglelefteq w$. By induction hypothesis we get $u\mathchoice{\mathrel{\overset{*}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}}}{\longrightarrow^{*}}{\longrightarrow^{*}}w\mathchoice{\mathrel{\overset{}{\underset{}{\longrightarrow}}}}{\mathrel{\longrightarrow}}{\longrightarrow}{\longrightarrow}v$ and we conclude by Proposition 33.

\opt

draftnotes

5.3. Strategies

We now turn to strategies. Our strategies are Functions from the positions of one player to the set of labels of the other player, which must respect the edge relations. This implies that all our strategies are, by definition, positional.

{defi}

[Strategies]

Let $\mathcal{G}=(\mathsf{P},\mathsf{O},E_{\mathsf{P}},E_{\mathsf{O}})$ where $\mathsf{P},\mathsf{O}$ are $\mathrm{HF}$-variables and where $E_{\mathsf{P}},E_{\mathsf{O}}$ are Function variables.

1. (1)

A $\mathsf{P}$-strategy on $\mathcal{G}$ is a Function $\sigma$ which satisfies the formula

[TABLE] 2. (2)

An $\mathsf{O}$-strategy on $\mathcal{G}$ is a Function $\sigma$ which satisfies the formula

[TABLE]

Strategies naturally induce subgames in the sense of Definition 31. This will allow us to lift to strategies notions which are more naturally defined at the level of games.

{defi}

[Subgame induced by a Strategy] Given a player $\mathsf{J}$ (either $\mathsf{P}$ or $\mathsf{O}$) and a $\mathsf{J}$-strategy $\sigma$ on $\mathcal{G}$, we let

[TABLE]

where

[TABLE]

and where $\{\sigma\}_{\mathsf{J}}\subseteq E(\mathcal{G})_{\mathsf{J}}$ is defined by $\mathrm{HF}$-Bounded Choice to be the Function taking $u\in\mathscr{D}^{*}\times\mathcal{G}_{\mathsf{J}}$ to the singleton $\{\sigma(u)\}$.

Whenever possible, we write $\mathcal{G}\mathord{\upharpoonright}\{\sigma\}$ or even just $\sigma$ for $\mathcal{G}\mathord{\upharpoonright}\{\sigma\}_{\mathsf{J}}$, when it is unambiguous.

Lemma 40**.**

${\mathsf{FSO}_{\mathscr{D}}}$* proves the following, where $\mathsf{J}$ is a player (either $\mathsf{P}$ or $\mathsf{O}$):*

[TABLE]

This in particular allows us to speak of the infinite plays of a strategy $\sigma$ on $\mathcal{G}$ simply as infinite plays of the game $\mathcal{G}\mathord{\upharpoonright}\{\sigma\}$.

5.4. Winning

In order to deal with acceptance for automata, we equip games with a notion of winning. Given a game $\mathcal{G}$, a winning condition on $\mathcal{G}$ is a formula $\mathcal{W}(U)$ where $U$ is intended to range over the infinite plays of $\mathcal{G}$. As usual a $\mathsf{P}$-strategy $\sigma$ on $(\mathcal{G},\mathcal{W})$ is winning from a position $v$ whenever all the infinite plays $U$ of $\sigma$ from $v$ satisfy $\mathcal{W}(U)$. Dually, an $\mathsf{O}$-strategy is winning from $v$ when all its infinite plays $U$ from $v$ satisfy $\lnot\mathcal{W}(U)$.

We formally proceed as follows.

{defi}

Let $\mathcal{G}=(\mathsf{P},\mathsf{O},E_{\mathsf{P}},E_{\mathsf{O}})$ where $\mathsf{P},\mathsf{O}$ are $\mathrm{HF}$-variables and $E_{\mathsf{P}},E_{\mathsf{O}}$ are Function variables. Let $\mathcal{W}(U)$ be a given $\mathsf{FSO}$-formula where $U$ is a Function variable.

1. (1)

We define the following formulae.

[TABLE] 2. (2)

Given a player $\mathsf{J}$ (either $\mathsf{P}$ or $\mathsf{O}$), we say that a $\mathsf{J}$-strategy $\sigma$ is winning in $(\mathcal{G},\mathcal{W})$ from $v$ if the game $(\mathcal{G}\mathord{\upharpoonright}\{\sigma\}_{\mathsf{J}},\mathcal{W})$ is won by $\mathsf{J}$ from $v$, i.e. if the following formula holds

[TABLE]

Strictly speaking, in Definition 5.4 above, $\mathrm{WonGame}_{\mathsf{J}}$ and $\mathrm{WinStrat}_{\mathsf{J}}$ are actually families of $\mathsf{FSO}$ formulae, parametrized by the choice of $\mathsf{FSO}$-formula $\mathcal{W}$.

As expected, a game position cannot be winning for both players.

Lemma 41**.**

${\mathsf{FSO}_{\mathscr{D}}}$* proves the following.*

[TABLE]

Proof 5.7**.**

Assume for contradiction that for some $v$ we have

[TABLE]

that is

[TABLE]

Consider the game

[TABLE]

Note that $\mathcal{G}^{\prime}$ is a subgame of both $\mathcal{G}\mathord{\upharpoonright}\{\sigma_{\mathsf{P}}\}$ and $\mathcal{G}\mathord{\upharpoonright}\{\sigma_{\mathsf{O}}\}$. We thus get

[TABLE]

which implies that there is no $U$ such that $\mathrm{Play}(\mathcal{G}^{\prime},v,U)$, contradicting Lemma 38.

5.5. Parity Conditions

In this paper, we mostly consider winning conditions expressed as parity conditions. Parity conditions are defined from colorings of game positions by natural numbers from a given finite interval. We represent natural numbers and the operations and relations on them using the Functions on $\mathrm{HF}$-Sets of $\mathsf{FSO}$ and the axioms of §3.4.4.

{conv}

In order to conveniently manipulate colorings and parity conditions, we will use the following functions on finite ordinals (a.k.a. natural numbers), obtained from the Axioms on $\mathrm{HF}$-Functions (see §3.4.4). We rely on the well-known fact that “$n$ is an ordinal” can be expressed by an $\mathrm{HF}$-formula $\mathrm{Ord}(n)$ (see e.g. [Jec06, Lemma 12.10]).

1. (1)

We consider unary $\mathrm{HF}$-Functions

[TABLE]

such that for all finite ordinals $n$, we have

[TABLE] 2. (2)

We consider binary $\mathrm{HF}$-Functions

[TABLE]

such that for finite ordinals $n,m$

[TABLE]

In $\mathsf{FSO}$-formulae, we write $n\leq m$ for the formula $\dot{\mathsf{g}}_{\leq}(n,m)\mathrel{\doteq}1$, and so on. 3. (3)

We consider a unary $\mathrm{HF}$-Function

[TABLE]

such that for each ordinal $n$, $\dot{\mathsf{even}}(n)$ is the set of ordinals $m\in[0,n]$ such that $m$ represents an even number. 4. (4)

We consider $\mathrm{HF}$-Functions $\mathsf{max}(-,-)$ and $(-)+1$, computing respectively the maximum of two finite ordinals and the successor ordinal of an ordinal.

Remark 42**.**

Even if “$n$ is an ordinal*” can be expressed by an $\mathrm{HF}$-formula, quantification over all finite ordinals cannot be expressed in $V_{\omega}$ by an $\mathrm{HF}$-formula, since for each finite ordinal $n>0$ we have $n\in V_{n}\setminus V_{n-1}$. In particular, induction over finite $\mathrm{HF}$-ordinals cannot be expressed by an $\mathrm{HF}$-formula.*

{defi}

[Parity Conditions]

Let $\mathcal{G}=(\mathsf{P},\mathsf{O},E_{\mathsf{P}},E_{\mathsf{O}})$ where $\mathsf{P},\mathsf{O}$ are $\mathrm{HF}$-variables and $E_{\mathsf{P}},E_{\mathsf{O}}$ are Function variables.

1. (1)

A coloring is given by a Function $C$ and an $\mathrm{HF}$-term $n$ satisfying the following formula

[TABLE] 2. (2)

We define the following formula:

[TABLE]

Remark 43**.**

The formula $\mathrm{Par}(\mathcal{G},C,n,U)$ will be used to say that an infinite play $U$ satisfies the (min) parity condition induced by the coloring $C:\mathcal{G}\mathrel{\mathfrak{to}}[0,n]$. In the standard model $\mathfrak{T}$, if $U$ is an infinite play in $\mathcal{G}$, then $\mathrm{Par}(\mathcal{G},C,n,U)$ holds if and only if there is an even $m\leq n$ such that $U$ has infinitely many positions colored by $m$, and $U$ has only finitely many positions colored by any $k<m$. Also, notice that any $U$ (not necessarily a play) satisfying $\mathrm{Par}(\mathcal{G},C,n,U)$ in $\mathfrak{T}$ is infinite.

Remark 44**.**

Assume that $\mathcal{G}^{\prime}$ is a subgame of $\mathcal{G}$ (in the sense of Definition 31). Note that $\mathsf{FSO}$ proves

[TABLE]

Furthermore, as noted earlier, every infinite play in $\mathcal{G}^{\prime}$ is an infinite play in $\mathcal{G}$. It follows that $\mathsf{FSO}$ proves

[TABLE]

Remark 45**.**

When considering parity automata in §6, it will actually be convenient to define acceptance via the formula $\mathrm{Par}$ for games of the form $\mathcal{G}(\trianglelefteq)$ in the sense of Remark 32. It follows from Remarks 32 and 44 that $\mathsf{FSO}$ proves

[TABLE]

We use the following more succinct notation for winning in the case parity games.

{nota}

[Winning in Parity Games] Let $\mathcal{G}=(\mathsf{P},\mathsf{O},E_{\mathsf{P}},E_{\mathsf{O}})$ where $\mathsf{P},\mathsf{O}$ are $\mathrm{HF}$-variables and $E_{\mathsf{P}},E_{\mathsf{O}}$ are Function variables. Let $C$ be a Function variable and $n$ be an $\mathrm{HF}$-variable. We write the following, where $\mathsf{J}$ is a player (either $\mathsf{P}$ or $\mathsf{O}$).

[TABLE]

5.6. The Axiom of Positional Determinacy of Parity Games.

We now formulate the axiom scheme $(\mathit{PosDet})$, which states the (positional) determinacy of parity games. Intuitively $(\mathit{PosDet})$ should consist of all formulae of the form

[TABLE]

But note that these formulae are open, and in particular

[TABLE]

contain free Function variables. On the other hand, when formulating our completeness results in §8, it will be interesting to have translations of instances of $(\mathit{PosDet})$ in $\mathsf{MSO}$, based on the map $\langle-\rangle:\mathsf{FSO}\to\mathsf{MSO}$ of §3.6. However, the translation $\langle-\rangle$ only handles $\mathrm{HF}$-closed formulae without free Function variables. We therefore officially let $(\mathit{PosDet})$ consist of all formulae $\mathrm{PosDet}(\mathsf{P},\mathsf{O},n)$, for $\mathsf{P}$, $\mathsf{O}$ and $n$ ranging over $\mathrm{HF}$-terms (see §3.2), where $\mathrm{PosDet}(\mathsf{P},\mathsf{O},n)$ is the formula

[TABLE]

It follows from the positional determinacy of parity games [EJ91] (see also [Tho97, Wal02, PP04]) that all instances of $(\mathit{PosDet})$ hold in the standard model $\mathfrak{T}$ of $\mathsf{FSO}$. We can thus extend Proposition 14 to the following.

Proposition 46**.**

For each closed $\mathsf{FSO}$-formula $\varphi$,

[TABLE]

5.6.1. The Axiom of Positional Determinacy in $\mathsf{MSO}$.

In order to obtain a complete axiomatization of ${\mathsf{MSO}_{\mathscr{D}}}$ from the completeness of ${\mathsf{FSO}_{\mathscr{D}}}+(\mathit{PosDet})$ (see §8), we extend the axioms of ${\mathsf{MSO}_{\mathscr{D}}}$ with sufficiently many translated instantiations $\langle\mathrm{PosDet}(\mathsf{P},\mathsf{O},n)\rangle$ for $\mathsf{P}$, $\mathsf{O}$ and $n$ closed $\mathrm{HF}$-terms. However, in general these terms may contain arbitrary $\mathrm{HF}$-Functions symbols, which make the translation $\langle\mathrm{PosDet}(\mathsf{P},\mathsf{O},n)\rangle$ in general uncomputable from $\mathsf{P}$, $\mathsf{O}$ and $n$ (see Remark 16 and §3.4.4). However, for each closed $\mathrm{HF}$-terms $\mathsf{P}$, $\mathsf{O}$ and $n$, there are constant symbols for $\mathrm{HF}$-sets $\dot{\mathsf{\mathsf{P}}}$, $\dot{\mathsf{\mathsf{O}}}$ and $\dot{\mathsf{n}}$ such that the formulae $\langle\mathrm{PosDet}(\mathsf{P},\mathsf{O},n)\rangle$ and $\langle\mathrm{PosDet}(\dot{\mathsf{\mathsf{P}}},\dot{\mathsf{\mathsf{O}}},\dot{\mathsf{n}})\rangle$ are syntactically identical. We therefore officially take the following version of $(\mathit{PosDet})$ in ${\mathsf{MSO}_{\mathscr{D}}}$.

{defi}

[The Axiom of Positional Determinacy in $\mathsf{MSO}$]

We let $\langle\mathit{PosDet}\rangle$ consist of all formulae of the form $\langle\mathrm{PosDet}(\dot{\mathsf{\mathsf{P}}},\dot{\mathsf{\mathsf{O}}},\dot{\mathsf{n}})\rangle$, for $\dot{\mathsf{\mathsf{P}}}$, $\dot{\mathsf{\mathsf{O}}}$ and $\dot{\mathsf{n}}$ ranging over constant symbols for $\mathrm{HF}$-sets.

6. Alternating Tree Automata

We detail in this Section a representation of alternating tree automata in $\mathsf{FSO}$. We closely follow the presentation of [Wal02]. Our main motivation to consider alternating automata is that when formulating acceptance with (parity) games (of the kind of §5), complementation follows from (positional) determinacy (i.e. in our setting from the Axiom $(\mathit{PosDet})$). Let us recall the main ideas underlying alternating automata. The original formulation, as in e.g. [MS87, MS95], is for an automaton $\mathcal{A}$ with state set $Q$ to have transitions with values in the free distributive lattice over $\mathscr{D}\times Q$ (in other words, transitions have positive Boolean formulae over $\mathscr{D}\times Q$ as values). Actually, following [Wal02] we can simply assume that transitions are of the form

[TABLE]

and we read $\partial(q,\mathtt{a})$ as the disjunctive normal form

[TABLE]

This results in acceptance games where intuitively $\mathsf{P}$ plays from disjunctions while $\mathsf{O}$ plays from conjunctions. In the following we often call the $\gamma\in\partial(q,\mathtt{a})$ conjunctions.

We begin by giving basic definitions in 6.1. Because our setting is restricted to only describe positional strategies, and because parity games are positionally determined, we give a special emphasis to parity automata, whose acceptance conditions are parity conditions generated from a coloring of their states. We then present a series of operations on automata, on which we rely in §8.3 for the interpretation of $\mathsf{MSO}$ formulae as automata. We recapitulate them in Table 1. First, §6.2 and §6.3 present two simple constructions implementing respectively a substitution and a disjunction operation. We discuss in §6.4 and §6.5 the important special case of non-deterministic automata. Non-deterministic automata are important because they allow us, via the usual projection operation (§6.5), to interpret the existential quantifier of $\mathsf{MSO}$ (see §8.3). To this end, an important result of the theory of automata on infinite trees is the Simulation Theorem [EJ91, MS95], which states that each alternating automaton is equivalent to a non-deterministic one. The formalization of this result in $\mathsf{FSO}$ is deferred to §9. This is the only part of this paper where we shall (momentarily) use automata with acceptance conditions which are not parity conditions. This result moreover relies on the complete axiomatization of $\mathsf{MSO}$ on $\omega$-words for paths of $\mathsf{FSO}$ (to be discussed in §7). Finally, in §6.6 we discuss complementation in the setting of $\mathsf{FSO}$, and show that alternating automata can be complemented in $\mathsf{FSO}$ when we assume the Axiom $(\mathit{PosDet})$ of Positional Determinacy of Parity Games.

\opt

draftnotes

6.1. Alternating Tree Automata in ${\mathsf{FSO}_{\mathscr{D}}}$

We present here a representation of alternating tree automata in $\mathsf{FSO}$.

{defi}

[Alternating Tree Automata]

Given an $\mathrm{HF}$-Set $\Sigma$, an Alternating Tree Automaton (or simply Automaton) $\mathcal{A}$ on $\Sigma$ (notation $\mathcal{A}:\Sigma$) is given by $\mathrm{HF}$-terms $Q_{\mathcal{A}}$, ${q}^{\iota}_{\mathcal{A}}$ and $\partial_{\mathcal{A}}$ together with an $\mathsf{FSO}$-formula $\Omega_{\mathcal{A}}(U)$ of a Function variable $U$, which are required to satisfy the following formula:

[TABLE]

where $\mathcal{P}_{*}(-)$ is the $\mathrm{HF}$-Function of §3.4.4.4. We write

[TABLE]

and adopt the following terminology: $\Sigma$ is the input alphabet of $\mathcal{A}$, $Q_{\mathcal{A}}$ is its set of states (with ${q}^{\iota}_{\mathcal{A}}$ initial), $\partial_{\mathcal{A}}$ is the transition function of $\mathcal{A}$ and $\Omega_{\mathcal{A}}$ is its acceptance condition.

We often write $\mathrm{Aut}(\mathcal{A}:\Sigma)$ or even $\mathrm{Aut}(\mathcal{A})$ for $\mathrm{Aut}(\Sigma,Q_{\mathcal{A}},{q}^{\iota}_{\mathcal{A}},\partial_{\mathcal{A}})$.

An automaton $\mathcal{A}:\Sigma$ is intended to run over $\Sigma$-labeled $\mathscr{D}$-ary trees, represented as Functions $F:\Sigma$ (equivalently $F:\mathscr{D}^{*}\mathrel{\mathfrak{to}}\Sigma$, following §3.7). As usual, acceptance is modeled using games, which we formalize in the setting of §5.

{defi}

[Acceptance Games]

Given an automaton $\mathcal{A}:\Sigma$ and a Function $F:\Sigma$ we define the acceptance game $\mathcal{G}(\mathcal{A},F)$ as follows:

[TABLE]

and $E(\mathcal{G}(\mathcal{A},F))_{\mathsf{P}}$, $E(\mathcal{G}(\mathcal{A},F))_{\mathsf{O}}$ are defined by $\mathrm{HF}$-Bounded Choice for Product Types (Theorem 22) and Comprehension for $\mathrm{HF}$-Sets (Remark 3.7.3) as

[TABLE]

Remark 47**.**

Note that $\mathrm{Aut}(\mathcal{A})$ implies $\mathrm{Game}(\mathcal{G}(\mathcal{A},F))$ for $F:\Sigma$. The edge relations of $\mathcal{G}(\mathcal{A},F)$ (in the sense of Definition 5.1) are given by

[TABLE]

Note also that an $\mathsf{O}$-position $(x,(q,\gamma))$ is equipped with the information $(x,q)\in\mathcal{G}(\mathcal{A},F)_{\mathsf{P}}$. It follows that an $\mathsf{O}$-position has at most one predecessor. This is useful when complementing automata (§6.6).

{conv}

It the rest of this paper, unless explicitly stated otherwise, when speaking of an infinite play in an acceptance game $\mathcal{G}(\mathcal{A},F)$ (including infinite plays in strategies in such games), we always mean an infinite play from position $(\dot{\varepsilon},{q}^{\iota}_{\mathcal{A}})$.

Given $\mathcal{A}:\Sigma$ and $F:\Sigma$, the acceptance condition $\Omega_{\mathcal{A}}(-)$ of $\mathcal{A}$ induces a winning condition in the sense of §5.4 in the game $\mathcal{G}(\mathcal{A},F)$. This gives the following notions of tree acceptance and language generated by an automaton.

{defi}

[Language of an Automaton]

Given an automaton $\mathcal{A}:\Sigma$, a winning condition $\Omega_{\mathcal{A}}$ (in the sense of Definition 5.4) and $F:\Sigma$, we say that $\mathcal{A}$ accepts $F$ when the following formula $F\in\mathcal{L}(\mathcal{A})$ holds.

[TABLE]

Recall that the formulae $\mathrm{Strat}$ and $\mathrm{WinStrat}$ are defined in Def. 5.3 (§5.3) and Def. 5.4 (§5.4) respectively. In words, the formula $F\in\mathcal{L}(\mathcal{A})$ of Definition 47 states that $\mathsf{P}$ has a winning strategy from position $(\dot{\varepsilon},{q}^{\iota}_{\mathcal{A}})$ in the game $\mathcal{G}(\mathcal{A},F)$.

Except for the Simulation Theorem in §9, we shall only consider automata whose acceptance conditions are given by parity conditions in the sense of §5.5. Recall from Definition 42 that a parity condition on a game $\mathcal{G}$ is given by the formula

[TABLE]

which depends on $\mathcal{G}$. However, it is desirable that automata come, as in Definition 6.1, with acceptance conditions which are independent from any particular acceptance game. Note that for a given automaton $\mathcal{A}$, all acceptance games $\mathcal{G}(\mathcal{A},F)$ have the same sets of $\mathsf{P}$ and $\mathsf{O}$ labels and positions; the input trees $F$ can only induce different edge relations. Recall now the games $\mathcal{G}(\trianglelefteq)$ from Remark 32. The game $\mathcal{G}(\trianglelefteq)$ has the same labels and positions as $\mathcal{G}$, but its edge relation is exactly the partial order $\trianglelefteq$ discussed in §4. It follows that for a fixed automaton $\mathcal{A}:\Sigma$, all acceptance games $\mathcal{G}(\mathcal{A},F)$ for $F:\Sigma$ induce the same $\mathcal{G}(\mathcal{A},F)(\trianglelefteq)$, that we shall write

[TABLE]

{defi}

[Parity Automata]

Let

[TABLE]

We say that $\mathcal{A}$ is a parity automaton if $\mathcal{A}$ comes equipped with $\mathrm{HF}$-terms $n_{\mathcal{A}}$ and $C_{\mathcal{A}}$ such that the two following conditions are satisfied.

1. (1)

The following formula holds

[TABLE] 2. (2)

The formula $\Omega_{\mathcal{A}}(U)$ is $\mathrm{Par}(\mathcal{G}(\mathcal{A})(\trianglelefteq),\hat{C}_{\mathcal{A}},n_{\mathcal{A}})$, where

[TABLE]

We write

[TABLE]

for a parity automaton $\mathcal{A}$ with $C_{\mathcal{A}}$ and $n_{\mathcal{A}}$ as above. Furthermore, we write $\mathrm{Par}(\mathcal{A},\hat{C}_{\mathcal{A}},n_{\mathcal{A}},U)$ or even $\mathrm{Par}(\mathcal{A},U)$ for the formula $\mathrm{Par}(\mathcal{G}(\mathcal{A})(\trianglelefteq),\hat{C}_{\mathcal{A}},n_{\mathcal{A}},U)$.

In Definition 47, the purpose of the coloring $\hat{C}_{\mathcal{A}}$ is to equip the game $\mathcal{G}(\mathcal{A})(\trianglelefteq)$ with a coloring in the sense of Def. 42 (§5.5), namely a coloring of the positions of the game, while the coloring $C_{\mathcal{A}}$ only colors the states of $\mathcal{A}$.

Note that it follows from Remarks 32 and 45 that $\mathsf{FSO}$ proves

[TABLE]

where the formula $\mathrm{Sub}(\mathcal{G},\mathcal{G}^{\prime})$ (stating that $\mathcal{G}$ is a subgame of $\mathcal{G}^{\prime}$) is defined in Def. 31 (§5.1), and

[TABLE]

The following simple fact will be useful when proving the Simulation Theorem in §9.

Remark 48**.**

Given two plays $U$ and $V$ in $\mathcal{G}(\mathcal{A})(\trianglelefteq)$, if $U_{\mathsf{P}}=V_{\mathsf{P}}$ then

[TABLE]

Other than the Simulation Theorem in §9, all constructions we need on automata can be performed on automata $\mathcal{A}:\Sigma$ where $\Sigma$, $Q_{\mathcal{A}}$, ${q}^{\iota}_{\mathcal{A}}$, $\partial_{\mathcal{A}}$, $C_{\mathcal{A}}$ and $n_{\mathcal{A}}$ are given by arbitrary $\mathrm{HF}$-terms. However, our completeness result (§8) ultimately relies, via Proposition 78, on the completeness of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ over $\omega$-words (§7) and requires automata to be given by closed $\mathrm{HF}$-terms. In addition, our proof of the Simulation Theorem uses McNaughton’s Theorem [McN66], and imports it into $\mathsf{FSO}$ by Proposition 78, which also requires automata to be closed objects. This leads to the following.

{defi}

A parity automaton $\mathcal{A}:\Sigma$ is $\mathrm{HF}$-closed if $\Sigma$, $Q_{\mathcal{A}}$, ${q}^{\iota}_{\mathcal{A}}$, $\partial_{\mathcal{A}}$, $C_{\mathcal{A}}$ and $n_{\mathcal{A}}$ are closed $\mathrm{HF}$-terms.

Remark 49**.**

For each of our constructions on automata (see Table 1), the alphabets, states and colorings of new automata will be obtained by composing simple Functions on $\mathrm{HF}$-Sets from §3.4.4 and Convention 5.5. In particular this means that the obtained automata have $\mathrm{HF}$-closed alphabets, states and coloring provided we started from $\mathrm{HF}$-closed ones.

On the other hand, transition functions may be more complex (see §6.6 or §9), and we often present them in a way suggesting the use of the Axiom of $\mathrm{HF}$-Bounded Choice for $\mathrm{HF}$-Sets (§3.4.5). This is unproblematic when $\mathrm{HF}$-closedness is not at issue. To preserve $\mathrm{HF}$-closedness, starting from $\mathrm{HF}$-closed automata, the transition functions of the newly built automata must always be read as being constructed from concrete $\mathrm{HF}$-sets.

{conv}

In the rest of this paper, whenever we speak of a (parity) automaton $\mathcal{A}$ in formal statements, we always mean that the formula $\mathrm{Aut}(\mathcal{A})$ (resp. $\mathrm{PAut}(\mathcal{A})$) holds. (By contrast, $\mathrm{HF}$-closedness is an external notion.)

6.2. Substitution

Let $\mathcal{A}:\Sigma$ be an automaton and let $\Gamma$ and $f:\Gamma\mathrel{\mathfrak{to}}\Sigma$ be $\mathrm{HF}$-sets. The automaton $\mathcal{A}[f]:\Gamma$ is defined to have the same states and acceptance condition as $\mathcal{A}:\Sigma$, and its transitions are given by

[TABLE]

Note that $\mathrm{Aut}(\mathcal{A})\land(\exists\mathtt{b}\mathrel{\dot{\in}}\Gamma)$ implies $\mathrm{Aut}(\mathcal{A}[f])$. Also, $\mathcal{A}[f]$ is a parity automaton whenever $\mathcal{A}$ is. Furthermore, it follows from Remark 49 that $\mathcal{A}[f]:\Gamma$ is $\mathrm{HF}$-closed when $\mathcal{A}:\Gamma$ is $\mathrm{HF}$-closed and in addition $\Gamma$ and $f$ are closed $\mathrm{HF}$-terms. A typical use of substitution, on which we rely when translating formulae to automata in §8.3, is to enlarge the input alphabet of an automaton. For instance, given $\mathrm{HF}$-closed $\Sigma_{1},\dots,\Sigma_{n}$ and an $\mathrm{HF}$-closed $\mathcal{A}:\Sigma_{i}$, we obtain an $\mathrm{HF}$-closed

[TABLE]

where

[TABLE]

is a projection $\mathrm{HF}$-Function of §3.4.4.6.

Lemma 50**.**

Given $\Gamma$, $f$ and $\mathcal{A}$ as above, ${\mathsf{FSO}_{\mathscr{D}}}$ proves the following.

[TABLE]

Note that by $\mathrm{HF}$-Bounded Choice, ${\mathsf{FSO}_{\mathscr{D}}}$ proves that

[TABLE]

so the above Lemma could have equivalently been stated with an existentially bound $F$.

6.3. Disjunction

We use here the $\mathrm{HF}$-Functions from §3.4.4.8 and Convention 5.5.4. Given parity automata $\mathcal{A}_{0},\mathcal{A}_{1}:\Sigma$, the parity automaton $\mathcal{A}_{0}\oplus\mathcal{A}_{1}:\Sigma$ has state set

[TABLE]

with ${q}^{\iota}$ initial, transitions given by

[TABLE]

and coloring $C:Q_{\mathcal{A}_{0}\oplus\mathcal{A}_{1}}\mathrel{\mathfrak{to}}[0,n]$ (where $n=\mathsf{max}(n_{\mathcal{A}_{0}},n_{\mathcal{A}_{1}})$) given by

[TABLE]

We have

[TABLE]

Moreover, if follows from Remark 49 that $\mathcal{A}_{0}\oplus\mathcal{A}_{1}:\Sigma$ is $\mathrm{HF}$-closed whenever $\mathcal{A}_{0}:\Sigma$ and $\mathcal{A}_{1}:\Sigma$ are.

Remark 51**.**

Even in our positional setting, strictly speaking the automaton $\mathcal{A}_{0}\oplus\mathcal{A}_{1}$ does not require $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$ to be parity automata (see Table 1). However, the acceptance condition of $\mathcal{A}_{0}\oplus\mathcal{A}_{1}$ is actually simpler to define when both $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$ are parity automata. Since we shall only need $\mathcal{A}_{0}\oplus\mathcal{A}_{1}$ for parity automata, we only formally define disjunction in this setting.

Lemma 52**.**

${\mathsf{FSO}_{\mathscr{D}}}$* proves the following.*

[TABLE]

Proof 6.2**.**

Assume first that $F\in\mathcal{L}(\mathcal{A}_{0}\oplus\mathcal{A}_{1})$ for $F:\Sigma$, and consider a winning $\mathsf{P}$-strategy $\sigma$ in the acceptance game $\mathcal{G}(\mathcal{A}_{0}\oplus\mathcal{A}_{1},F)$. We first look at the move of $\sigma$ on the initial position $(\dot{\varepsilon},{q}^{\iota})$. By definition of $\mathcal{A}_{0}\oplus\mathcal{A}_{1}$ we have

[TABLE]

Assume $\gamma\in\partial_{\mathcal{A}_{i}}({q}^{\iota}_{\mathcal{A}_{i}},F(\dot{\varepsilon}))$. Then $\sigma$ induces a $\mathsf{P}$-strategy $\sigma_{i}$ in $\mathcal{G}(\mathcal{A}_{i},F)$. The strategy $\sigma_{i}$ is defined using $\mathrm{HF}$-Bounded Choice for Product Types (Theorem 22) by putting

[TABLE]

It remains to show that $\sigma_{i}$ is winning, that is

[TABLE]

for $\iota=(\dot{\varepsilon},{q}^{\iota}_{\mathcal{A}_{i}})$. Consider an infinite play $V$ of $\sigma_{i}$ from $\iota$. Then by Comprehension for Product Types (Theorem 23), define $U:\mathcal{G}(\mathcal{A}_{0}\oplus\mathcal{A}_{1},F)\mathrel{\mathfrak{to}}2$ as the set of all $(x,\ell)$ such that either $(x,\ell)=(\dot{\varepsilon},{q}^{\iota})$ or $(x,\ell)\in V$. It is clear that

[TABLE]

The converse is proved similarly.

6.4. Non-Deterministic Automata

We turn to the important class of alternating automata known as non-deterministic automata. Non-deterministic automata are important because they allow us, via the usual projection operation (§6.5), to interpret the existential quantifier of $\mathsf{MSO}$ (see §8). An important result in the theory of automata on infinite trees is the Simulation Theorem [EJ91, MS95] (addressed in §9), stating that each alternating automata can be simulated by a non-deterministic one.

Intuitively, an automaton $\mathcal{A}$ is non-deterministic if in acceptance games $\mathsf{O}$ can only explicitly choose tree directions but not states.

{defi}

[Non-Deterministic Automata]

An automaton $(\mathcal{A}:\Sigma)$ in the sense of Definition 6.1, with

[TABLE]

is non-deterministic if for every $q\in Q_{\mathcal{A}}$, every $\mathtt{a}\in\Sigma$, every $\gamma\in\partial_{\mathcal{A}}(q,\mathtt{a})$, and every tree direction $d\in\mathscr{D}$, there is at most one $q^{\prime}\in Q_{\mathcal{A}}$ such that $(d,q^{\prime})\in\gamma$.

The key property of non-deterministic automata is that in each play of a $\mathsf{P}$-strategy $\sigma$ in an acceptance game, the sequence of states is uniquely determined from the tree positions. We formally state this as follows.

Lemma 53**.**

Consider a non-deterministic automaton $\mathcal{A}:\Sigma$, and let $F:\Sigma$. Furthermore let $\sigma$ be a $\mathsf{P}$-strategy in $\mathcal{G}(\mathcal{A},F)$. Then ${\mathsf{FSO}_{\mathscr{D}}}$ proves that for all $x\in\mathscr{D}^{*}$ and all infinite plays $V$ and $V^{\prime}$ of $\sigma$, if

[TABLE]

then for all $y\mathrel{\dot{\leq}}x$, all $q\in Q_{\mathcal{A}}$, and all $\gamma\in\mathcal{P}_{*}(\mathscr{D}\times Q_{\mathcal{A}})$, we have

[TABLE]

Proof 6.3**.**

Fix $\sigma$ and $V,V^{\prime}$ as in the statement of the Lemma and let $x\in\mathscr{D}^{*}$. First, note that for every $y\mathrel{\dot{\leq}}x$ we have

Claim 54**.**

[TABLE]

Proof 6.4** (Proof of Claim 54).**

We use the Induction Axiom of $\mathsf{FSO}$ (§3.4.2). The property holds for $\dot{\varepsilon}\mathrel{\dot{\leq}}x$ since $(\dot{\varepsilon},{q}^{\iota}_{\mathcal{A}})$ belongs to both $V$ and $V^{\prime}$. Assume now the property for $y\mathrel{\dot{\leq}}x$, and consider some tree direction $d\in\mathscr{D}$ such that $\mathsf{S}_{d}(y)\mathrel{\dot{\leq}}x$. By assumption, we have some $q\in Q_{\mathcal{A}}$ such that $(y,q)\in V$, and by using $\mathrm{Game}(\sigma)$ twice, we get some $q^{\prime}\in Q_{\mathcal{A}}$ and some $d^{\prime}\in\mathscr{D}$ such that

[TABLE]

But since $V$ is a play of $\sigma$, by Proposition 33 we must have $\mathsf{S}_{d^{\prime}}(y)\trianglelefteq x$, so that $d^{\prime}=d$ and we are done. The same reasoning gives the result for $V^{\prime}$.

Using the Induction Axiom of $\mathsf{FSO}$ (§3.4.2), we now show that

[TABLE]

First, we have

[TABLE]

Assume now the property for $y\mathrel{\dot{\leq}}x$ and let us prove it for $\mathsf{S}_{d}(y)$ with $\mathsf{S}_{d}(y)\mathrel{\dot{\leq}}x$. It follows from the induction hypothesis and Claim 54 that we have $(y,q)\in V$ and $(y,q)\in V^{\prime}$ for some $q\in Q_{\mathcal{A}}$. Again by Claim 54, let $q^{\prime},q^{\prime\prime}\in Q_{\mathcal{A}}$ such that $(\mathsf{S}_{d}(y),q^{\prime})\in V$ and $(\mathsf{S}_{d}(y),q^{\prime\prime})\in V^{\prime}$. Now since $V$ and $V^{\prime}$ are plays of $\sigma$, there are $\gamma$, $\gamma^{\prime}$ such that $(y,(q,\gamma))\in V$ and $(y,(q,\gamma^{\prime}))\in V^{\prime}$, and we necessarily have

[TABLE]

so that $\gamma=\gamma^{\prime}$. Moreover, we have $(d,q^{\prime}),(d,q^{\prime\prime})\in\gamma$, but this implies $q^{\prime}=q^{\prime\prime}$ since $\mathcal{A}$ is non-deterministic.

This concludes the proof of Lemma 53.

\opt

draftnotes

Corollary 55**.**

Given $\mathcal{A}$, $F$ and $\sigma$ as in Lemma 53, ${\mathsf{FSO}_{\mathscr{D}}}$ proves that for each $x\in\mathscr{D}^{*}$ there is at most one $q\in Q_{\mathcal{A}}$ such that

[TABLE]

We now state the Simulation Theorem [EJ91, MS95]. Its proof in ${\mathsf{FSO}_{\mathscr{D}}}$, requiring $\mathrm{HF}$-closedness of automata, is deferred to §9.

Theorem 56** (Simulation).**

For each $\mathrm{HF}$-closed parity automaton $\mathcal{A}:\Sigma$ there is a non-deterministic $\mathrm{HF}$-closed parity automaton $\mathrm{ND}(\mathcal{A}):\Sigma$ such that

[TABLE]

6.5. Projection

We now discuss the usual operation of projection, which allows us to interpret (existential) quantification in $\mathsf{MSO}$ (see §8.3). This operation is defined on arbitrary alternating automata, but it only correctly computes the appropriate projection for non-deterministic ones.

Given an automaton $\mathcal{A}:\Sigma\times\Gamma$ as in Definition 6.1, we define its projection on $\Sigma$ to be the automaton $\exists_{\Gamma}\mathcal{A}:\Sigma$ with

[TABLE]

where

[TABLE]

is given by

[TABLE]

Note that $\mathrm{Aut}(\mathcal{A}:\Sigma\times\Gamma)$ implies $\mathrm{Aut}(\exists_{\Gamma}\mathcal{A})$. Moreover, $\exists_{\Gamma}\mathcal{A}:\Sigma$ is an ($\mathrm{HF}$-closed) parity automaton whenever so is $\mathcal{A}:\Sigma\times\Gamma$.

We shall now prove that $\exists_{\Gamma}\mathcal{A}:\Sigma$ indeed implements the projection of $\mathcal{A}:\Sigma\times\Gamma$. This involves a notion of pairing for trees. Given $F:\Sigma$ and $G:\Gamma$, we let ${\langle F,G\rangle}:\Gamma\times\Sigma$ be given (using the axiom of $\mathrm{HF}$-Bounded Choice for $\mathrm{HF}$-Functions (§3.4.5)) by

[TABLE]

Proposition 57**.**

Consider a non-deterministic $\mathcal{A}:\Sigma\times\Gamma$ and let $\exists_{\Gamma}\mathcal{A}:\Sigma$ be as defined above. Then ${\mathsf{FSO}_{\mathscr{D}}}$ proves the following.

[TABLE]

Proof 6.6**.**

Given $G:\Gamma$ and a winning $\mathsf{P}$-strategy $\sigma$ on $\mathcal{G}(\mathcal{A},{\langle F,G\rangle})$, it is easy to see that $\sigma$ is also a winning strategy on $\mathcal{G}(\exists_{\Gamma}\mathcal{A},F)$.

Conversely, assume that $\sigma$ is a winning $\mathsf{P}$-strategy on $\mathcal{G}(\exists_{\Gamma}\mathcal{A},F)$. We define a tree $G:\Gamma$ by $\mathrm{HF}$-Bounded Choice for $\mathrm{HF}$-Functions (§3.4.5) as follows:

• •

For $x\in\mathscr{D}^{*}$, if there is some infinite play $U$ of $\sigma$ such that $(x,q)\in U$ for some state $q\in Q_{\mathcal{A}}$, then we let $G(x)$ be some $\mathtt{b}\in\Gamma$ such that $\sigma(x,q)\in\partial_{\mathcal{A}}(q,(F(x),\mathtt{b}))$.

• •

Otherwise, we let $G(x)$ be any element of $\Gamma$.

We now define a $\mathsf{P}$-strategy $\sigma_{G}$ on $\mathcal{G}(\mathcal{A},{\langle F,G\rangle})$ as follows, again using $\mathrm{HF}$-Bounded Choice for $\mathrm{HF}$-Functions (§3.4.5).

• •

If $(x,q)\in U$ for some infinite play $U$ of $\sigma$, then we let $\sigma_{G}(x,q)\mathrel{:=}\sigma(x,q)$.

• •

Otherwise, we let $\sigma_{G}(x,q)=(q,\gamma)$, where $\gamma\in\partial_{\mathcal{A}}(q,{\langle F,G\rangle}(x))$.

We first check that $\sigma_{G}$ is indeed a strategy on $\mathcal{G}(\mathcal{A},{\langle F,G\rangle})$, namely that for all $(x,q)\in\mathscr{D}^{*}\times Q_{\mathcal{A}}$, if $\sigma_{G}(x,q)=(q,\gamma)$ then $\gamma\in\partial_{\mathcal{A}}(q,{\langle F,G\rangle}(x))$. If $(x,q)$ belongs to no infinite play of $\sigma$, then the result follows by definition of $\sigma_{G}$. Otherwise, by Corollary 55, $q$ is unique in $Q_{\mathcal{A}}$ such that $(x,q)$ belongs to an infinite play of $\sigma$, and we are done since

[TABLE]

In order to show that $\sigma_{G}$ is winning, we show that any infinite play of $\sigma_{G}$ is also an infinite play of $\sigma$. So let $U:\mathcal{G}(\mathcal{A},{\langle F,G\rangle})\mathrel{\mathfrak{to}}2$ such that

[TABLE]

We are done if we show that

[TABLE]

which follows from the fact that

Claim 58**.**

[TABLE]

Proof 6.7** (Proof of Claim 58).**

We apply the Induction Scheme of ${\mathsf{FSO}_{\mathscr{D}}}$ (§3.4.2). In the base case $x=\dot{\varepsilon}$, and we conclude by Lemma 38.

For the induction step consider the case of $\mathsf{S}_{d}(x)$, assuming the property for $x$. So let $q^{\prime}\in Q_{\mathcal{A}}$ such that $(\mathsf{S}_{d}(x),q^{\prime})\in U$. First, by applying twice the Predecessor Lemma 37 for Infinite Plays, we get some $q\in Q_{\mathcal{A}}$ such that $(x,q)\in U$, and by induction hypothesis, there is some infinite play $W$ of $\sigma$ such that $(x,q)\in W$. But then, by definition of $\sigma_{G}$, we have $\sigma(x,q)=\sigma_{G}(x,q)$. We thus have $(d,q^{\prime})\in\gamma$, where $(q,\gamma)=\sigma(x,q)$. Using Lemma 38, let now $W^{\prime}$ be an infinite play of $\sigma$ from position $(\mathsf{S}_{d}(x),q^{\prime})$. By Comprehension for Product Types (Theorem 23), we define an infinite play $W^{\prime\prime}$ of $\sigma$ from position $(\dot{\varepsilon},{q}^{\iota}_{\mathcal{A}})$ as follows:

• •

Given $u$ a position of $\mathcal{G}(\mathcal{A},{\langle F,G\rangle})$, if $u\in W^{\prime}$ then $u\in W^{\prime\prime}$. Otherwise, we let $u\in W^{\prime\prime}$ iff $u\in W$ and $u\mathchoice{\mathrel{\overset{*}{\underset{\sigma}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}_{\sigma}}}{\longrightarrow^{*}_{\sigma}}{\longrightarrow^{*}_{\sigma}}(\mathsf{S}_{d}(x),q^{\prime})$.

It is then easy to check that $W^{\prime\prime}$ is an infinite play of $\sigma$.

This concludes the proof of Proposition 57.

6.6. Complementation

It is known that, assuming the determinacy of acceptance games, alternating tree automata are closed under complement [MS87]. On the other hand, our setting only allows us to manipulate positional strategies on acceptance games, which leads us to formulate complementation for parity automata, since their acceptance games are always positionally determined. Thus, in this section, we formalize the fact that, assuming the axiom $(\mathit{PosDet})$, each alternating parity automaton has a complement in $\mathsf{FSO}$. More precisely, we prove the following.

Theorem 59** (Complementation of Tree Automata).**

For each ($\mathrm{HF}$-closed) parity automaton $\mathcal{A}:\Sigma$, there is an ($\mathrm{HF}$-closed) parity automaton ${\mathord{\sim}}\mathcal{A}:\Sigma$ such that

[TABLE]

Alternating automata may be directly complemented in a locally syntactic fashion. For an automaton $\mathcal{A}:\Sigma$ we may define a complement automaton ${\mathord{\sim}}\mathcal{A}:\Sigma$ with the same states as $\mathcal{A}$, and such that $\mathsf{P}$-strategies in acceptance games for ${\mathord{\sim}}\mathcal{A}$ correspond (w.r.t. the visited states in infinite plays) to $\mathsf{O}$-strategies in acceptance games for $\mathcal{A}$, and vice-versa. Closely following [Wal02], the basic idea is to see the transition function of $\mathcal{A}$

[TABLE]

as taking $(q,\mathtt{a})$ to the disjunctive normal form

[TABLE]

Then, for the complement ${\mathord{\sim}}\mathcal{A}:\Sigma$ of $\mathcal{A}$, we can let

[TABLE]

take $(q,\mathtt{a})$ to the De Morgan dual of $\partial_{\mathcal{A}}(q,\mathtt{a})$.

We now proceed to the formal definition.

{defi}

Given a parity automaton $\mathcal{A}:\Sigma$, we define the parity automaton ${\mathord{\sim}}\mathcal{A}:\Sigma$ as follows. The automaton ${\mathord{\sim}}\mathcal{A}$ has the same states and initial state as $\mathcal{A}$. Its transitions are defined as

[TABLE]

Its coloring is given as follows, using Convention 5.5.4:

[TABLE]

Note that by Remark 49, ${\mathord{\sim}}\mathcal{A}:\Sigma$ is $\mathrm{HF}$-closed whenever so is $\mathcal{A}:\Sigma$. We are now going to prove Theorem 59. To this end, fix a parity automaton $\mathcal{A}:\Sigma$ and let ${\mathord{\sim}}\mathcal{A}:\Sigma$ be as in Definition 6.6. Fix also some $F:\Sigma$. We split Theorem 59 into the following statements.

Proposition 60**.**

$\mathsf{FSO}+(\mathit{PosDet})\vdash\leavevmode\nobreak\ \leavevmode\nobreak\ F\notin\mathcal{L}(\mathcal{A})\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ F\in\mathcal{L}({\mathord{\sim}}\mathcal{A})$.

Proposition 61**.**

$\mathsf{FSO}\vdash\leavevmode\nobreak\ \leavevmode\nobreak\ F\in\mathcal{L}({\mathord{\sim}}\mathcal{A})\leavevmode\nobreak\ \leavevmode\nobreak\ \Longrightarrow\leavevmode\nobreak\ \leavevmode\nobreak\ F\notin\mathcal{L}(\mathcal{A})$.

The key is that $\mathsf{P}$-strategies on $\mathcal{G}({\mathord{\sim}}\mathcal{A},F)$ correspond to $\mathsf{O}$-strategies on $\mathcal{G}(\mathcal{A},F)$, and vice-versa. We make this formal in §6.6.1 and §6.6.2 below. First, notice that $Q_{{\mathord{\sim}}\mathcal{A}}=Q_{\mathcal{A}}$, so that the games $\mathcal{G}(\mathcal{A},F)$ and $\mathcal{G}({\mathord{\sim}}\mathcal{A},F)$ have the same sets of labels

[TABLE]

In the following, we let

[TABLE]

be the set of positions of the games $\mathcal{G}(\mathcal{A},F)$ and $\mathcal{G}({\mathord{\sim}}\mathcal{A},F)$, and we let $\iota\mathrel{:=}(\dot{\varepsilon},{q}^{\iota}_{\mathcal{A}})$ be their (common) initial position.

6.6.1. Proof of Proposition 60.

We are going to show that $\mathsf{FSO}+(\mathit{PosDet})$ proves

[TABLE]

First, given an $\mathsf{O}$-strategy $\sigma_{\mathsf{O}}$ on $\mathcal{G}(\mathcal{A},F)$, we define a $\mathsf{P}$-strategy $\sigma_{\mathsf{P}}$ on $\mathcal{G}({\mathord{\sim}}\mathcal{A},F)$. Assuming that $\sigma_{\mathsf{O}}$ satisfies $\mathrm{Strat}_{\mathsf{O}}(\mathcal{G}(\mathcal{A},F),\sigma_{\mathsf{O}})$, the strategy $\sigma_{\mathsf{P}}$ will satisfy $\mathrm{Strat}_{\mathsf{P}}(\mathcal{G}({\mathord{\sim}}\mathcal{A},F),\sigma_{\mathsf{P}})$. Recall that this in particular means

[TABLE]

By $\mathrm{HF}$-Bounded Choice for Product Types (Theorem 22) we are going to define $\sigma_{\mathsf{P}}$ such that $\sigma_{\mathsf{P}}(x,q)\in\partial_{{\mathord{\sim}}\mathcal{A}}(q,F(x))$ for each $(x,q)\in\mathscr{D}^{*}\times Q_{\mathcal{A}}$. Assume fixed $(x,q)\in\mathscr{D}^{*}\times Q_{\mathcal{A}}$. For all $\gamma\in\mathcal{P}_{*}(\mathscr{D}\times Q_{\mathcal{A}})$ such that $\gamma\in\partial_{\mathcal{A}}(q,F(x))$, we have $\sigma_{\mathsf{O}}(x,(q,\gamma))\in\gamma$. By $\mathrm{HF}$-Comprehension (Remark 3.7.3), let

[TABLE]

By construction, we thus have $\overline{\gamma}\in\partial_{{\mathord{\sim}}\mathcal{A}}(q,F(x))$, and we let

[TABLE]

We trivially have $\mathrm{Strat}_{\mathsf{P}}(\mathcal{G}({\mathord{\sim}}\mathcal{A},F),\sigma_{\mathsf{P}})$.

Lemma 62**.**

Consider $\sigma_{\mathsf{O}}$ and $\sigma_{\mathsf{P}}$ as above. For every infinite play $V$ of $\sigma_{\mathsf{P}}$ in $\mathcal{G}({\mathord{\sim}}\mathcal{A},F)$ there is some infinite play $U$ of $\sigma_{\mathsf{O}}$ in $\mathcal{G}(\mathcal{A},F)$ with $V_{\mathsf{P}}=U_{\mathsf{P}}$.

Proof 6.8**.**

We define $U$ by Comprehension for Product Types (Theorem 23) as follows.

• •

First, for $(x,k)\in\mathcal{G}_{\mathsf{P}}$, if $(x,k)\in V_{\mathsf{P}}$ then we let $(x,k)\in U_{\mathsf{P}}$.

• •

Consider $(x,(q,\gamma))\in\mathcal{G}_{\mathsf{O}}$. Using Remark 12, let $\prec$ be a well-order on $\mathcal{P}_{*}(\mathscr{D}\times Q_{\mathcal{A}})$. Then we let $(x,(q,\gamma))\in U_{\mathsf{O}}$ iff $(x,q)\in V_{\mathsf{P}}$ and $\gamma$ is $\preceq$-minimal in $\partial_{\mathcal{A}}(q,F(x))$ such that $(\mathsf{S}_{d}(x),q^{\prime})\in V_{\mathsf{P}}$ for $(d,q^{\prime})=\sigma_{\mathsf{O}}(x,(q,\gamma))$.

Note that consecutive $\mathsf{P}$-positions in $U_{\mathsf{P}}$ are indeed connected by the edge relation of $\mathcal{G}(\mathcal{A},F)$:

Claim 63**.**

[TABLE]

Proof 6.9** (Proof of Claim 63).**

We first show uniqueness. Let $(y_{0},(q_{0},\gamma_{0})),(y_{1},(q_{1},\gamma_{1}))\in U_{\mathsf{O}}$ be between $(x,q)$ and $(\mathsf{S}_{d}(x),q^{\prime})$. Then we must have $y_{0}=y_{1}=x$ and $q_{0}=q_{1}=q$. Hence, $\gamma_{0}$ and $\gamma_{1}$ are both $\preceq$-minimal in $\partial_{\mathcal{A}}(q,F(x))$ such that $\sigma_{\mathsf{O}}(x,(q,\gamma_{0}))=\sigma_{\mathsf{O}}(x,(q,\gamma_{1}))=(d,q^{\prime})$, yielding $\gamma_{0}=\gamma_{1}$ as required.

We now show the existence of an appropriate $(x,(q,\gamma))\in U_{\mathsf{O}}$. Since $\mathrm{Play}(\sigma_{\mathsf{P}},\iota,V)$, we have $(d,q^{\prime})\in\overline{\gamma}$ with $(\ell,\overline{\gamma})\in\sigma_{\mathsf{P}}(y,\ell)$ for some $(y,\ell)\in V_{\mathsf{P}}$. But $\mathrm{Play}(\sigma_{\mathsf{P}},\iota,V)$ moreover implies that either $(y,\ell)\vartriangleleft(x,q)$ or $(x,q)\trianglelefteq(y,\ell)$, from which follows that $(y,\ell)=(x,q)$ and $(q,\overline{\gamma})\in\sigma_{\mathsf{P}}(x,q)$. Since

[TABLE]

it follows that $(d,q^{\prime})\in\sigma_{\mathsf{O}}(x,\gamma)$ for some $\gamma\in\partial_{\mathcal{A}}(q,F(x))$, and we are done.

We now check that $U$ is indeed an infinite play of $\sigma_{\mathsf{O}}$, i.e. that $\mathrm{Play}(\sigma_{\mathsf{O}},\iota,U)$ holds. First, we have $\iota\in U$. Moreover,

Claim 64**.**

[TABLE]

Proof 6.10** (Proof of Claim 64).**

We reason by induction on $\mathchoice{\mathrel{\overset{}{\underset{\sigma_{\mathsf{O}}}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\sigma_{\mathsf{O}}}}}{\longrightarrow_{\sigma_{\mathsf{O}}}}{\longrightarrow_{\sigma_{\mathsf{O}}}}{}$ (Corollary 34). First, if $u\in U_{\mathsf{O}}$, then $u$ is of the form $(x,(q,\gamma))$. By definition of $U_{\mathsf{O}}$ we have $(x,q)\in U_{\mathsf{P}}$ with $(x,q)\mathchoice{\mathrel{\overset{}{\underset{\sigma_{\mathsf{O}}}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\sigma_{\mathsf{O}}}}}{\longrightarrow_{\sigma_{\mathsf{O}}}}{\longrightarrow_{\sigma_{\mathsf{O}}}}(x,(q,\gamma))$ and we conclude by induction hypothesis.

Consider now the case of $u\in U_{\mathsf{P}}=V_{\mathsf{P}}$. In this case, $u$ of the form $(x,q)$. We apply Proposition 6.(5), stating that either $x\mathrel{\doteq}\dot{\varepsilon}$ or $x=\mathsf{S}_{d}(y)$ for some $d$ and $y$. In the former case, since $V$ is a play, we have $\iota\mathchoice{\mathrel{\overset{*}{\underset{\sigma_{\mathsf{P}}}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}_{\sigma_{\mathsf{P}}}}}{\longrightarrow^{*}_{\sigma_{\mathsf{P}}}}{\longrightarrow^{*}_{\sigma_{\mathsf{P}}}}(x,q)$, and Proposition 33.(9) implies $u=\iota$. In the latter case, assume $x$ is $\mathsf{S}_{d}(y)$. We apply twice the Predecessor Lemma 37 for Infinite Plays, which gives some $(y,q^{\prime})\in V_{\mathsf{P}}$ such that

[TABLE]

By induction hypothesis we get $\iota\mathchoice{\mathrel{\overset{*}{\underset{\sigma_{\mathsf{O}}}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}_{\sigma_{\mathsf{O}}}}}{\longrightarrow^{*}_{\sigma_{\mathsf{O}}}}{\longrightarrow^{*}_{\sigma_{\mathsf{O}}}}(y,q^{\prime})$ and we conclude by Claim 63.

Also,

Claim 65**.**

[TABLE]

Proof 6.11** (Proof of Claim 65).**

The case of $u\in U_{\mathsf{P}}=V_{\mathsf{P}}$ follows directly from the definition of $U_{\mathsf{O}}$ and the fact that $\sigma_{\mathsf{O}}:\mathcal{G}_{\mathsf{O}}\mathrel{\mathfrak{to}}\mathscr{D}\times\mathsf{P}$ and $\mathrm{Play}(\sigma_{\mathsf{P}},\iota,V)$. Consider now the case of $u\in U_{\mathsf{O}}$. By definition of $U_{\mathsf{O}}$ there is some $v\in U_{\mathsf{P}}$ such that $u\mathchoice{\mathrel{\overset{}{\underset{\sigma_{\mathsf{O}}}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\sigma_{\mathsf{O}}}}}{\longrightarrow_{\sigma_{\mathsf{O}}}}{\longrightarrow_{\sigma_{\mathsf{O}}}}v$. Uniqueness follows from the fact that $U_{\mathsf{P}}=V_{\mathsf{P}}$ and $\mathrm{Play}(\sigma_{\mathsf{P}},\iota,V)$.

In order to obtain $\mathrm{Play}(\sigma_{\mathsf{O}},\iota,U)$, we invoke Proposition 35 and it remains to show:

Claim 66**.**

[TABLE]

Proof 6.12** (Proof of Claim 66).**

The case of $u\in U_{\mathsf{O}}$ follows from the definition of $U_{\mathsf{O}}$. The case of $u\in U_{\mathsf{P}}$ directly follow from Claim 63 (together with Proposition 33.(9)) and $\mathrm{Play}(\sigma_{\mathsf{P}},\iota,V)$.

This concludes the proof of Lemma 62.

We use the following simple fact in order to obtain from Lemma 62 that $\sigma_{\mathsf{P}}$ is winning in $\mathcal{G}({\mathord{\sim}}\mathcal{A},F)$ whenever $\sigma_{\mathsf{O}}$ is winning in $\mathcal{G}(\mathcal{A},F)$.

Lemma 67**.**

Given plays $U,V:\mathcal{G}\mathrel{\mathfrak{to}}2$ as in Lemma 62, we have $\mathrm{Par}(\mathcal{A},U)\leavevmode\nobreak\ \Leftrightarrow\leavevmode\nobreak\ \lnot\mathrm{Par}({\mathord{\sim}}\mathcal{A},V)$.

We now have everything we need to obtain Proposition 60, namely

[TABLE]

Assume $F\notin\mathcal{L}(\mathcal{A})$. By Definition 47, there is no winning $\mathsf{P}$-strategy in $\mathcal{G}(\mathcal{A},F)$. By the axiom of positional determinacy of parity games $(\mathit{PosDet})$ there is a winning $\mathsf{O}$-strategy $\sigma_{\mathsf{O}}$ in $\mathcal{G}(\mathcal{A},F)$, so that

[TABLE]

Consider now the $\mathsf{P}$-strategy $\sigma_{\mathsf{P}}$ on $\mathcal{G}({\mathord{\sim}}\mathcal{A},F)$ as defined above. We claim that $\sigma_{\mathsf{P}}$ is winning, that is

{clm}

[TABLE]

Proof 6.13** (Proof of Claim \theclm).**

Given an infinite play $V$ of $\sigma_{\mathsf{P}}$, by Lemma 62 we can build an infinite play $U$ of $\sigma_{\mathsf{O}}$, which by (13) satisfies $\lnot\mathrm{Par}(\mathcal{A},-)$, so that $V$ satisfies $\mathrm{Par}({\mathord{\sim}}\mathcal{A},-)$ thanks to Lemma 67.

We thus have $F\in\mathcal{L}({\mathord{\sim}}\mathcal{A},F)$. This concludes the proof of Proposition 60.

6.6.2. Proof of Proposition 61.

We are now going to show that $\mathsf{FSO}$ proves

[TABLE]

We associate a (winning) $\mathsf{O}$-strategy $\sigma_{\mathsf{O}}$ on $\mathcal{G}(\mathcal{A},F)$ to each (winning) $\mathsf{P}$-strategy $\sigma_{\mathsf{P}}$ on $\mathcal{G}({\mathord{\sim}}\mathcal{A},F)$. Assuming that the $\mathsf{P}$-strategy satisfies $\mathrm{Strat}_{\mathsf{P}}(\mathcal{G}({\mathord{\sim}}\mathcal{A},F),\sigma_{\mathsf{P}})$, the $\mathsf{O}$-strategy will satisfy $\mathrm{Strat}_{\mathsf{O}}(\mathcal{G}(\mathcal{A},F),\sigma_{\mathsf{O}})$. Note that

[TABLE]

We define $\sigma_{\mathsf{O}}(x,(q,\gamma))$ for each position

[TABLE]

By definition of $\partial_{{\mathord{\sim}}\mathcal{A}}(q,F(p))$, we have $\sigma_{\mathsf{P}}(p,q)=(q,\overline{\gamma})$ where $\overline{\gamma}$ intersects all $\gamma\in\partial_{\mathcal{A}}(q,F(p))$. So if $\gamma\in\partial_{\mathcal{A}}(q,F(p))$, by $\mathrm{HF}$-Bounded Choice for Product Types (Theorem 22) we let $\sigma_{\mathsf{O}}(p,(q,\gamma))$ be some $(d,q^{\prime})$ such that $(d,q^{\prime})\in\gamma\cap\overline{\gamma}$. Otherwise, since $\gamma\neq\varnothing$, we let $\sigma_{\mathsf{O}}(p,(q,\gamma))$ be some $(d,q^{\prime})$ such that $(d,q^{\prime})\in\gamma$.

We also trivially have that $\mathrm{Strat}_{\mathsf{O}}(\mathcal{G}(\mathcal{A},F),\sigma_{\mathsf{O}})$.

Lemma 68**.**

Consider a $\mathsf{P}$-strategy $\sigma_{\mathsf{P}}$ and an $\mathsf{O}$-strategy $\sigma_{\mathsf{O}}$ as in above. For every infinite play $V$ of $\sigma_{\mathsf{O}}$ on $\mathcal{G}(\mathcal{A},F)$ there is some infinite play $U$ of $\sigma_{\mathsf{P}}$ on $\mathcal{G}({\mathord{\sim}}\mathcal{A},F)$ with $V_{\mathsf{P}}=U_{\mathsf{P}}$.

Proof 6.14**.**

We define $U$ by Comprehension for Product Types (Theorem 23) as follows.

• •

Definition of $U$.* For $(x,k)\in\mathcal{G}_{\mathsf{P}}$, if $(x,k)\in V_{\mathsf{P}}$ then we let $(x,k)\in U_{\mathsf{P}}$, and for $(x,(q,\overline{\gamma}))\in\mathcal{G}_{\mathsf{O}}$, we let $(x,(q,\overline{\gamma}))\in U_{\mathsf{O}}$ iff $(q,\overline{\gamma})=\sigma_{\mathsf{P}}(x,q)$ for $(x,q)\in U_{\mathsf{P}}$.*

Similarly as in Lemma 62, we have

Claim 69**.**

[TABLE]

Proof 6.15** (Proof of Claim 69).**

Uniqueness directly follows from the fact that $u=(x,\sigma_{\mathsf{P}}(x,q))$. As for existence, we directly have $(x,q)\mathchoice{\mathrel{\overset{}{\underset{\sigma_{\mathsf{P}}}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\sigma_{\mathsf{P}}}}}{\longrightarrow_{\sigma_{\mathsf{P}}}}{\longrightarrow_{\sigma_{\mathsf{P}}}}u$, so it remains to show $u\mathchoice{\mathrel{\overset{}{\underset{\sigma_{\mathsf{P}}}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\sigma_{\mathsf{P}}}}}{\longrightarrow_{\sigma_{\mathsf{P}}}}{\longrightarrow_{\sigma_{\mathsf{P}}}}(\mathsf{S}_{d}(x),q^{\prime})$, which amounts to $(d,q^{\prime})\in\overline{\gamma}$ for $(q,\overline{\gamma})=\sigma_{\mathsf{P}}(x,q)$. But $(\mathsf{S}_{d}(x),q^{\prime})\in V_{\mathsf{P}}$ with $\mathrm{Play}(\sigma_{\mathsf{O}},\iota,V)$ imply that $(d,q^{\prime})=\sigma_{\mathsf{O}}(x,(\ell,\gamma))$ for some $\ell$ such that $(x,\ell)\in V_{\mathsf{P}}$ and some $\gamma\in\partial_{\mathcal{A}}(\ell,F(x))$. Moreover, $\mathrm{Play}(\sigma_{\mathsf{O}},\iota,V)$ implies $\ell=q$. By definition of $\sigma_{\mathsf{O}}$, we thus have $(d,q^{\prime})\in\gamma\cap\overline{\gamma}$ and we are done.

We now check that $\mathrm{Play}(\sigma_{\mathsf{P}},\iota,U)$. Note that $\iota\in U$. Moreover, proceeding as in Lemma 62, we have

Claim 70**.**

[TABLE]

Proof 6.16** (Proof of Claim 70).**

By induction on $\mathchoice{\mathrel{\overset{}{\underset{\sigma_{\mathsf{P}}}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\sigma_{\mathsf{P}}}}}{\longrightarrow_{\sigma_{\mathsf{P}}}}{\longrightarrow_{\sigma_{\mathsf{P}}}}$ (Corollary 34). The case of $u\in U_{\mathsf{O}}$ follows directly form the induction hypothesis and the definition of $U_{\mathsf{O}}$. As for $u\in U_{\mathsf{P}}$, we proceed as in Lemma 62, using Claim 69 and Lemma 37.

Continuing as in Lemma 62, we now invoke Proposition 35 and we are left with showing

Claim 71**.**

[TABLE]

Proof 6.17** (Proof of Claim 71).**

The cases of $u\in U_{\mathsf{P}}$ follow from the definition of $U_{\mathsf{O}}$, and from Claim 63 (together with Proposition 33.(9)) and $\mathrm{Play}(\sigma_{\mathsf{O}},\iota,V)$. Consider now $u\in U_{\mathsf{O}}$. The predecessor property follows from the definition of $U_{\mathsf{O}}$. The unique successor property is obtained from Claim 69 together with $\mathrm{Play}(\sigma_{\mathsf{O}},\iota,V)$.

This concludes the proof of Lemma 68.

Similarly as in §6.6.1, we use the following simple fact.

Lemma 72**.**

Given plays $U,V:\mathcal{G}\mathrel{\mathfrak{to}}2$ as in Lemma 68, we have $\mathrm{Par}({\mathord{\sim}}\mathcal{A},U)\leavevmode\nobreak\ \Leftrightarrow\leavevmode\nobreak\ \lnot\mathrm{Par}(\mathcal{A},V)$

It is now easy to obtain Proposition 61, namely

[TABLE]

Assume that $F\in\mathcal{L}({\mathord{\sim}}\mathcal{A})$. By Definition 47, we thus have a winning $\mathsf{P}$-strategy $\sigma_{\mathsf{P}}$ in $\mathcal{G}({\mathord{\sim}}\mathcal{A},F)$, so that

[TABLE]

Consider now the $\mathsf{O}$-strategy $\sigma_{\mathsf{O}}$ on $\mathcal{G}(\mathcal{A},F)$ as defined above. Reasoning as in the case $F\notin\mathcal{L}(\mathcal{A})$ (§6.6.1), Lemmas 68 and 72 imply

[TABLE]

It then follows from Lemma 41 that there is no winning $\mathsf{P}$-strategy on $\mathcal{G}(\mathcal{A},F)$, so that $F\notin\mathcal{L}(\mathcal{A})$.

This concludes the proof of Proposition 61.

7. $\mathsf{MSO}$ on Infinite Words in Paths of ${\mathsf{FSO}_{\mathscr{D}}}$

We discuss here the theory of $\mathsf{MSO}$ over $\omega$-words for the infinite paths of ${\mathsf{FSO}_{\mathscr{D}}}$. Since $\mathsf{MSO}$ on $\omega$-words admits a complete axiomatization [Sie70], this will allow us to freely import results on $\mathsf{MSO}$ over $\omega$-words for the paths of ${\mathsf{FSO}_{\mathscr{D}}}$. In particular, our completeness argument (§8) relies on a version of the Büchi-Landweber’s Theorem [BL69] formulated with $\mathsf{MSO}$ over $\omega$-words, that we lift for free to ${\mathsf{FSO}_{\mathscr{D}}}$. Also, to prove the Simulation Theorem 56 in §9, we use McNaughton’s Theorem [McN66], and similarly obtain it for free in ${\mathsf{FSO}_{\mathscr{D}}}$.

An obvious way to obtain $\mathsf{MSO}$ over $\omega$-words is to consider the system $\mathsf{MSO}_{1}$ (that is ${\mathsf{MSO}_{\mathscr{D}}}$ for $\mathscr{D}=1$). However, recall that we want to see each path of ${\mathsf{FSO}_{\mathscr{D}}}$ (in the sense of (16) below) as a model of $\mathsf{MSO}$ on $\omega$-words. This is technically simpler if, following [Rib12], one uses a version of $\mathsf{MSO}$ on $\omega$-words over a purely relational vocabulary with only the strict order $\mathrel{\dot{<}}$ on numbers as atomic relation (besides equality $\mathrel{\doteq}$).

{defi}

[The Theory $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$]

The language of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ is the language of ${\mathsf{FSO}_{\mathscr{D}}}$ with the following restriction:

• •

the only Individual terms of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ are the constant $\dot{\varepsilon}$ and the individual variables ($x,y,z$ etc.)

The deduction rules of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ are the same as the rules of ${\mathsf{FSO}_{\mathscr{D}}}$. The axioms of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ are the Equality Axioms of §3.4.1, the Axioms on $\mathrm{HF}$-Sets of §3.4.4, the Functional Choice Axioms of §3.4.5, together with the axioms displayed on Figure 6, stating that $\mathrel{\dot{<}}$ is a discrete unbounded strict linear order with $\dot{\varepsilon}$ as its minimal element (see e.g. [Rib12]), and with the following induction scheme.

• •

Well-Founded Induction. For each formula $\varphi$, the axiom

[TABLE]

Remark 73**.**

Note that all Individuals of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ are Individuals of ${\mathsf{FSO}_{\mathscr{D}}}$, but not conversely. As a consequence, all $\mathrm{HF}$-terms of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ are $\mathrm{HF}$-terms of ${\mathsf{FSO}_{\mathscr{D}}}$, but not conversely. Also, note that it may have seemed more natural not to include the individual constant $\dot{\varepsilon}$ in the language of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$. We have included it because this eases our concrete uses of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ in §8.4 and §9.2.

Similarly to the case of $\mathscr{D}$-ary trees (§2), the theory $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ is intended to be interpreted in a theory $\mathsf{MSO}[\mathrel{\dot{<}}]^{\omega}$. Intuitively, $\mathsf{MSO}[\mathrel{\dot{<}}]^{\omega}$ is to $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ what ${\mathsf{MSO}_{\mathscr{D}}}$ is to ${\mathsf{FSO}_{\mathscr{D}}}$.

{defi}

[The Theory $\mathsf{MSO}[\mathrel{\dot{<}}]^{\omega}$]

The language of $\mathsf{MSO}[\mathrel{\dot{<}}]^{\omega}$ is the language of ${\mathsf{MSO}_{\mathscr{D}}}$ with the following restriction:

• •

the only Individual terms of $\mathsf{MSO}[\mathrel{\dot{<}}]^{\omega}$ are individual variables ($x,y,z$ etc.).

The axioms of $\mathsf{MSO}[\mathrel{\dot{<}}]^{\omega}$ are the equality axioms and the comprehension scheme of ${\mathsf{MSO}_{\mathscr{D}}}$ (§2.2), together with the induction scheme and the axioms on $\mathrel{\dot{<}}$ of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ displayed in Figure 6.

We write $\mathfrak{N}$ both for the standard model of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ and for the standard model of $\mathsf{MSO}[\mathrel{\dot{<}}]^{\omega}$. In the case of $\mathsf{MSO}[\mathrel{\dot{<}}]^{\omega}$, formulae are interpreted in $\mathfrak{N}$ as expected: individual variables range over $\mathbb{N}$, monadic predicate variables range over $\mathcal{P}(\mathbb{N})$ and $\mathrel{\dot{<}}$ is the standard order $<$ on $\mathbb{N}$. The interpretation of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$-formulae in $\mathfrak{N}$ is similar, with the obvious changes w.r.t. §3.5 for the interpretation of terms, and where Functions range over

[TABLE]

The key property of $\mathsf{MSO}[\mathrel{\dot{<}}]^{\omega}$ we rely on is that it completely axiomatizes the theory of the standard model $\mathfrak{N}$ of $\omega$-words [Sie70] (see also [Rib12]).

Theorem 74** ([Sie70]).**

For every closed $\mathsf{MSO}[\mathrel{\dot{<}}]^{\omega}$-formula $\varphi$,

[TABLE]

The formula translation from ${\mathsf{FSO}_{\mathscr{D}}}$ to ${\mathsf{MSO}_{\mathscr{D}}}$ of §3.6 restricts to a translation of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$-formulae to $\mathsf{MSO}[\mathrel{\dot{<}}]^{\omega}$-formulae. This easily extends to theories, and we get the following version of Proposition 20.

Proposition 75**.**

For every closed $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$-formula $\varphi$,

[TABLE]

Thanks to (15), the completeness of $\mathsf{MSO}[\mathrel{\dot{<}}]^{\omega}$ directly gives the completeness of $\mathsf{MSO}[\mathrel{\dot{<}}]^{\omega}$ w.r.t. the translation closed of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$-formulae $\varphi$:

[TABLE]

Our goal now is to prove that if a closed $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ formula holds in the standard model $\mathfrak{N}$ of $\omega$-words, then ${\mathsf{FSO}_{\mathscr{D}}}$ proves its relativization to any rooted tree path. Given a formula $\varphi$ of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ and a Function variable $P$, write $\varphi^{P}$ for the ${\mathsf{FSO}_{\mathscr{D}}}$ formula obtained from $\varphi$ by relativizing all individual quantifications to $P$ and by replacing all Function quantifications $F:K$ by $F:P\mathrel{\mathfrak{to}}K$. Moreover, we say that $P:\mathscr{D}^{*}\mathrel{\mathfrak{to}}2$ is a rooted path when the following formula $\mathrm{TPath}(P)$ holds:

[TABLE]

where $\mathrm{S}(x,y)$ stands for

[TABLE]

We can now formally state the property we are targeting:

[TABLE]

The proof of (17) is deferred to Proposition 78. It relies on two lemmas. The first one is an adaptation of Lemma 27 (§4.3) to rooted tree paths, which will give the last axiom of Figure 6 for rooted tree paths. The second one is a weakening of (17) where $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}\vdash\varphi$ is assumed instead of $\mathfrak{N}\models\varphi$.

Lemma 76**.**

${\mathsf{FSO}_{\mathscr{D}}}$* proves the following, assuming $P:\mathscr{D}^{*}\mathrel{\mathfrak{to}}2$ and $\mathrm{TPath}(P)$:*

[TABLE]

Lemma 77**.**

For all closed $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$-formula $\varphi$, we have

[TABLE]

Proof 7.1**.**

The proof is by induction on derivations of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$-formulae. For formulae $\bm{\psi},\varphi$ with free Function variables $\bm{F}=F_{1},\dots,F_{p}$ and free Individual variables $\bm{x}=x_{1},\dots,x_{q}$ (and possibly further free $\mathrm{HF}$-variables), we show that for all $\mathrm{HF}$-terms $\bm{K}=K_{1},\dots,K_{p}$ of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ we have

[TABLE]

The cases for each inference rule are immediate from their respective induction hypothesis, and we also easily obtain the Equality Axioms (§3.4.1), the Axioms of $\mathrm{HF}$-Sets (§3.4.4) and the Axiom of $\mathrm{HF}$-Bounded Choice for $\mathrm{HF}$-Sets (§3.4.5). We resort on Theorem 22 for the axioms of $\mathrm{HF}$-Bounded Choice for Functions and of Iterated $\mathrm{HF}$-Bounded Choice. Moreover, the Induction axiom of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ on the formula $\varphi(x)$ directly follows from Well-Founded Induction in ${\mathsf{FSO}_{\mathscr{D}}}$ (Theorem 7) on the formula

[TABLE]

It remains to deal with the $\mathrel{\dot{<}}$-axioms of Figure 6. The first five axioms (stating that $\mathrel{\dot{<}}$ is an unbounded linear order) directly follow from the Tree axioms of ${\mathsf{FSO}_{\mathscr{D}}}$ (Figure 3) and from relativization to $P$ with $\mathrm{TPath}(P)$. Finally, we have to show that ${\mathsf{FSO}_{\mathscr{D}}}$ proves that the translation of the predecessor axiom holds within $P$ whenever $\mathrm{TPath}(P)$ is assumed:

[TABLE]

*This is handled by Lemma 76. *

We have now everything we need to prove (17).

Proposition 78**.**

Consider a closed formula $\varphi$ of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$. Then

[TABLE]

Proof 7.2**.**

Assume $\mathfrak{N}\models\varphi$. By (15) we have $\mathfrak{N}\models\langle{\varphi}\rangle$. Theorem 74 then implies $\mathsf{MSO}[\mathrel{\dot{<}}]^{\omega}\vdash\langle{\varphi}\rangle$, and by (14) we have that $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}\vdash\varphi$. We conclude by Lemma 77.

8. Completeness

This Section is devoted to the proof of our main result, the completeness of $\mathsf{FSO}+(\mathit{PosDet})$.

Theorem 79** (Main Theorem).**

For each closed formula $\varphi$ of $\mathsf{FSO}$,

[TABLE]

8.1. Overview

The two main ingredients of Theorem 79 are the following.

1. (1)

The translations

[TABLE]

providing faithful mutual interpretations of $\mathsf{FSO}$ and $\mathsf{MSO}$ (§3.6, recapitulated in Table 2). 2. (2)

The translation of $\mathsf{MSO}$-formulae to automata, that we detail in §8.2 and §8.3 below. This translation relies on the correctness of the constructions on automata of §6, which are recapitulated in Table 1. In particular, we require the Axiom $(\mathit{PosDet})$ of positional determinacy of parity games (§5.6) for the complementation of tree automata (Theorem 59).

The mutual interpretability results of Table 2 also allows us to obtain a completeness result for $\mathsf{MSO}$. Recall that $\langle\mathit{PosDet}\rangle$ is defined in Definition 5.6.1, §5.6.1. We then get the following corollary to Theorem 79.

Corollary 80**.**

For each closed formula $\varphi$ of $\mathsf{MSO}$,

[TABLE]

Proof 8.1**.**

Consider a closed $\mathsf{MSO}$-formula $\varphi$. Assume $\mathsf{FSO}+(\mathit{PosDet})\vdash\varphi^{\circ}$. Let $\mathrm{PosDet}(\mathsf{P}_{i},\mathsf{O}_{i},n_{i})$ ($i=1,\dots,k$) be the instances of $(\mathit{PosDet})$ used in the proof, so that

[TABLE]

By (6) (Proposition 20), we get

[TABLE]

and since $(-)^{\circ}$ commutes over propositional connectives, by Theorem 19 we obtain

[TABLE]

Moreover, since $\varphi^{\circ}$ is $\mathrm{HF}$-closed, we can assume the $\mathrm{HF}$-terms $\mathsf{P}_{i}$, $\mathsf{O}_{i}$ and $n_{i}$ to be closed. It follows that there are constants for $\mathrm{HF}$-sets $\dot{\mathsf{\mathsf{P}}}_{i}$, $\dot{\mathsf{\mathsf{O}}}_{i}$ and $\dot{\mathsf{n}}_{i}$ ($i=1\dots,k$) such that each formula $\langle\mathrm{PosDet}(\mathsf{P}_{i},\mathsf{O}_{i},n_{i})\rangle$ is syntactically identical to $\langle\mathrm{PosDet}(\dot{\mathsf{\mathsf{P}}}_{i},\dot{\mathsf{\mathsf{O}}}_{i},\dot{\mathsf{n}}_{i})\rangle$. We thus obtain

[TABLE]

which implies that $\mathsf{MSO}+\langle\mathit{PosDet}\rangle$ proves $\varphi$.

If $\mathsf{FSO}+(\mathit{PosDet})$ does not prove $\varphi^{\circ}$, Theorem 79 gives $\mathsf{FSO}+(\mathit{PosDet})\vdash\lnot(\varphi^{\circ})$ and we conclude similarly.

In particular, it follows from Proposition 46 that $\mathsf{FSO}+(\mathit{PosDet})$ completely axiomatizes the standard model $\mathfrak{T}$ of $\mathscr{D}$-ary trees.

Corollary 81**.**

• •

For each closed formula $\varphi$ of $\mathsf{FSO}$,

[TABLE]

• •

For each closed formula $\varphi$ of $\mathsf{MSO}$,

[TABLE]

Remark 82**.**

Note that it follows from Remark 9 that Theorem 79 together with Corollary 81 implies the decidability of $\mathsf{FSO}$ over its standard model $\mathfrak{T}$. By Lemma 18 (see Table 2) we thus obtain a proof of Rabin’s Tree Theorem [Rab69], namely the decidability of $\mathsf{MSO}$ over $\mathfrak{T}$. However, even if provability in $\mathsf{FSO}$ is semi-recursive, the axiom set of $\mathsf{FSO}$ is not recursive and the interpretation of $\mathrm{HF}$-Functions is not computable (see Remarks 9 and 10 in §3.4.4, as well as Remark 16 in §3.6.1). We further elaborate on this in §8.5.

We will actually deduce Theorem 79 via Proposition 20, (6) (see Table 2) from the following.

Theorem 83**.**

For each closed formula $\varphi$ of $\mathsf{MSO}$,

[TABLE]

The proof of Theorem 83 proceeds as expected via a translation of $\mathsf{MSO}$-formulae to automata. As usual, such translations are easier to define when one starts from a version of $\mathsf{MSO}$ with a purely relational and individual-free language. We perform a translation of $\mathsf{MSO}$ to such a language in §8.2. Then, the translation of formulae to automata is presented in §8.3. It relies on the constructions of §6. We thus arrive at Proposition 87, namely that for each closed formula $\varphi$ of $\mathsf{MSO}$ there is an $\mathrm{HF}$-closed parity automaton $\mathcal{A}$ over the singleton alphabet $1$ such that

[TABLE]

In order to obtain Theorem 83, it remains to show that $\mathsf{FSO}$ actually decides the emptiness of such automata:

[TABLE]

This is Proposition 88. Its proof relies on the fact that the acceptance games of $(\mathcal{A}:1)$ are actually generated from closed $\mathrm{HF}$-Sets. We call such games reduced parity games. Section 8.4 is devoted to defining reduced parity games and to showing that $\mathsf{FSO}$ decides winning for them (Theorem 93). This essentially amounts to a version of the Büchi-Landweber Theorem [BL69] (see also e.g. [Tho97, PP04]), the effective determinacy of parity games on finite graphs, which is obtained thanks to the completeness of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ (§7). Theorem 93 then follows from the lifting of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ to the paths of $\mathsf{FSO}$ (Proposition 78).

8.2. Restricted Languages for ${\mathsf{MSO}_{\mathscr{D}}}$

For the translation of formulae to automata, it is useful and customary to work with formulae in a slightly different syntax, based on a purely relational, individual-free vocabulary.

8.2.1. Restriction to a Relational Language

We first restrict to a purely relational vocabulary, based on the defined formulae

[TABLE]

The relational formulae $\varphi,\psi\in\bm{\Lambda}^{R}_{\mathscr{D}}$ are built from atomic formulae $Xy$ and $\mathrm{S}_{d}(x,y)$ by means of $\lnot$, $\lor$, $\exists x$ and $\exists X$. To each $\mathsf{MSO}$-formula $\varphi\in\bm{\Lambda}$ we associate a formula $\varphi^{R}$ as follows. For $t$ a term of $\mathsf{MSO}$, define the formula $(z\eqcirc t)$ by structural induction on $t$:

[TABLE]

Note that

[TABLE]

Then, $\varphi^{R}$ is obtained from $\varphi$ by replacing each atomic formula $Xt$, where $t$ is not a variable, by $(\exists z)[(z\eqcirc t)\land Xz]$, where $z$ is a fresh variable.

Lemma 84**.**

For every $\mathsf{MSO}$-formula $\varphi$, we have $\mathsf{MSO}\vdash\varphi\Leftrightarrow\varphi^{R}$.

8.2.2. Restriction to an Individual-Free Language

The next step is to get rid of individual quantifiers. Consider the defined formulae:

[TABLE]

The individual-free formulae $\varphi,\psi\in{\bm{\Lambda}}^{\textit{IF}}_{\mathscr{D}}$ are built from atomic formulae $(X\mathrel{\dot{\subseteq}}Y)$ and $\mathrm{S}_{d}(X,Y)$ by means of negation, disjunction and second-order monadic quantification $\exists X$ only. Let $\varphi\in\bm{\Lambda}^{R}$ with free variables among $x_{1},\dots,x_{p},Y_{1},\dots,Y_{q}$. We inductively associate to $\varphi$ a formula ${\varphi}^{\textit{IF}}\in{\bm{\Lambda}}^{\textit{IF}}$ with free variables among $X_{1},\dots,X_{p},Y_{1},\dots,Y_{q}$ as follows. Let

[TABLE]

where

[TABLE]

The other inductive cases are given as follows:

[TABLE]

Lemma 85**.**

For every formula $\varphi\in\bm{\Lambda}^{R}$ with free variables among $\bm{x},\bm{Y}$, we have

[TABLE]

By composing the translations $(-)^{R}:\bm{\Lambda}\to\bm{\Lambda}^{R}$ and ${(-)}^{\textit{IF}}:\bm{\Lambda}^{R}\to{\bm{\Lambda}}^{\textit{IF}}$, we obtain:

Corollary 86**.**

For every closed $\mathsf{MSO}$-formula $\varphi$, there is a closed formula $\psi\in{\bm{\Lambda}}^{\textit{IF}}$ such that $\mathsf{MSO}\vdash\varphi\Leftrightarrow\psi$.

8.3. From Formulae to Automata

We are now going to associate to each formula $\varphi\in{\bm{\Lambda}}^{\textit{IF}}$ with free variables among $X_{1},\dots,X_{p}$ an $\mathrm{HF}$-closed parity automaton $\mathcal{A}(\varphi):2^{p}$ such that

[TABLE]

Note that the correctness of $\mathcal{A}(\varphi)$ w.r.t. $\varphi$ is proved in $\mathsf{FSO}$ using the translation $(-)^{\circ}:\mathsf{MSO}\to\mathsf{FSO}$ of §3.6.2. Recall that $(-)^{\circ}$ replaces each monadic variable $X_{i}$ of $\varphi$ by a Function variable $(F_{X_{i}}:2)$. The construction of $\mathcal{A}(\varphi)$ from $\varphi$ is done by induction on $\varphi$ using the operations on automata devised in §6 (see Table 1). The base cases are provided by the automata $\mathcal{A}(X_{i}\mathrel{\dot{\subseteq}}X_{j})$ and $\mathcal{A}(\mathrm{S}_{d}(X_{i},X_{j}))$ discussed in §8.3.1 below for the atomic formulae of ${\bm{\Lambda}}^{\textit{IF}}$. The inductive cases are performed as follows, where we implicitly apply substitutions (cf. §6.2) when necessary:

[TABLE]

In particular, if $\varphi$ is closed then $\mathcal{A}(\varphi)$ is an automaton over the singleton alphabet $1$, whence by Corollary 86 we have:

Proposition 87**.**

For each closed formula $\varphi$ of $\mathsf{MSO}$ there is an $\mathrm{HF}$-closed parity automaton $(\mathcal{A}:1)$ such that

[TABLE]

In order to obtain Theorem 83 from Proposition 87, it remains to show that $\mathsf{FSO}$ actually decides the emptiness of $\mathcal{L}(\mathcal{A})$ for an $\mathrm{HF}$-closed parity automaton $\mathcal{A}$ over the singleton alphabet $1$.

Proposition 88**.**

Given an $\mathrm{HF}$-closed parity automaton $(\mathcal{A}:1)$,

[TABLE]

Proposition 88 is proved in §8.4 below.

8.3.1. Automata for Atomic Formulae

We provide $\mathrm{HF}$-closed parity automata for the atomic formulae $(X_{1}\mathrel{\dot{\subseteq}}X_{2})$ and $\mathrm{S}_{d}(X_{1},X_{2})$ of the individual-free syntax ${\bm{\Lambda}}^{\textit{IF}}$ of $\mathsf{MSO}$.

• •

The automaton $\mathcal{A}(X_{1}\mathrel{\dot{\subseteq}}X_{2})$ over $2\times 2$ has state set $\mathbb{B}=\{\mathtt{tt},\mathtt{ff}\}$, with $\mathtt{tt}$ initial, transitions given by

[TABLE]

and coloring $C:\mathbb{B}\mathrel{\mathfrak{to}}2$ given by

[TABLE]

• •

For $d\in\mathscr{D}$, the automaton $\mathcal{A}(\mathrm{S}_{d}(X_{1},X_{2}))$ over $2\times 2$ has state set $Q_{\mathrm{S}}\mathrel{:=}\mathbb{B}+\{\mathtt{w}\}$, with $\mathtt{ff}$ initial, transitions given by

[TABLE]

and with coloring given by

[TABLE]

Remark 89**.**

Recall from §8.2.2 that the formula $\mathrm{S}_{d}(X,Y)$ of the individual-free syntax ${\bm{\Lambda}}^{\textit{IF}}$ amounts in $\mathsf{MSO}$ to the formula $(\exists x)(\exists y)\left[Xx\land Yy\land y\mathrel{\doteq}\mathsf{S}_{d}(x)\right]$. So the automaton $\mathcal{A}(\mathrm{S}_{d}(X,Y))$ only looks for some $x\in X$ and $y\in Y$ such that $y$ is the $d$-successor of $x$, but is does not check whether $X$ and $Y$ are singletons.

Lemma 90**.**

$\mathsf{FSO}$* proves that*

[TABLE]

8.4. Reduced Parity Games

The goal of this Section is to prove Proposition 88, namely that for an $\mathrm{HF}$-closed parity automaton $\mathcal{A}$ over the singleton alphabet $1$,

[TABLE]

Consider an $\mathrm{HF}$-closed automaton $\mathcal{A}$ over the singleton alphabet $1=\{0\}$. Then for any $(F:1)$ the game $\mathcal{G}\mathrel{:=}\mathcal{G}(\mathcal{A},F)$ has edge relations induced by functions

[TABLE]

given (following Remark 49) by

[TABLE]

So in particular the edge relations of $\mathcal{G}(\mathcal{A},F)$ are independent from $F$. But also, since

[TABLE]

the whole game $\mathcal{G}(\mathcal{A},F)$ is actually generated from $\mathrm{HF}$-Sets.

In this Section, we discuss games generated from $\mathrm{HF}$-Sets, that we call reduced games. We show that for reduced parity games, winning can actually be defined within $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$. Thanks to the completeness of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ w.r.t. its standard model (§7), this implies that $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ itself decides winning in such games. This essentially amounts to a version of the Büchi-Landweber Theorem [BL69] using the completeness of $\mathsf{MSO}[\mathrel{\dot{<}}]^{\omega}$ over its standard model. Using Proposition 78 we can then lift this result to $\mathsf{FSO}$.

In §8.4.1 and §8.4.2 we repeat some material of §5, but for the slightly different setting of reduced games. We then obtain that $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ decides winning in reduced parity games, and we lift this to $\mathsf{FSO}$ in Theorem 93, §8.4.3. This directly entails Proposition 88.

8.4.1. Reduced Games as $\mathrm{HF}$-Sets

The purpose of this Section is to give adaptations of the notions of §5 to those parity games which are entirely generated from $\mathrm{HF}$-Sets. All the formulae of this Section are $\mathrm{HF}$-formulae in the sense of Definition 3.4.4. Hence, thanks to the Axioms of $\mathrm{HF}$-Sets (Remark 11, §3.4.4) their closed instances are provable (both in $\mathsf{FSO}$ and $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$) if and only if they hold in $V_{\omega}$.

{defi}

[Reduced Games]

A reduced game $G$ is given by $\mathrm{HF}$-terms $\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}}$ which satisfy the following formula

[TABLE]

We often write $\mathrm{Game}_{0}(G)$ for $\mathrm{Game}_{0}(\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}})$. Moreover, when no ambiguity arises, we abbreviate $G=(\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}})$ as $G=(\mathsf{P},\mathsf{O},e(G))$.

{defi}

[Reduced Subgame]

We say that $G^{\prime}=(\mathsf{P}^{\prime},\mathsf{O}^{\prime},e_{\mathsf{P}}^{\prime},e_{\mathsf{O}}^{\prime})$ is a reduced subgame of $G=(\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}})$ whenever the following formula holds

[TABLE]

{defi}

[Reduced Strategies]

Let $G=(\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}})$ where $\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}}$ are $\mathrm{HF}$-variables.

1. (1)

A reduced $\mathsf{P}$-strategy on $G$ is an $\mathrm{HF}$-set $s$ which satisfies the formula

[TABLE] 2. (2)

A reduced $\mathsf{O}$-strategy on $G$ is an $\mathrm{HF}$-set $s$ which satisfies the formula

[TABLE]

{defi}

[Reduced Subgame induced by a Reduced Strategy] Given a player $\mathsf{J}$ (either $\mathsf{P}$ or $\mathsf{O}$) and a $\mathsf{J}$-strategy $s$ on $G$, we let

[TABLE]

where

[TABLE]

and where $\{s\}_{\mathsf{J}}\subseteq e_{\mathsf{J}}$ is defined (following the method of Remark 49) to be the function taking $k\in G_{\mathsf{J}}$ to the singleton $\{s(k)\}$.

Whenever possible, we write $G\mathord{\upharpoonright}\{s\}$ or even just $s$ for $G\mathord{\upharpoonright}\{s\}_{\mathsf{J}}$.

8.4.2. Reduced Games in $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$

In §8.4.1 we gave notions of reduced parity games and reduced strategies. In this Section, we work within $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ and show that this setting suffices to define winning for reduced parity games. Thanks to the completeness of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ w.r.t. the standard model of $\omega$-words (§7), we obtain that $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ decides winning in such games. This is essentially the Büchi-Landweber Theorem [BL69].

We use the following $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$-formula:

[TABLE]

{defi}

[Infinite Plays in Reduced Games]

Working in $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$, let $G=(\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}})$, where $\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}}$ are $\mathrm{HF}$-variables. Given an $\mathrm{HF}$ set $K$ and a Function $(\tilde{V}:\mathsf{P})$, we say that $\tilde{V}$ is an infinite play in $G$ from $K$ when the following formula $\mathrm{Play}[\mathrel{\dot{<}}](G,K,\tilde{V})$ holds:

[TABLE]

Note that in Definition 8.4.2 above, we use the notation $\tilde{V}$ for a play in a reduced games, to mark the difference with the notion of plays (and more generally sets of game positions) in the setting of §4.

{defi}

[Parity Conditions for Reduced Games]

Working in $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$, let $G=(\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}})$, where $\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}}$ are $\mathrm{HF}$-variables.

1. (1)

A coloring is given by Function $C$ and an $\mathrm{HF}$-set $n$ satisfying the following formula

[TABLE] 2. (2)

We define the following formula:

[TABLE]

{defi}

[Winning of Reduced Parity Games]

Working in $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$, let $G=(\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}})$, where $\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}}$ are $\mathrm{HF}$-variables. Furthermore let $C$ be a Function variable and $n$ be an $\mathrm{HF}$-variable.

1. (1)

We define the following formulae.

[TABLE] 2. (2)

Given a player $\mathsf{J}$ (either $\mathsf{P}$ or $\mathsf{O}$), we say that a $\mathsf{J}$-strategy $s$ is winning in $(G,C,n)$ from $K$ if the game $(G\{s\}_{\mathsf{J}},\mathrm{Par}[\mathrel{\dot{<}}](C,n,-))$ is won by $\mathsf{J}$ from $K$, i.e. if the following formula holds

[TABLE]

Note that in Definition 8.4.2, we have denoted strategies in reduced games with a lower case roman $s$. This notation contrasts with our notation $\sigma$ for games in the sense of §5 in order to insist on the fact that strategies on reduced games are $\mathrm{HF}$-sets.

Consider now $G=(\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}})$ where $\mathsf{P}$, $\mathsf{O}$, $e_{\mathsf{P}}$ and $e_{\mathsf{O}}$ are closed $\mathrm{HF}$-terms such that

[TABLE]

Assume also given closed $\mathrm{HF}$-terms $n$ and $C$ such that

[TABLE]

Then the positional determinacy of parity games (cf. [EJ91]) implies that for every $\mathrm{HF}$-set $\kappa\in\mathsf{P}$, the following holds in the standard model $\mathfrak{N}$ of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$:

• •

For some player $\mathsf{J}$ (either $\mathsf{P}$ or $\mathsf{O}$) there an $\mathrm{HF}$-set $\mathtt{s}$ such that

[TABLE]

Thanks to the completeness of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ w.r.t. $\mathfrak{N}$ (Theorem 74 and Proposition 75), we obtain the following result, that may be viewed as a formulation of the Büchi-Landweber Theorem [BL69] (see also e.g. [Tho97, PP04]). Recall that $\mathrm{Strat}^{0}_{\mathsf{J}}(G,\mathtt{s})$ holds in $\mathfrak{N}$ (resp. $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$, $\mathsf{FSO}$) if and only if it holds in $V_{\omega}$.

Proposition 91**.**

Assume given closed $\mathrm{HF}$-terms $G=(\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}})$, $n$ and $C$ such that

[TABLE]

Then for every $\kappa\in\mathsf{P}$, there is a player $\mathsf{J}$ (either $\mathsf{P}$ or $\mathsf{O}$) and an $\mathrm{HF}$-set $\mathtt{s}$ such that

[TABLE]

Proposition 78, namely

[TABLE]

(for $\varphi$ a closed $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$-formula) moreover gives the following.

Proposition 92**.**

Assume given closed $\mathrm{HF}$-terms $G=(\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}})$, $n$ and $C$ such that

[TABLE]

Then for every $\kappa\in\mathsf{P}$, there is a player $\mathsf{J}$ (either $\mathsf{P}$ or $\mathsf{O}$) and an $\mathrm{HF}$-set $\mathtt{s}$ such that

[TABLE]

8.4.3. Reduced Games in $\mathsf{FSO}$

We now come back to $\mathsf{FSO}$. In this Section, we show, using Proposition 92, that $\mathsf{FSO}$ decides winning for parity games induced from reduced parity games (Theorem 93). This directly gives Proposition 88.

A reduced game $G=(\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}})$ induces a game $\mathcal{G}=(\mathsf{P},\mathsf{O},E_{\mathsf{P}},E_{\mathsf{O}})$ in the sense of Definition 5.1, where

[TABLE]

are defined using $\mathrm{HF}$-Bounded Choice for Product Types (Theorem 22) as

[TABLE]

Similarly, a strategy $s$ in a reduced game $G$ induces a strategy $\sigma$ in $\mathcal{G}$ in the sense of Definition 5.3, with

[TABLE]

As for colorings, from $(C:\mathsf{P}\mathrel{\mathfrak{to}}[0,n])$ we define $\hat{C}:\mathcal{G}\mathrel{\mathfrak{to}}[0,n]$ as in Definition 47:

[TABLE]

We clearly have the following:

[TABLE]

Theorem 93**.**

Assume given closed $\mathrm{HF}$-terms $G=(\mathsf{P},\mathsf{O},e_{\mathsf{P}},e_{\mathsf{O}})$, $n$ and $C$ such that

[TABLE]

Then for every $\kappa\in\mathsf{P}$,

[TABLE]

In the statement of Theorem 93, $\mathcal{G}(\trianglelefteq)$ refers to the the game of Remark 32 (see also Remark 45).

Remark 94**.**

The crucial differences between Theorem 93 and the axiom $(\mathit{PosDet})$ are the following. On one hand, Theorem 93 allows us to derive $(\mathit{PosDet})$ for games on finite graphs only, while $(\mathit{PosDet})$ speaks about arbitrary $\mathsf{FSO}$-definable games (in the sense of §5). On the other hand, Theorem 93 says that $\mathsf{FSO}$ decides winning for games on finite graphs, while $(\mathit{PosDet})$ is a statement of determinacy, i.e. that one of the players wins, but not which player wins.

Proof 8.2** (Proof of Theorem 93).**

Fix $G$, $n$, $C$ and $\kappa$ as in the statement. Let $\mathsf{J}$ and $\mathtt{s}$ be given by Proposition 92, and let $\sigma$ be induced from $\mathtt{s}$ as above. We are going to show that $\sigma$ is winning in $\mathcal{G}$ from position $\kappa$:

[TABLE]

So let $V:\mathcal{G}\mathrel{\mathfrak{to}}2$ be an infinite play of $\sigma$ from $\kappa$. Our plan is to obtain $\mathrm{Par}(\mathcal{G},\hat{C},n,V)$ from Proposition 92. By Comprehension for Product Types (Theorem 23), let $|V|:\mathscr{D}^{*}\mathrel{\mathfrak{to}}2$ be the set of all $x\in\mathscr{D}^{*}$ such that $(x,k)\in V$ for some $k\in\mathsf{P}$. Note that $\mathrm{TPath}(|V|)$ holds in $\mathsf{FSO}$. Proposition 92 then gives

[TABLE]

Note that

[TABLE]

and similarly for $\mathrm{WinStrat}[\mathrel{\dot{<}}]^{|V|}_{\mathsf{O}}(G,\mathtt{s},\kappa,C,n)$. By $\mathrm{HF}$-Bounded Choice for Functions (§3.4.5), let $\tilde{V}:|V|\mathrel{\mathfrak{to}}\mathsf{P}$ take $x\in|V|$ to the unique $k\in\mathsf{P}$ such that $(x,k)\in V$. Then we are done as soon as we show

Claim 95**.**

[TABLE]

Proof 8.3** (Proof of Claim 95).**

The property on parity conditions follows from the fact that for all $m\in[0,n]$ we have

[TABLE]

As for $\mathrm{Play}[\mathrel{\dot{<}}]^{|V|}(\mathtt{s},\kappa,\tilde{V})$, note that it unfolds to

[TABLE]

where

[TABLE]

But this directly follows from the definition of $\sigma$ from $\mathtt{s}$ together with the fact that $V$ is a play of $\sigma$ from $\kappa$.

This concludes the proof of Theorem 93.

We are now ready to prove Proposition 88, thus completing the proof of Theorem 83.

Proof 8.4** (Proof of Proposition 88).**

We have to show that for an $\mathrm{HF}$-closed parity automaton $(\mathcal{A}:1)$,

[TABLE]

For any $(F:1)$, the game $\mathcal{G}(\mathcal{A},F)$ is generated as above from the edge relations (18). Moreover, recall from Definition 47 that the winning condition of $\mathcal{G}(\mathcal{A},F)$ is generated, as in the statement of Theorem 93, by the game $\mathcal{G}(\mathcal{A},F)(\trianglelefteq)$ of Remark 32. We then conclude by Theorem 93, and this completes the proof of Proposition 88.

8.5. Remarks on Recursiveness

We noted in Remark 82 that the completeness of $\mathsf{FSO}+(\mathit{PosDet})$ indeed allows us to decide $\mathsf{FSO}$ and $\mathsf{MSO}$ formulae in the standard model $\mathfrak{T}$ of §3.5. This however comes with two apparent defects. The first one is that the interpretation $\llbracket-\rrbracket$ of $\mathrm{HF}$-terms fixed in Convention 3.4.4 is not computable (see Remarks 10 and 15), because provability in $\mathbf{Sk}(\mathsf{ZFC}^{-})$ is not decidable (as this theory contains the $\Pi^{0}_{1}$ fragment of arithmetic). The second one is that, although the axiom set $\mathsf{MSO}+\langle\mathit{PosDet}\rangle$ is even polynomial-time recognizable (recall that $\langle\mathit{PosDet}\rangle$ is defined in Definition 5.6.1, §5.6.1), the interpretation $\langle-\rangle$ for $\mathrm{HF}$-terms relies on Convention 3.4.4 (fixing the interpretation of $\mathrm{HF}$-Functions), and is thus not computable. We discuss here a workaround for this involving a slightly different setting for $\mathsf{FSO}$. We chose to not officially work in that setting because we found it less uniform and elegant than the current presentation of $\mathsf{FSO}$, which nonetheless still allows us to derive Rabin’s Tree Theorem [Rab69].

Rather than taking all the axioms on $\mathrm{HF}$-sets of §3.4.4, in particular considering the whole theory $\mathbf{Sk}(\mathsf{ZFC}^{-})$ there, we may work in systems parametrized by chosen sets of $\mathrm{HF}$-Functions. A way to implement this would be to consider systems $\mathsf{FSO}(\mathbf{SK})$, where the parameter $\mathbf{SK}$ specifies some interpretations $\mathtt{g}_{n,m}$ for constants $\dot{\mathsf{g}}_{n,m}$ such that (2) is assumed to hold. Concretely, a specification $\mathbf{SK}$ consists of a set $\textrm{SK}\subseteq\mathbb{N}\times\mathbb{N}$ together with functions

[TABLE]

Given a set $\textrm{SK}\subseteq\mathbb{N}\times\mathbb{N}$, we let $\mathsf{ZFC}^{-}(\textrm{SK})$ consist of $\mathsf{ZFC}^{-}$ augmented with the axioms

[TABLE]

We say that $\mathbf{SK}$ is a specification if

[TABLE]

where, for each $(n,m)\in\textrm{SK}$,

• •

$\mathtt{g}_{n,m}$ is a computable function $V^{n}_{\omega}\to V_{\omega}$, and

• •

for each each $\dot{\mathsf{g}}_{n^{\prime},m^{\prime}}$ occurring in $\varphi_{n,m}$, we have $(n^{\prime},m^{\prime})\in\textrm{SK}$, and

• •

$\mathsf{ZFC}^{-}(\textrm{SK})\vdash(\forall k_{1},\dots,k_{n})(\exists!\ell)\varphi_{n,m}$, and

• •

$V_{\omega}\models(\forall k_{1},\dots,k_{n})\varphi_{n,m}[\mathtt{g}_{n,m}(k_{1},\dots,k_{n})/\ell]$

Given a specification $\mathbf{SK}$, one can fix the interpretation of all constants $(\dot{\mathsf{g}}_{n,m})_{n,m\in\mathbb{N}}$ by taking for $\dot{\mathsf{g}}_{n,m}$ with $(n,m)\notin\textrm{SK}$ the function $V^{n}_{\omega}\to V_{\omega}$ with constant value $\varnothing$.

For the formal definition of $\mathsf{FSO}(\mathbf{SK})$, instead of the Axioms on $\mathrm{HF}$-Sets of §3.4.4, one has the following.

• •

For each $(n,m)\in\textrm{SK}$, and for all $\mathrm{HF}$-terms $\bm{K}=K_{1},\dots,K_{n}$, the axiom

[TABLE]

• •

For each closed $\mathrm{HF}$-formula $\varphi$ such that $V_{\omega}\models\varphi$, the axiom

[TABLE]

Given a specification $\mathbf{SK}$, the interpretations $\llbracket-\rrbracket$ and $\langle-\rangle$ are computable. All results of this paper hold for sufficiently large specifications.

Theorem 96**.**

Let $\mathbf{SK}$ be a specification defining all the $\mathrm{HF}$-Functions of 1–8, §3.4.4, as well as those of Convention 5.5, §5.5. Then all the results stated in §8 hold for $\mathsf{FSO}(\mathbf{SK})$ instead of $\mathsf{FSO}$.

9. The Simulation Theorem

This Section is devoted to the proof of the Simulation Theorem, cf. [EJ91, MS95].

Theorem 97** (Simulation Theorem 56).**

For each $\mathrm{HF}$-closed parity automaton $\mathcal{A}:\Sigma$ there is a non-deterministic $\mathrm{HF}$-closed parity automaton $\mathrm{ND}(\mathcal{A}):\Sigma$ such that

[TABLE]

We assume that $\mathcal{A}$ is $\mathrm{HF}$-closed in Theorem 97 because we rely on McNaughton’s Theorem [McN66], in the standard model for $\omega$-words, which we import into $\mathsf{FSO}$ thanks to Proposition 78.

Before a detailed exposition, let us explain the main idea behind Theorem 97. We momentarily work in the usual mathematical universe (i.e. not in the formal theory $\mathsf{FSO}$). Recall that in a non-deterministic automaton $\mathcal{N}$, $\mathsf{O}$ can only explicitly choose tree directions, since for each possible $\gamma_{\mathcal{N}}$ in the image of $\partial_{\mathcal{N}}$, if $(d,q),(d,q^{\prime})\in\gamma_{\mathcal{N}}$ then $q=q^{\prime}$, by definition. In order to obtain a non-deterministic automaton $\mathcal{N}$ from an alternating automaton $\mathcal{A}$, the idea is to perform a subset construction, such that each $\gamma_{\mathcal{N}}$ in the image of $\partial_{\mathcal{N}}$ is of the form

[TABLE]

where each $S^{\prime}_{d}$ gathers states $q$ such that $(d,q)\in\gamma_{\mathcal{A}}$ with $\gamma_{\mathcal{A}}$ in the image of $\partial_{\mathcal{A}}$.

More precisely, assuming $S\in\mathcal{P}_{*}(Q_{\mathcal{A}})$, one may consider functions

[TABLE]

Each such $f$ induces

[TABLE]

and we can let

[TABLE]

Then, for each $\gamma_{\mathcal{N}}(f)$ in the image of $\partial_{\mathcal{N}}$ and for each tree direction $d\in\mathscr{D}$, the set $S^{\prime}_{d}$ is unique such that $(d,S^{\prime}_{d})\in\gamma_{\mathcal{N}}(f)$, and $\mathcal{N}$ satisfies the property asked in Definition 6.4 to non-deterministic automata.

There is however a difficulty in the definition of the acceptance condition of $\mathcal{N}$. We follow here the construction of [Wal02] where the states of $\mathcal{N}$, rather than being simply sets of states, are sets of pairs of states $S\in\mathcal{P}(Q_{\mathcal{A}}\times Q_{\mathcal{A}})$. Then an infinite sequence of states $S_{0},S_{1},\ldots\in Q_{\mathcal{N}}$ induces a set of traces $q_{0},q_{1},\ldots\in Q_{\mathcal{A}}$ with $(q_{i},q_{i+1})\in S_{i+1}$. For $(S_{n})_{n\in\mathbb{N}}\in Q_{\mathcal{N}}^{\omega}$ to be accepting, one may then require all its traces $(q_{n})_{n\in\mathbb{N}}\in Q_{\mathcal{A}}^{\omega}$ to be accepting. We may obtain a parity condition for $\mathcal{N}$ by noticing that its acceptance condition is $\omega$-regular (i.e. definable in $\mathsf{MSO}$ over $\omega$-words). This allows us to apply McNaughton’s Theorem [McN66], and to obtain a deterministic $\omega$-word parity automaton $\mathcal{D}$ whose language is the set of accepting sequences $(S_{n})_{n\in\mathbb{N}}\in Q_{\mathcal{N}}^{\omega}$. A suitable product of $\mathcal{N}$ with $\mathcal{D}$ then gives a non-deterministic parity automaton equivalent to $\mathcal{A}$.

The organization of this Section follows the above construction. Working in $\mathsf{FSO}$, consider a parity automaton $\mathcal{A}:\Sigma$. We will build a non-deterministic automaton $\mathrm{ND}(\mathcal{A}):\Sigma$ with the same language. The automaton $\mathrm{ND}(\mathcal{A})$ will be defined in three steps:

1. (1)

We first define in §9.1 a non-deterministic automaton $\oc\mathcal{A}$ in the sense of Definition 6.1. The acceptance condition of $\oc\mathcal{A}$ will be given by an $\mathsf{FSO}$-formula with a free Function variable (intended to be range over infinite plays) rather than a parity condition. 2. (2)

For an $\mathrm{HF}$-closed parity automaton $\mathcal{A}$, the formula describing the acceptance condition of $\oc\mathcal{A}$ is then transformed in §9.2 to a $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$ formula relativized to infinite rooted tree paths (see §7). This construction relies, via Proposition 78, on Proposition 75 (i.e. Proposition 20) which requires the manipulation of closed (and in particular $\mathrm{HF}$-closed) objects. 3. (3)

Using the tools of §7, and relying on McNaughton’s Theorem [McN66] (see also e.g. [Tho97, PP04]), in §9.3 we will then turn $\oc\mathcal{A}$ into an equivalent non-deterministic parity automaton $\mathrm{ND}(\mathcal{A})$, in the sense of Definition 47.

\opt

draftnotes

In this Section, it is convenient to work with the following games.

{defi}

Given an automaton $\mathcal{A}:\Sigma$, we let $\mathcal{G}(\mathcal{A})$ be the game with

[TABLE]

and with transitions defined by $\mathrm{HF}$-Bounded Choice for Product Types (Theorem 22) and Comprehension for $\mathrm{HF}$-Sets (Remark 3.7.3) as

[TABLE]

As for winning, we will consider the game $\mathcal{G}(\mathcal{A})$ as being equipped with the winning condition $\Omega_{\mathcal{A}}$ in the sense of §5.4. Note that for $F:\Sigma$, the acceptance game $\mathcal{G}(\mathcal{A},F)$ is a subgame of $\mathcal{G}(\mathcal{A})$ in the sense of Def. 31.

Remark 98**.**

Note that if $\mathrm{Aut}(\mathcal{A})$ then $\Sigma$ is non-empty, so we indeed have $\mathrm{Game}(\mathcal{G}(\mathcal{A}))$. For each $F:\Sigma$, the acceptance game $\mathcal{G}(\mathcal{A},F)$ is a subgame of $\mathcal{G}(\mathcal{A})$ (in the sense of Def. 31). In particular infinite plays in $\mathcal{G}(\mathcal{A},F)$ are infinite plays in $\mathcal{G}(\mathcal{A})$. Moreover, it is easy to see that (winning) strategies on $\mathcal{G}(\mathcal{A},F)$ are (winning) strategies on $\mathcal{G}(\mathcal{A})$.

Furthermore, note that the game $\mathcal{G}(\mathcal{A})(\trianglelefteq)$ induced by Remark 31 from Definition 9 is precisely the game $\mathcal{G}(\mathcal{A})(\trianglelefteq)$ of (12). It follows that in the case of a parity automaton $\mathcal{A}$, we unambiguously extend the notation of Definition 47 and write $\mathrm{Par}(\mathcal{A},\hat{C}_{\mathcal{A}},n_{\mathcal{A}},U)$ or $\mathrm{Par}(\mathcal{A},U)$ for the formula $\mathrm{Par}(\mathcal{G}(\mathcal{A})(\trianglelefteq),\hat{C}_{\mathcal{A}},n_{\mathcal{A}},U)$.

9.1. The Construction of $\oc\mathcal{A}$.

Consider an alternating parity automaton $\mathcal{A}$, in the sense of Definition 47. So we have $\mathcal{A}=(Q_{\mathcal{A}},\,{q}^{\iota}_{\mathcal{A}},\,\partial_{\mathcal{A}},\,C_{\mathcal{A}},\,n_{\mathcal{A}})$ where

[TABLE]

We define the state set and the initial state of $\oc\mathcal{A}$ as:

[TABLE]

The transition function of $\oc\mathcal{A}$ is defined as follows, using Remark 49. For $\mathtt{a}\in\Sigma$ and $S\in Q_{\oc\mathcal{A}}$ we let $\oc\gamma\in\partial_{\oc\mathcal{A}}(S,\mathtt{a})$ if and only if there is some $\mathrm{HF}$-set

[TABLE]

such that $\oc\gamma=\{(d,S^{\prime}_{d})\leavevmode\nobreak\ |\leavevmode\nobreak\ d\in\mathscr{D}\land S^{\prime}_{d}\neq\varnothing\}$, where

[TABLE]

and where $\pi_{2}$ is a projection $\mathrm{HF}$-Function of §3.4.4.6.

Remark 99**.**

We indeed have

[TABLE]

since for $S\in Q_{\oc\mathcal{A}}=\mathcal{P}_{*}(Q_{\mathcal{A}}\times Q_{\mathcal{A}})$, by $\mathrm{HF}$-Bounded Choice for $\mathrm{HF}$-Sets (§3.4.5) there is always some $f\in\mathcal{P}_{*}(\mathscr{D}\times Q_{\mathcal{A}})^{\pi_{2}(S)}$ with $\forall q\in\pi_{2}(S)(f(q)\in\partial_{\mathcal{A}}(q,\mathtt{a}))$, and moreover such that $S^{\prime}_{d}$ is non-empty for at least one $d\in\mathscr{D}$.

Note our unusual choice of taking non-empty sets as states of $\oc\mathcal{A}$. It would have been more natural to allow the empty set as a state, in particular because it would have allowed us to strengthen Corollary 55 to an “exists unique” statement. This could also have worked in our setting where games are assumed to have no dead ends, and in which transitions of alternating automata range over non-empty sets of non-empty subsets of $\mathscr{D}\times Q_{\mathcal{A}}$. However, the empty state would have appeared in the transitions of $\oc\mathcal{A}$ only in case there is some tree direction $d\in\mathscr{D}$ which is not available to $\mathsf{O}$ at some stage. Since the empty state of $\oc\mathcal{A}$ would have been unconditionally winning for $\mathsf{P}$, this would have lead to an additional case to handle in the proof of completeness of $\oc\mathcal{A}$ (Proposition 101 below).

So far we have defined for $\oc\mathcal{A}$ a state set (with an initial state) and a transition function. As explained above, we will not directly equip it with a parity condition. Instead, we will define its acceptance condition via an $\mathsf{FSO}$-formula $\mathcal{W}_{\oc\mathcal{A}}$, in the sense of Definition 6.1. Consider first $V:\mathcal{G}(\oc\mathcal{A})\mathrel{\mathfrak{to}}2$ and $T:\mathcal{G}(\mathcal{A})\mathrel{\mathfrak{to}}2$. We say that $T$ is a trace in $V$ if the following formula $\mathrm{Trace}(T,V)$ holds,

[TABLE]

where we use the following formula:

[TABLE]

The formula $\mathcal{W}_{\oc\mathcal{A}}(V)$ is defined to be:

[TABLE]

Recall our notation $\mathrm{Par}(\mathcal{A},\hat{C}_{\mathcal{A}},n_{\mathcal{A}},-)$ from Remark 98. Note that $\mathcal{W}_{\oc\mathcal{A}}$ requires no condition w.r.t. the transitions of $\mathcal{G}(\mathcal{A})$. We are now going to show that $\oc\mathcal{A}$ has the same language as $\mathcal{A}$.

Theorem 100**.**

Fix a parity automaton $\mathcal{A}:\Sigma$ and consider the automaton $\oc\mathcal{A}:\Sigma$ as defined above. Then ${\mathsf{FSO}_{\mathscr{D}}}$ proves that for all $F:\Sigma$, $\oc\mathcal{A}$ accepts $F$ if and only if $\mathcal{A}$ accepts $F$.

The proof of Theorem 100 is split into Propositions 101 and 102 below.

{conv}

In Propositions 101 and 102, for fixed automata $\mathcal{A}$ and $\oc\mathcal{A}$, we let

[TABLE]

Proposition 101**.**

Fix a parity automaton $\mathcal{A}:\Sigma$ and consider the automaton $\oc\mathcal{A}:\Sigma$ as defined above. Then ${\mathsf{FSO}_{\mathscr{D}}}$ proves that for all $F:\Sigma$, if $\mathcal{A}$ accepts $F$ then $\oc\mathcal{A}$ accepts $F$.

Proof 9.2**.**

Let $\sigma$ be a winning $\mathsf{P}$-strategy in $\mathcal{G}(\mathcal{A},F)$. We define a winning $\mathsf{P}$-strategy $\tau$ in $\mathcal{G}(\oc\mathcal{A},F)$. Note that

[TABLE]

First, given a $\mathsf{P}$-position $(x,S)$ in $\mathcal{G}(\oc\mathcal{A},F)$, we define a conjunction

[TABLE]

as follows.

• •

Definition of $\gamma_{(x,S)}$. For each $q\in\pi_{2}(S)$, $\sigma(x,q)$ gives some $\gamma_{q}\in\partial_{\mathcal{A}}(q,F(x))$. $\mathrm{HF}$-Bounded $\mathrm{HF}$-Choice (§3.4.5) then gives

[TABLE]

By $\mathrm{HF}$-Comprehension (Remark 3.7.3), we then let $\gamma_{(x,S)}$ be $\{(d,S^{\prime}_{d})\leavevmode\nobreak\ |\leavevmode\nobreak\ d\in\mathscr{D}\land S^{\prime}_{d}\neq\varnothing\}$ where each $S^{\prime}_{d}$ is defined as in (20).

We now define the $\mathsf{P}$-strategy $\tau$ on $\mathcal{G}(\oc\mathcal{A},F)$. By $\mathrm{HF}$-Bounded Choice for Functions (§3.4.5), we let

[TABLE]

We have $\mathrm{Strat}_{\mathsf{P}}(\mathcal{G}(\oc\mathcal{A},F),\tau)$ directly by definition of $\partial_{\oc\mathcal{A}}$. It remains to check that $\tau$ is winning in $\mathcal{G}(\oc\mathcal{A},F)$. Consider an infinite play of $\tau$, that is some $V:\mathcal{G}(\oc\mathcal{A})\mathrel{\mathfrak{to}}2$ such that $\mathrm{Play}(\tau,\iota_{\oc},V)$. Since $\sigma$ is winning in $\mathcal{G}(\mathcal{A},F)$, by definition of $\mathcal{W}_{\oc\mathcal{A}}$ and by Remarks 48 and 98, we are done if we show that:

[TABLE]

Assume $\mathrm{Trace}(T,V)$. By $\mathrm{HF}$-Comprehension for Product Types, we let $U:\mathcal{G}(\mathcal{A})\mathrel{\mathfrak{to}}2$ be such that $U_{\mathsf{P}}=T_{\mathsf{P}}$ and such that $U_{\mathsf{O}}$ consists of the $\{(x,\sigma(x,q))\}$ for $(x,q)\in U_{\mathsf{P}}$. Note that we actually have $U:\sigma\mathrel{\mathfrak{to}}2$. It remains to check that

[TABLE]

We apply Lemma 39, whence it remains to show:

[TABLE]

Note that $\mathrm{Path}(\mathcal{G}(\mathcal{A}),\iota,T)$ since $\mathrm{Trace}(T,V)$.

• •

Proof 9.3** (Proof of (21)).**

We obviously have $\iota\in U_{\mathsf{P}}=T_{\mathsf{P}}$. Also, given $u\in U$, if $u\in U_{\mathsf{P}}=T_{\mathsf{P}}$ then $\iota\trianglelefteq_{\mathcal{G}(\mathcal{A})}u$, and if $u\in U_{\mathsf{O}}$, then $v\mathchoice{\mathrel{\overset{}{\underset{\sigma}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\sigma}}}{\longrightarrow_{\sigma}}{\longrightarrow_{\sigma}}u$ for some $v\in U_{\mathsf{P}}=T_{\mathsf{P}}$, so that $\iota\trianglelefteq_{\mathcal{G}(\mathcal{A})}v\vartriangleleft_{\mathcal{G}(\mathcal{A})}u$. Moreover, for each $u\in U_{\mathsf{P}}$, we have $u\mathchoice{\mathrel{\overset{}{\underset{\sigma}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\sigma}}}{\longrightarrow_{\sigma}}{\longrightarrow_{\sigma}}v$ for some $v\in U_{\mathsf{O}}$, and we get $\mathsf{s}^{\vartriangleleft}_{\mathcal{G}(\mathcal{A})}(u,v)$ by Proposition 33.

It remains to show that $U$ is linearly ordered w.r.t. $\trianglelefteq_{\mathcal{G}(\mathcal{A})}$. For $U_{\mathsf{P}}$ this follows from the same property for $T_{\mathsf{P}}$. Now let $u\in U_{\mathsf{P}}$ and $v^{\prime}\in U_{\mathsf{O}}$. Hence $v^{\prime}$ is of the form $(x,\sigma(x,q))$ with $v\mathrel{:=}(x,q)\in U_{\mathsf{P}}=T_{\mathsf{P}}$. If $u\trianglelefteq_{\mathcal{G}(\mathcal{A})}v$ then $u\vartriangleleft_{\mathcal{G}(\mathcal{A})}v^{\prime}$ and we are done. Otherwise, $v\vartriangleleft_{\mathcal{G}(\mathcal{A})}u$. But by definition of $\trianglelefteq_{\mathcal{G}(\mathcal{A})}$, this implies $u=(y,q^{\prime})$ with $x\mathrel{\dot{<}}y$, so that $v^{\prime}\vartriangleleft_{\mathcal{G}(\mathcal{A})}u$. Consider now $u^{\prime},v^{\prime}\in U_{\mathsf{O}}$ and let $u,v\in U_{\mathsf{P}}$ be their immediate predecessors. If $u\vartriangleleft_{\mathcal{G}(\mathcal{A})}v$ then $u^{\prime}\vartriangleleft_{\mathcal{G}(\mathcal{A})}v^{\prime}$ and we are done. Otherwise, without loss of generality we have that $u=v$. But then $u^{\prime}=v^{\prime}$ by definition of $U$.

• •

Proof 9.4** (Proof of (22)).**

Assume first $u\in U_{\mathsf{P}}$. In this case, $u$ is of the form $(x,q)$ with $(x,q)\in T_{\mathsf{P}}$, and $u^{\prime}$ is of the form $(x,\sigma(x,q^{\prime}))$ with $(x,q^{\prime})\in T_{\mathsf{P}}$. But $T$ is linearly ordered w.r.t. $\trianglelefteq_{\mathcal{G}(\mathcal{A})}$, so that $q=q^{\prime}$. It follows that $u\mathchoice{\mathrel{\overset{}{\underset{\sigma}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\sigma}}}{\longrightarrow_{\sigma}}{\longrightarrow_{\sigma}}u^{\prime}$.

Assume now that $u\in U_{\mathsf{O}}$. In this case, $u$ is of the form $(x,(q,\gamma_{\mathcal{A}}))$ with $(x,q)\in U_{\mathsf{P}}=T_{\mathsf{P}}$ and $(q,\gamma_{\mathcal{A}})=\sigma(x,q)$. Moreover, $u^{\prime}\in U_{\mathsf{P}}=T_{\mathsf{P}}$ is of the form $(\mathsf{S}_{d}(x),q^{\prime})$. We thus get $u\mathchoice{\mathrel{\overset{}{\underset{\sigma}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\sigma}}}{\longrightarrow_{\sigma}}{\longrightarrow_{\sigma}}u^{\prime}$ as soon as

[TABLE]

Since $\mathrm{Trace}(T,V)$ and since $V$ is a play of $\tau$, there are unique $S,S^{\prime}$ with $(x,S),(\mathsf{S}_{d}(x),S^{\prime})\in V_{\mathsf{P}}$ and such that $q\in\pi_{2}(S)$ and $q^{\prime}\in\pi_{2}(S^{\prime})$. Moreover, we necessarily have $(d,S^{\prime})\in\gamma_{(x,S)}$ for $(S,\gamma_{(x,S)})=\tau(x,S)$. But $\mathrm{Trace}(T,V)$ implies $(q,q^{\prime})\in S^{\prime}$, and it follows that $(d,q^{\prime})\in\gamma_{\mathcal{A}}$ by definition of $\gamma_{(x,S)}$.

*This concludes the proof of Proposition 101. *

Proposition 102**.**

Fix a parity automaton $\mathcal{A}:\Sigma$ and consider the automaton $\oc\mathcal{A}:\Sigma$ as defined above. Then ${\mathsf{FSO}_{\mathscr{D}}}$ proves that for all $F:\Sigma$, if $\oc\mathcal{A}$ accepts $F$ then $\mathcal{A}$ accepts $F$.

Proof 9.5**.**

Let $\tau$ be a winning $\mathsf{P}$-strategy in $\mathcal{G}(\oc\mathcal{A},F)$. We will define a winning $\mathsf{P}$-strategy $\sigma$ in $\mathcal{G}(\mathcal{A},F)$. To this end, we invoke Corollary 55, which tells us that since $\oc\mathcal{A}$ is non-deterministic, for each $x\in\mathscr{D}^{*}$ there is at most one $S\in Q_{\oc\mathcal{A}}$ such that $(x,S)$ belongs to an infinite play of $\tau$. Moreover, using Remark 12, for each $S\in Q_{\oc\mathcal{A}}$ we fix a well-order $\preceq$ on $\mathcal{P}_{*}(\mathscr{D}\times Q_{\mathcal{A}})^{\pi_{2}(S)}$.

We now define the strategy $\sigma$.

• •

Definition of $\sigma$. We apply $\mathrm{HF}$-Bounded Choice for Product Types (Theorem 22). Consider $(x,q)\in\mathcal{G}(\mathcal{A},F)_{\mathsf{P}}$. We first assign to $(x,q)$ an $S\in Q_{\oc\mathcal{A}}$ such that $q\in\pi_{2}(S)$. If there exists such an $S$ where furthermore $(x,S)$ belongs to an infinite play of $\tau$, then this $S$ is unique and we choose that one. Otherwise, by Comprehension for $\mathrm{HF}$-Sets (Remark 3.7.3), we define an ad hoc $S\in Q_{\oc\mathcal{A}}$ with $q\in\pi_{2}(S)$.

Let now $(S,\gamma_{(x,S)})\mathrel{:=}\tau(x,S)$. By definition of $\gamma_{(x,S)}$ there is some

[TABLE]

such that $\gamma_{(x,S)}=\{(d,S^{\prime}_{d})\leavevmode\nobreak\ |\leavevmode\nobreak\ d\in\mathscr{D}\land S^{\prime}_{d}\neq\varnothing\}$ where each $S^{\prime}_{d}$ is as in (20). Consider the $\prec$-least such $f$. We let

[TABLE]

It remains to show that $\sigma$ is winning. To this end, given an infinite play $T$ of $\sigma$, we will define an infinite play $V$ of $\tau$ such that:

[TABLE]

Since $\tau$ is assumed to be winning, thanks to Remarks 48 and 98, this will imply that $\sigma$ is also winning. Assume $\mathrm{Play}(\sigma,\iota,T)$. We define $V$ using the Recursion Theorem (Proposition 26). Let $\varphi(V,v)$ be a $\mathsf{FSO}$-formula stating that:

• •

either $v=\iota_{\oc}$,

• •

or $v=(x,\tau(x,S))$ with $(x,S)\in V$,

• •

or $v=(\mathsf{S}_{d}(x),S^{\prime}_{d})$ and

• –

for some $q^{\prime}\in Q_{\mathcal{A}}$ we have $(\mathsf{S}_{d}(x),q^{\prime})\in T$,

• –

and for some $S\in Q_{\oc\mathcal{A}}$, we have $(x,S)\in V$ and $\tau(x,S)=(S,\gamma_{(x,S)})$ with $(d,S^{\prime}_{d})\in\gamma_{(x,S)}$.

Note that $\varphi(V,v)$ indeed satisfies the assumptions of the Recursion Theorem (Proposition 26), since

• •

in the second clause we always have $(x,S)\vartriangleleft_{\mathcal{G}(\oc\mathcal{A})}(x,\tau(x,S))$, and $\tau(x,S)$ is uniquely determined from $(x,S)$;

• •

in the last clause, we always have $(x,S)\vartriangleleft_{\mathcal{G}(\oc\mathcal{A})}(\mathsf{S}_{d}(x),S^{\prime}_{d})$.

Note also that since $T$ is a play of $\sigma$, there is at most one $d\in\mathscr{D}$ such that $(\mathsf{S}_{d}(x),q)\in T$ for some $q\in Q_{\mathcal{A}}$, and $S^{\prime}_{d}$ is uniquely determined from $d$ and $\tau(x,S)$ by construction of $\oc\mathcal{A}$. So by the Recursion Theorem (Proposition 26) we indeed let $V:\mathcal{G}(\oc\mathcal{A})\mathrel{\mathfrak{to}}2$ be unique such that

[TABLE]

We begin with a series of easy claims on $V$.

Claim 103**.**

For every $x\in\mathscr{D}^{*}$, there is at most one $S\in Q_{\oc\mathcal{A}}$ such that $(x,S)\in V$.

Proof 9.6** (Proof of Claim 103).**

We apply the Induction Axiom of $\mathsf{FSO}$ (§3.4.2). The property holds for $\dot{\varepsilon}$, since $(\dot{\varepsilon},S)\in V$ implies $S={q}^{\iota}_{\oc\mathcal{A}}$ by definition of $V$. Now assume the property for $x$ and let us show it for $\mathsf{S}_{d}(x)$. So assume $(\mathsf{S}_{d}(x),S^{\prime}_{d}),(\mathsf{S}_{d}(x),\tilde{S}^{\prime}_{d})\in V$. By definition of $V$, there are $(x,S),(x,\tilde{S})\in V$ such that $(d,S^{\prime}_{d})\in\gamma_{(x,S)}$ and $(d,\tilde{S}^{\prime}_{d})\in\gamma_{(x,\tilde{S})}$ where $(S,\gamma_{(x,S)})=\tau(x,S)$ and $(\tilde{S},\gamma_{(x,\tilde{S})})=\tau(x,\tilde{S})$. But by induction hypothesis we get $S=\tilde{S}$, which implies $\gamma_{(x,S)}=\gamma_{(x,\tilde{S})}$. This in turn implies $S^{\prime}_{d}=\tilde{S}^{\prime}_{d}$ by construction of $\oc\mathcal{A}$.

Claim 104**.**

For every $u\in V$, the set $\{v\in V\leavevmode\nobreak\ |\leavevmode\nobreak\ v\trianglelefteq_{\mathcal{G}(\oc\mathcal{A})}u\}$ is linearly ordered w.r.t. $\mathchoice{\mathrel{\overset{*}{\underset{\tau}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}_{\tau}}}{\longrightarrow^{*}_{\tau}}{\longrightarrow^{*}_{\tau}}$.

Proof 9.7** (Proof of Claim 104).**

We reason by $\vartriangleleft$-Induction (Theorem 25). So let $u\in V$ be such that the property holds for all $w\vartriangleleft_{\mathcal{G}(\oc\mathcal{A})}u$.

Assume first that $u\in V_{\mathsf{O}}$. In this case, we must have $u=(x,\tau(x,S))$ with $(x,S)\in V$. By induction hypothesis, the set $\{v\in V\leavevmode\nobreak\ |\leavevmode\nobreak\ v\trianglelefteq_{\mathcal{G}(\oc\mathcal{A})}(x,S)\}$ is linearly ordered w.r.t. $\mathchoice{\mathrel{\overset{*}{\underset{\tau}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}_{\tau}}}{\longrightarrow^{*}_{\tau}}{\longrightarrow^{*}_{\tau}}$. On the other hand, it follows from Claim 103 that $(x,S)$ is the only immediate $\mathchoice{\mathrel{\overset{}{\underset{\tau}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\tau}}}{\longrightarrow_{\tau}}{\longrightarrow_{\tau}}$-predecessor of $u$ in $V$. Since $(x,S)\mathchoice{\mathrel{\overset{}{\underset{\tau}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\tau}}}{\longrightarrow_{\tau}}{\longrightarrow_{\tau}}u$, we get the result by Proposition 33.

Assume now that $u\in V_{\mathsf{P}}$. If $u=\iota_{\oc}$ then the result is trivial. Otherwise, $u$ is of the form $(\mathsf{S}_{d}(x),S^{\prime}_{d})$ and its membership to $V$ is given by the last clause defining $V$. Let $S$ be such that $(x,S)\in V$ and such that $\tau(x,S)=(S,\gamma_{(x,S)})$ with $(d,S^{\prime}_{d})\in\gamma_{(x,S)}$. Since $(x,\tau(x,S))\mathchoice{\mathrel{\overset{}{\underset{\tau}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\tau}}}{\longrightarrow_{\tau}}{\longrightarrow_{\tau}}u$ with $(x,\tau(x,S))\in V$, by induction hypothesis the set $\{v\leavevmode\nobreak\ |\leavevmode\nobreak\ v\trianglelefteq_{\mathcal{G}(\oc\mathcal{A})}(x,\tau(x,S))\}$ is linearly ordered w.r.t. $\mathchoice{\mathrel{\overset{*}{\underset{\tau}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}_{\tau}}}{\longrightarrow^{*}_{\tau}}{\longrightarrow^{*}_{\tau}}$. In order to obtain the result for $\{v\leavevmode\nobreak\ |\leavevmode\nobreak\ v\trianglelefteq_{\mathcal{G}(\oc\mathcal{A})}(\mathsf{S}_{d}(x),S^{\prime}_{d})\}$ we need to show that $(x,\tau(x,S))$ is the unique immediate $\mathchoice{\mathrel{\overset{}{\underset{\tau}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\tau}}}{\longrightarrow_{\tau}}{\longrightarrow_{\tau}}$-predecessor of $(\mathsf{S}_{d}(x),S^{\prime}_{d})$ in $V$. But if $(x,\tau(x,\tilde{S}))\in V$ then we should have $(x,\tilde{S})\in V$, so that $\tilde{S}=S$ by Claim 103.

Claim 105**.**

For every $u\in V$, there is an infinite play $U$ of $\tau$ such that:

[TABLE]

Proof 9.8** (Proof of Claim 105).**

Let $u\in V$. First, by Lemma 38 there is an infinite play $U_{0}$ in the game $\mathcal{G}(\oc\mathcal{A})\mathord{\upharpoonright}\{\tau\}$ such that $u\in U_{0}$ and $u\mathchoice{\mathrel{\overset{*}{\underset{\tau}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}_{\tau}}}{\longrightarrow^{*}_{\tau}}{\longrightarrow^{*}_{\tau}}v$ for all $v\in U_{0}$. By Comprehension for Product Types (Theorem 23) we let

[TABLE]

We then get $\mathrm{Play}(\tau,\iota_{\oc},U)$ from Claim 104 and $\mathrm{Play}(\tau,u,U_{0})$.

Claim 106**.**

Let $(x,S)\in V$, and assume $(x,q),(\mathsf{S}_{d}(x),q^{\prime})\in T$ with $q\in\pi_{2}(S)$. Then there is some $S^{\prime}_{d}\in Q_{\oc\mathcal{A}}$ such that $(\mathsf{S}_{d}(x),S^{\prime}_{d})\in V$ and $(q,q^{\prime})\in S^{\prime}_{d}$. Moreover, we have $(d,S^{\prime}_{d})\in\gamma_{(x,S)}$ for $(S,\gamma_{(x,S)})=\tau(x,S)$.

Proof 9.9** (Proof of Claim 106).**

Since $T$ is a play of $\sigma$, we have $(d,q^{\prime})\in\gamma$ for $(q,\gamma)=\sigma(x,q)$. Moreover, by Claim 105, $(x,S)$ belongs to an infinite play of $\tau$. Since $q\in\pi_{2}(S)$, by definition of $\sigma$ this implies that there is some $S^{\prime}_{d}$ such that $(d,S^{\prime}_{d})\in\gamma_{(x,S)}$ for $(S,\gamma_{(x,S)})=\tau(x,S)$ and $(q,q^{\prime})\in S^{\prime}_{d}$. We then obtain $(\mathsf{S}_{d}(x),S^{\prime}_{d})\in V$ by definition of $V$.

We now proceed to show:

[TABLE]

We begin with $\mathrm{Trace}(T,V)$. First, we have $\mathrm{Path}(\mathcal{G}(\mathcal{A}),\iota,T)$ since $T$ is a play of $\sigma$. Moreover

Claim 107**.**

[TABLE]

Proof 9.10** (Proof of Claim 107).**

Using the Induction Axiom of $\mathsf{FSO}$ (§3.4.2), we show

[TABLE]

*For the base case $x=\dot{\varepsilon}$, if $(x,q)\in T$ then we must have $q={q}^{\iota}_{\mathcal{A}}$, so $q\in\pi_{2}({q}^{\iota}_{\oc\mathcal{A}})$. Assume now the property for $x$, and consider $\mathsf{S}_{d}(x)$ and $q,q^{\prime}\in Q_{\mathcal{A}}$ such that $(x,q)\in T$ and $(\mathsf{S}_{d}(x),q^{\prime})\in T$. Furthermore, by induction hypothesis, let $S\in Q_{\oc\mathcal{A}}$ such that $(x,S)\in V$ and $q\in\pi_{2}(S)$. By Claim 106, we then get $(\mathsf{S}_{d}(x),S^{\prime}_{d})\in V$ for some $S^{\prime}_{d}\in Q_{\oc\mathcal{A}}$ with $q^{\prime}\in\pi_{2}(S^{\prime}_{d})$. *

We can now show the last required property for $\mathrm{Trace}(T,V)$, namely:

Claim 108**.**

[TABLE]

Proof 9.11** (Proof of Claim 108).**

*Let $(x,q),(y,q^{\prime})\in T$ and $S^{\prime}\in Q_{\oc\mathcal{A}}$ such that $(x,q)\vartriangleleft^{\mathsf{P};\mathsf{O}}_{\mathcal{G}(\mathcal{A})}(y,q^{\prime})$ and $(y,S^{\prime})\in V$. Then by definition of $\vartriangleleft_{\mathcal{G}(\mathcal{A})}$ we must have $y=\mathsf{S}_{d}(x)$ for some $d\in\mathscr{D}$. Moreover, by Claim 107 there is some $S\in Q_{\oc\mathcal{A}}$ such that $(x,S)\in V$ and $q\in\pi_{2}(S)$. By Claim 106, we then have $(\mathsf{S}_{d}(x),S^{\prime}_{d})\in V$ for some $S^{\prime}_{d}\in Q_{\oc\mathcal{A}}$ with $(q,q^{\prime})\in S^{\prime}_{d}$. It follows from Claim 103 that $S^{\prime}=S^{\prime}_{d}$ so that $(q,q^{\prime})\in S^{\prime}$ and we are done. *

We now turn to showing $\mathrm{Play}(\tau,\iota_{\oc},V)$. Since $\iota_{\oc}\in V$, thanks to Proposition 35 it remains to show:

[TABLE]

First, we easily have:

Claim 109**.**

[TABLE]

Proof 9.12** (Proof of Claim 109).**

The result follows from Claim 105, but it can be proved directly, without the inductions underlying Claim 105. Indeed, if $v=(x,\tau(x,S))$, with $(x,S)\in V$, then the result directly follows from the definitions of $V$ and of the game $\mathcal{G}(\oc\mathcal{A})\mathord{\upharpoonright}\{\tau\}$. Otherwise, we have $v=(\mathsf{S}_{d}(x),S^{\prime}_{d})$, and there is $(x,S)\in V$ such that $\tau(x,S)=(S,\gamma_{(x,S)})$ with $(d,S^{\prime}_{d})\in\gamma_{(x,S)}$. But $(x,S)\in V$ implies $(x,\tau(x,S))\in V$, and again the result directly follows from the definition of $\mathcal{G}(\oc\mathcal{A})\mathord{\upharpoonright}\{\tau\}$.

It then easily follows that:

Claim 110**.**

[TABLE]

Proof 9.13** (Proof of Claim 110).**

First, we have $\iota_{\oc}\in V$ by definition of $V$. Moreover, given $u\in V$ we have either $\iota_{\oc}\mathchoice{\mathrel{\overset{*}{\underset{\tau}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}_{\tau}}}{\longrightarrow^{*}_{\tau}}{\longrightarrow^{*}_{\tau}}u$ or $u\mathchoice{\mathrel{\overset{*}{\underset{\tau}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}_{\tau}}}{\longrightarrow^{*}_{\tau}}{\longrightarrow^{*}_{\tau}}\iota_{\oc}$ by Claim 104. The result then follows from Proposition 33.

It remains to show:

[TABLE]

To this end, we first show:

Claim 111**.**

[TABLE]

Proof 9.14** (Proof of Claim 111).**

Using the Induction Axiom of $\mathsf{FSO}$ (§3.4.2), we show

[TABLE]

For the base case $x=\dot{\varepsilon}$, if $(x,S)\in V$ then we must have $S={q}^{\iota}_{\oc\mathcal{A}}$. Then we are done since $\iota\in T$ and ${q}^{\iota}_{\mathcal{A}}\in\pi_{2}(S)$. Assume the property for $x$, and consider $\mathsf{S}_{d}(x)$ and $S,S^{\prime}\in Q_{\oc\mathcal{A}}$ such that $(x,S),(\mathsf{S}_{d}(x),S^{\prime})\in V$. Furthermore, by induction hypothesis, let $q\in Q_{\mathcal{A}}$ such that $(x,q)\in T$ and $q\in\pi_{2}(S)$. By definition of $V$, we have $(\mathsf{S}_{d}(x),q^{\prime})\in T$ for some $q^{\prime}\in Q_{\mathcal{A}}$. It then follows from Claim 106 that $q^{\prime}\in\pi_{2}(S^{\prime}_{d})$ for some $S^{\prime}_{d}\in Q_{\oc\mathcal{A}}$ such that $(\mathsf{S}_{d}(x),S^{\prime}_{d})\in V$. But Claim 103 implies $S^{\prime}=S^{\prime}_{d}$ so that $q^{\prime}\in\pi_{2}(S^{\prime})$ and we are done.

We can now prove (23).

Proof 9.15** (Proof of (23)).**

If $u=(x,S)\in V$, then $v=(x,\tau(x,S))\in V$ and is unique such that $u\mathchoice{\mathrel{\overset{}{\underset{\tau}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\tau}}}{\longrightarrow_{\tau}}{\longrightarrow_{\tau}}v$. Otherwise, $u=(x,\tau(x,S))$ for some $(x,S)\in V$, and we have to show that there are some unique $d\in\mathscr{D}$ and $S^{\prime}_{d}\in Q_{\oc\mathcal{A}}$ such that $(\mathsf{S}_{d}(x),S^{\prime}_{d})\in V$. First, by Clim 111 there is some $q\in Q_{\mathcal{A}}$ such that $(x,q)\in T$ and $q\in\pi_{2}(S)$. Moreover, since $T$ is a play of $\sigma$, with have $(\mathsf{S}_{d}(x),q^{\prime})\in T$ for some unique $d\in\mathscr{D}$ and $q^{\prime}\in Q_{\mathcal{A}}$. It then follows from Claim 106 that there is some $S^{\prime}_{d}\in Q_{\oc\mathcal{A}}$ such that $(\mathsf{S}_{d}(x),S^{\prime}_{d})\in V$ and $u\mathchoice{\mathrel{\overset{}{\underset{\tau}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\tau}}}{\longrightarrow_{\tau}}{\longrightarrow_{\tau}}(\mathsf{S}_{d}(x),S^{\prime}_{d})$. The uniqueness of $S^{\prime}_{d}$ follows from Claim 103.

This concludes the proof of Proposition 102.

In the proof of Proposition 102 above, we have used Claim 111 in order to show that $V$ is a play of $\sigma$. Let us state here for the record that this has a more general converse: Claim 111 holds whenever $T$ is a trace in $V$ for $V$ a play in $\mathcal{G}(\oc\mathcal{A})$:

Lemma 112**.**

Given $V:\mathcal{G}(\oc\mathcal{A})\mathrel{\mathfrak{to}}2$ and $T:\mathcal{G}(\mathcal{A})\mathrel{\mathfrak{to}}2$, in ${\mathsf{FSO}_{\mathscr{D}}}$ we have

[TABLE]

whenever

[TABLE]

Proof 9.16**.**

Using the Induction Axiom of $\mathsf{FSO}$ (§3.4.2), we show

[TABLE]

For the base case $x=\dot{\varepsilon}$, if $(x,S)\in V$, since $\mathrm{Play}(\mathcal{G}(\oc\mathcal{A}),(\dot{\varepsilon},{q}^{\iota}_{\oc\mathcal{A}}),V)$ we must have $S={q}^{\iota}_{\oc\mathcal{A}}$, so ${q}^{\iota}_{\mathcal{A}}\in\pi_{2}(S)$. Assume now the property for $x$, and consider $d\in\mathscr{D}$ and $S,S^{\prime}\in Q_{\oc\mathcal{A}}$ such that $(x,S)\in V$ and $(\mathsf{S}_{d}(x),S^{\prime})\in V$. Furthermore, by induction hypothesis, let $q\in Q_{\mathcal{A}}$ such that $(x,q)\in T$ and $q\in\pi_{2}(S)$. It follows from $\mathrm{Path}(\mathcal{G}(\mathcal{A}),(\dot{\varepsilon},{q}^{\iota}_{\mathcal{A}}),T)$ that $(\mathsf{S}_{d^{\prime}}(x),q^{\prime})\in T$ for some $d^{\prime}\in\mathscr{D}$ and some $q^{\prime}\in Q_{\mathcal{A}}$. Moreover, $\mathrm{Trace}(T,V)$ implies $q^{\prime}\in\pi_{2}(S^{\prime\prime})$ for some $S^{\prime\prime}$ such that $(\mathsf{S}_{d^{\prime}}(x),S^{\prime\prime})\in V$. But $\mathrm{Play}(\mathcal{G}(\oc\mathcal{A}),(\dot{\varepsilon},{q}^{\iota}_{\oc\mathcal{A}}),V)$ implies $d^{\prime}=d$ and $S^{\prime\prime}=S^{\prime}$ and we are done.

9.2. Reformulating the Acceptance Condition of $\oc\mathcal{A}$.

For an automaton $\mathcal{A}$ which we now assume to be $\mathrm{HF}$-closed (in the sense of Definition 6.1), we are going to formulate the $\mathsf{FSO}$-formula $\mathcal{W}_{\oc\mathcal{A}}$ as a parity condition, which will allow us to obtain a parity automaton $\mathrm{ND}(\mathcal{A})$ in §9.3. In order to obtain a parity condition from $\mathcal{W}_{\oc\mathcal{A}}$ we note (following [Wal02]) that (when read in the standard model) it defines an $\omega$-regular condition, which can thus by McNaughton’s Theorem [McN66] (see also e.g. [Tho97, PP04]) be formulated with a deterministic parity automaton on $\omega$-words. We are actually not going to formalize McNaughton’s Theorem in our setting. Rather, we will apply Proposition 78, which allows us to import in $\mathsf{FSO}$ any true $\mathsf{FSO}$-formula on the infinite paths of $\mathscr{D}^{*}$. Our way to the application of Proposition 78 proceeds with constructions similar to some of those in the proof of Theorem 93.

Consider some $V:\mathcal{G}(\oc\mathcal{A})\mathrel{\mathfrak{to}}2$ such that:

[TABLE]

By Comprehension for Product Types (Theorem 23), let $|V|:\mathscr{D}^{*}\mathrel{\mathfrak{to}}2$ be the set of all $x\in\mathscr{D}^{*}$ such that $(x,S)\in V$ for some $S\in Q_{\oc\mathcal{A}}$. Note that $\mathrm{TPath}(|V|)$ (recall that $\mathrm{TPath}$ is defined in (16)). Furthermore, by $\mathrm{HF}$-Bounded Choice for Functions (§3.4.5), let $\tilde{V}:|V|\mathrel{\mathfrak{to}}Q_{\oc\mathcal{A}}$ take $x\in|V|$ to the unique $S\in Q_{\oc\mathcal{A}}$ such that $(x,S)\in V$.

In $\mathsf{FSO}$ we have that $\mathcal{W}_{\oc\mathcal{A}}$ is equivalent to the following formula $\mathcal{W}[\mathrel{\dot{<}}]^{|V|}_{\oc\mathcal{A}}(\tilde{V})$:

[TABLE]

where

• •

the formula $\mathrm{Trace}[\mathrel{\dot{<}}]^{|V|}(\tilde{T},\tilde{V})$ is

[TABLE]

with

[TABLE]

• •

and, for $C:Q_{\mathcal{A}}\mathrel{\mathfrak{to}}[0,n]$, the formula $\mathrm{Par}[\mathrel{\dot{<}}]^{|V|}(C,n,\tilde{T})$ is (using Convention 5.5):

[TABLE]

Let us first note the following simple property. Recall from Definition 47 that $C_{\mathcal{A}}:Q_{\mathcal{A}}\mathrel{\mathfrak{to}}[0,n_{\mathcal{A}}]$ is a coloring of the states of $\mathcal{A}$, while $\hat{C}_{\mathcal{A}}$ colors the positions of $\mathcal{G}(\mathcal{A})$, by taking for $\mathsf{P}$-positions $(x,q)$ the color given by $C_{\mathcal{A}}$ to $q$ and for $\mathsf{O}$-positions the maximal color $n_{\mathcal{A}}$.

Lemma 113**.**

Assume given $V$ and $|V|$ as above. Let $T:\mathcal{G}(\mathcal{A})\mathrel{\mathfrak{to}}2$ and $\tilde{T}:|V|\mathrel{\mathfrak{to}}Q_{\mathcal{A}}$ such that

[TABLE]

Then:

[TABLE]

Lemma 114**.**

Given $V$, $|V|$ and $\tilde{V}$ as above, ${\mathsf{FSO}_{\mathscr{D}}}$ proves that

[TABLE]

Proof 9.17**.**

Recall that the formula $\mathcal{W}_{\oc\mathcal{A}}$ requires no condition w.r.t. the transitions of $\mathcal{G}(\mathcal{A})$. We proceed as follows:

• •

Assume first $\mathcal{W}_{\oc\mathcal{A}}(V)$, and let $\tilde{T}:|V|\mathrel{\mathfrak{to}}Q_{\mathcal{A}}$ such that $\mathrm{Trace}[\mathrel{\dot{<}}]^{|V|}(\tilde{T},\tilde{V})$. Using Remark 12, let $\prec$ be a well-order on $\mathcal{P}_{*}(\mathscr{D}\times Q_{\mathcal{A}})$. By Comprehension for Product Types (Theorem 23), let $T:\mathcal{G}(\mathcal{A})\mathrel{\mathfrak{to}}2$ such that for all $(x,q)\in\mathcal{G}(\mathcal{A})_{\mathsf{P}}$, we have

[TABLE]

and such that $(x,\gamma)\in T_{\mathsf{O}}$ iff $\gamma\in\mathcal{P}_{*}(\mathscr{D}\times Q_{\mathcal{A}})$ is $\preceq$-minimal such that $\tilde{T}(\mathsf{S}_{d}(x))=q$ for some $(d,q)\in\gamma$. Since $V$ is an infinite play of $\mathcal{G}(\oc\mathcal{A})$ from $(\dot{\varepsilon},{q}^{\iota}_{\oc\mathcal{A}})$, we may conclude by Lemma 113 as soon as we show:

[TABLE]

We obviously have $\mathrm{Path}(\mathcal{G}(\mathcal{A}),T)$ as well as $(\dot{\varepsilon},{q}^{\iota}_{\mathcal{A}})\trianglelefteq_{\mathcal{G}(\mathcal{A})}u$ for all $u\in T$. Moreover we have:

[TABLE]

To see this, let $(x,q)\in T$, so that $\tilde{T}(x)=q$. So by assumption we have $q\in\pi_{2}(\tilde{V}(x))$, and we are done since $(x,\tilde{V}(x))\in V(x)$.

Finally we have

[TABLE]

To see this, given $(x,q),(y,q^{\prime})\in T$ such that $(x,q)\vartriangleleft^{\mathsf{P};\mathsf{O}}_{\mathcal{G}(\mathcal{A})}(y,q^{\prime})$ we necessarily have $\mathsf{S}_{\mathrel{\dot{<}}}(x,y)$, so that $(q,q^{\prime})\in\tilde{V}(y)$ since $\tilde{T}(x)=q$ and $\tilde{T}(y)=q^{\prime}$. But then we are done since $\tilde{V}(y)$ is the unique $S\in Q_{\oc\mathcal{A}}$ such that $(y,S)\in V$.

• •

Conversely, assume $\mathcal{W}[\mathrel{\dot{<}}]^{|V|}_{\oc\mathcal{A}}(\tilde{V})$ and let $T:\mathcal{G}(\mathcal{A})\mathrel{\mathfrak{to}}2$ such that $\mathrm{Trace}(T,V)$. Since $V$ is an infinite play of $\mathcal{G}(\oc\mathcal{A})$ from $(\dot{\varepsilon},{q}^{\iota}_{\oc\mathcal{A}})$, Lemma 112 implies:

[TABLE]

It follows that for all $x\in|V|$ there is $q\in Q_{\mathcal{A}}$ such that $(x,q)\in T$, and this defines $\tilde{T}:|V|\mathrel{\mathfrak{to}}Q_{\mathcal{A}}$ by $\mathrm{HF}$-Bounded Choice for Functions (§3.4.5). Note that we have:

[TABLE]

We can then conclude by Lemma 113 as soon as we show:

[TABLE]

To see this, first, for all $x\in|V|$, we have $(x,\tilde{T}(x))\in T$, so that $T(x)\in\pi_{2}(\tilde{V}(x))$ by definition of $\tilde{V}$. Moreover, given $x,y\in|V|$ with $\mathsf{S}_{\mathrel{\dot{<}}}(x,y)$, we have

[TABLE]

*so that $(\tilde{T}(x),\tilde{T}(y))\in\tilde{V}(y)$ since $\mathrm{Trace}(T,V)$. *

This concludes the proof of Lemma 114.

We are now going to show that $\mathcal{W}[\mathrel{\dot{<}}]^{|V|}_{\oc\mathcal{A}}(\tilde{V})$ is equivalent in $\mathsf{FSO}$ to a parity automaton on $\omega$-words. This relies on McNaughton’s Theorem [McN66] applied in the usual standard model $\mathfrak{N}$ of $\omega$-words, and, via Proposition 78, on the completeness of $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$. In order to apply Proposition 78, we rewrite $\mathcal{W}[\mathrel{\dot{<}}]^{|V|}_{\oc\mathcal{A}}(\tilde{V})$ as the relativization to $|V|$ of the $\mathsf{FSO}[\mathrel{\dot{<}}]^{\omega}$-formula

[TABLE]

where $\mathrm{Par}[\mathrel{\dot{<}}](C,n,\tilde{T})$ is the formula of Definition 8.4.2, and where

[TABLE]

Note that $\mathcal{W}[\mathrel{\dot{<}}]^{|V|}_{\oc\mathcal{A}}(\tilde{V})$ is the relativization to $|V|$ of $\mathcal{W}[\mathrel{\dot{<}}]_{\oc\mathcal{A}}(\tilde{V})$:

[TABLE]

Since $\mathcal{A}$ is $\mathrm{HF}$-closed, the formula $\mathcal{W}[\mathrel{\dot{<}}]_{\oc\mathcal{A}}(\tilde{V})$ is also $\mathrm{HF}$-closed, and we can look at it in the standard model $\mathfrak{N}$ of $\omega$-words (see §7). By McNaughton’s Theorem [McN66] (see also e.g. [Tho97, PP04]), there is a deterministic parity $\omega$-word automaton $\mathcal{D}=(Q_{\mathcal{D}},\,{q}^{\iota}_{\mathcal{D}},\,\partial_{\mathcal{D}},\,c_{\mathcal{D}})$ over $Q_{\oc\mathcal{A}}$, which accepts $\tilde{V}$ exactly when:

[TABLE]

It then follows that in $\mathfrak{N}$, for all $\tilde{V}:Q_{\oc\mathcal{A}}$, the formula $\mathcal{W}[\mathrel{\dot{<}}]_{\oc\mathcal{A}}(\tilde{V})$ is equivalent to

[TABLE]

Proposition 78 then implies that $\mathsf{FSO}$ proves that for $\tilde{V}:|V|\mathrel{\mathfrak{to}}Q_{\oc\mathcal{A}}$, the formula $\mathcal{W}[\mathrel{\dot{<}}]_{\oc\mathcal{A}}^{|V|}(\tilde{V})$ is equivalent to

[TABLE]

9.3. Definition of the Parity Automaton $\mathrm{ND}(\mathcal{A})$.

Consider an alternating parity tree automaton $\mathcal{A}:\Sigma$ as in the beginning of §9, and assume it to be $\mathrm{HF}$-closed. Let $\oc\mathcal{A}:\Sigma$ be defined as in §9.1. Moreover, let $\mathcal{D}:Q_{\oc\mathcal{A}}$ be the parity deterministic $\omega$-word automaton of §9.2. We then let

[TABLE]

where:

• •

the transition function

[TABLE]

takes $((S,q),\mathtt{a})$ to the set of all $\gamma\in\mathcal{P}_{*}(\mathcal{P}_{*}(\mathscr{D}\times(Q_{\oc\mathcal{A}}\times Q_{\mathcal{D}})))$ such that for some $\gamma_{\oc\mathcal{A}}\in\partial_{\oc\mathcal{A}}(S,\mathtt{a})$,

[TABLE]

• •

the coloring $C_{\mathrm{ND}(\mathcal{A})}:Q_{\oc\mathcal{A}}\times Q_{\mathcal{D}}\mathrel{\mathfrak{to}}[0,n_{\mathcal{D}}]$ takes $(S,q)$ to $C_{\mathcal{D}}(q)$.

Note that $\mathrm{ND}(\mathcal{A}):\Sigma$ is $\mathrm{HF}$-closed by Remark 49. We shall now show that $\mathrm{ND}(\mathcal{A})$ is equivalent to $\mathcal{A}$, thus completing the proof of the Simulation Theorem 97. The proof is split into Propositions 115 and 116. As expected, we invoke Theorem 100, that ${\mathsf{FSO}_{\mathscr{D}}}$ proves the equivalence of $!\mathcal{A}$ and $\mathcal{A}$.

{conv}

In Propositions 115 and 116, for fixed automata $\oc\mathcal{A}$ and $\mathrm{ND}(\mathcal{A})$, we let

[TABLE]

Proposition 115**.**

Fix an $\mathrm{HF}$-closed automaton $\mathcal{A}:\Sigma$ and consider $\mathrm{ND}(\mathcal{A}):\Sigma$ as defined above. Then ${\mathsf{FSO}_{\mathscr{D}}}$ proves that for all $F:\Sigma$, if $\mathcal{A}$ accepts $F$ then $\mathrm{ND}(\mathcal{A})$ accepts $F$.

Proof 9.18**.**

Thanks to Theorem 100, we are done if we show that $\mathrm{ND}(\mathcal{A})$ accepts $F$ whenever $\oc\mathcal{A}$ accepts $F$. Let $\sigma:\mathcal{G}(\oc\mathcal{A},F)_{\mathsf{P}}\mathrel{\mathfrak{to}}\mathsf{O}_{\mathcal{G}(\oc\mathcal{A},F)}$ be the winning $\mathsf{P}$-strategy in $\mathcal{G}(\oc\mathcal{A},F)$. We define a strategy $\tau:\mathcal{G}(\mathrm{ND}(\mathcal{A}),F)_{\mathsf{P}}\mathrel{\mathfrak{to}}\mathsf{O}_{\mathcal{G}(\mathrm{ND}(\mathcal{A}),F)}$ as follows.

• •

Definition of $\tau$.* By $\mathrm{HF}$-Bounded Choice for Functions (§3.4.5), we let $\tau(x,(S,q_{\mathcal{D}}))$ be $((S,q_{\mathcal{D}}),\gamma)$, where $\gamma\in\mathcal{P}_{*}(\mathcal{P}_{*}(\mathscr{D}\times(Q_{\oc\mathcal{A}}\times Q_{\mathcal{D}})))$ is defined by Comprehension for $\mathrm{HF}$-Sets (Remark 3.7.3) as the set of all $(d,(S^{\prime}_{d},\partial_{\mathcal{D}}(q_{\mathcal{D}},S)))$ such that $(d,S^{\prime}_{d})\in\gamma_{\oc\mathcal{A}}$, where $\sigma(x,S)=(S,\gamma_{\oc\mathcal{A}})$.*

It remains to show that $\tau$ is winning. So let $T$ such that

[TABLE]

By Comprehension for Product Types (Theorem 23), let $||U||:\mathscr{D}^{*}\times Q_{\mathcal{A}}\mathrel{\mathfrak{to}}2$ consist of the $(x,q_{\oc\mathcal{A}})$ for which there is $q_{\mathcal{D}}\in Q_{\mathcal{D}}$ such that $(x,(q_{\oc\mathcal{A}},q_{\mathcal{D}}))\in T$. By $\mathrm{HF}$-Bounded Choice for Functions (§3.4.5), now let $\tilde{U}:||U||\mathrel{\mathfrak{to}}Q_{\mathcal{D}}$ take $(x,q_{\oc\mathcal{A}})\in||U||$ to (the necessarily unique) $q_{\mathcal{D}}$ such that $(x,(q_{\oc\mathcal{A}},q_{\mathcal{D}}))\in T$. We have:

[TABLE]

It then follows from Lemma 114 that we are done if we show that $||U||$ is the set of all $(x,q_{\mathcal{A}})\in V_{\mathsf{P}}$ for some $V:\mathcal{G}(\oc\mathcal{A})\mathrel{\mathfrak{to}}2$ such that:

[TABLE]

But this is immediate from Comprehension for Product Types (Theorem 23) by letting $V$ be the union of $||U||$ with the set of all $(x,\sigma(x,q_{\oc\mathcal{A}}))$ for $(x,q_{\oc\mathcal{A}})\in||U||$.

When proving that $\mathcal{L}(\mathrm{ND}(\mathcal{A}))\subseteq\mathcal{L}(\mathcal{A})$ in Proposition 116 below, in order to apply Proposition 102, we have to extract a $\mathsf{P}$-strategy on $\mathcal{G}(\oc\mathcal{A},F)$ from a $\mathsf{P}$-strategy on $\mathcal{G}(\mathrm{ND}(\mathcal{A}),F)$. But $\mathrm{ND}(\mathcal{A})$ has more states than $\oc\mathcal{A}$, so we have to resort to Corollary 55, stating that in plays of strategies on non-deterministic automata, states are uniquely determined from tree positions.

Proposition 116**.**

Fix an $\mathrm{HF}$-closed automaton $\mathcal{A}:\Sigma$ and consider $\mathrm{ND}(\mathcal{A}):\Sigma$ as defined above. Then ${\mathsf{FSO}_{\mathscr{D}}}$ proves that for all $F:\Sigma$, if $\mathrm{ND}(\mathcal{A})$ accepts $F$ then $\mathcal{A}$ accepts $F$.

Proof 9.19**.**

Thanks to Theorem 100, we are done if we show $\oc\mathcal{A}$ accepts $F$ whenever $\mathrm{ND}(\mathcal{A})$ accepts $F$.

Let $\tau:\mathcal{G}(\mathrm{ND}(\mathcal{A}),F)_{\mathsf{P}}\mathrel{\mathfrak{to}}\mathsf{O}_{\mathcal{G}}(\mathrm{ND}(\mathcal{A}),F)$ be the winning $\mathsf{P}$-strategy in $\mathcal{G}(\mathrm{ND}(\mathcal{A}),F)$. We are going to define a winning strategy $\sigma:\mathcal{G}(\oc\mathcal{A},F)_{\mathsf{P}}\mathrel{\mathfrak{to}}\mathsf{O}_{\mathcal{G}}(\oc\mathcal{A},F)$ in $\mathcal{G}(\oc\mathcal{A},F)$. Note that $\mathrm{ND}(\mathcal{A})$ has more states than $\oc\mathcal{A}$ and that,

[TABLE]

whereas we need to define:

[TABLE]

As mentioned, we resort to Corollary 55. The strategy $\sigma$ is defined by $\mathrm{HF}$-Bounded Choice for Functions (§3.4.5) as follows. Let $(x,S)\in\mathscr{D}^{*}\times Q_{\oc\mathcal{A}}$.

• •

Assume that there is a play $U$ of $\tau$ such that

[TABLE]

Then it follows from Corollary 55 there is a unique $q_{\mathcal{D}}$ such that

[TABLE]

In this case, we let $\sigma(x,S)$ be $(S,\gamma_{\oc\mathcal{A}})$ where, by Comprehension for $\mathrm{HF}$-Sets (Remark 3.7.3), $\gamma_{\oc\mathcal{A}}$ is the set of all $(d,S^{\prime}_{d})$ such that there is some $q^{\prime}_{\mathcal{D}}\in Q_{\mathcal{D}}$ with $(d,(S^{\prime}_{d},q^{\prime}_{\mathcal{D}}))\in\gamma_{\mathrm{ND}(\mathcal{A})}$ for $((S,q_{\mathcal{D}}),\gamma_{\mathrm{ND}(\mathcal{A})})=\tau(x,(S,q_{\mathcal{D}}))$.

• •

Otherwise, we let $\sigma(x,S)$ be $(S,\gamma_{\oc\mathcal{A}})$ where, by Comprehension for $\mathrm{HF}$-Sets (Remark 3.7.3), $\gamma_{\oc\mathcal{A}}$ is the set of all $(d,S^{\prime}_{d})$ such that there is some $q_{\mathcal{D}}\in Q_{\mathcal{D}}$ with $(d,(S^{\prime}_{d},q_{\mathcal{D}}))\in\gamma_{\mathrm{ND}(\mathcal{A})}$ for $((S,{q}^{\iota}_{\mathcal{D}}),\gamma_{\mathrm{ND}(\mathcal{A})})=\tau(x,(S,{q}^{\iota}_{\mathcal{D}}))$.

We are now going to show that $\sigma$ is winning. To this end, consider an infinite play of $\sigma$, that is some $V:\mathcal{G}(\oc\mathcal{A})\mathrel{\mathfrak{to}}2$ such that

[TABLE]

We are going to define an infinite play of $\tau$, that is some $U:\mathcal{G}(\mathrm{ND}(\mathcal{A}))\mathrel{\mathfrak{to}}2$ with

[TABLE]

First, note that we are done if $U$ satisfies the following property:

[TABLE]

Indeed, by Comprehension (Theorem 23), let $|V|:\mathscr{D}^{*}\mathrel{\mathfrak{to}}2$ be the set of all $x\in\mathscr{D}^{*}$ such that $(x,S)\in V$ for some (necessarily unique) $S\in Q_{\oc\mathcal{A}}$. Moreover, by $\mathrm{HF}$-Bounded Choice for Functions (§3.4.5), let $\tilde{V}:|V|\mathrel{\mathfrak{to}}Q_{\oc\mathcal{A}}$ take $x\in|V|$ to the unique $S\in Q_{\oc\mathcal{A}}$ such that $(x,S)\in V$. By $\mathrm{HF}$-Bounded Choice for Functions (§3.4.5), let now $\tilde{U}:|V|\mathrel{\mathfrak{to}}Q_{\mathcal{D}}$ take $x\in|V|$ to the unique $q_{\mathcal{D}}\in Q_{\mathcal{D}}$ such that $(x,(\tilde{V}(x),q_{\mathcal{D}}))\in U$. Since $\mathrm{Par}(\mathcal{G}(\mathrm{ND}(\mathcal{A})),\hat{C}_{\mathrm{ND}(\mathcal{A})},n_{\mathrm{ND}(\mathcal{A})},U)$, we have $\mathrm{Par}[\mathrel{\dot{<}}]^{|V|}(C_{\mathcal{D}},n_{\mathcal{D}},\tilde{U})$, so that $\mathcal{W}[\mathrel{\dot{<}}]^{|V|}_{\oc\mathcal{A}}(\tilde{V})$ and we conclude by Lemma 114.

We now define an infinite play $U$ of $\tau$ satisfying (24), for which we appeal to the Recursion Theorem (Proposition 26). Let $\varphi(U,u)$ be a $\mathsf{FSO}$ formula stating the disjunction of the following:

• •

$u=\iota_{\mathrm{ND}}$; or

• •

$u=(x,\tau(x,(S,q_{\mathcal{D}})))$* with $(x,(S,q_{\mathcal{D}}))\in U$; or*

• •

$u=(\mathsf{S}_{d}(x),(S^{\prime}_{d},q^{\prime}_{\mathcal{D}}))$, where

• –

$(\mathsf{S}_{d}(x),S^{\prime}_{d})\in V$, and

• –

for some $q_{\mathcal{D}}\in Q_{\mathcal{D}}$ and some $S\in Q_{\oc\mathcal{A}}$ such that $(x,S)\in V$ and $(x,(S,q_{\mathcal{D}}))\in U$, we have $q^{\prime}_{\mathcal{D}}=\partial_{\mathcal{D}}(q_{\mathcal{D}},S)$.

By the Recursion Theorem (Proposition 26) we let $U:\mathcal{G}(\mathrm{ND}(\mathcal{A}))\mathrel{\mathfrak{to}}2$ be unique such that:

[TABLE]

We need to show (24) and:

[TABLE]

We first show that $U$ is a play of $\tau$. Since $\iota_{\mathrm{ND}}\in U$, by Proposition 35 it suffices to show:

[TABLE]

We proceed similarly to the proof of Proposition 102. First, we prove:

Claim 117**.**

[TABLE]

Proof 9.20** (Proof of Claim 117).**

The result directly follows from the definition of $\varphi$ and the definition of $\mathcal{G}(\mathrm{ND}(\mathcal{A}))\mathord{\upharpoonright}\{\tau\}$ (Definition 5.3). If $v=(x,\tau(x,(S,q_{\mathcal{D}})))$, with $(x,(S,q_{\mathcal{D}}))\in U$, then the result is trivial. Otherwise, we have $v=(\mathsf{S}_{d}(x),(S^{\prime}_{d},q^{\prime}_{\mathcal{D}}))$, and there is $(x,(S,q_{\mathcal{D}}))\in U$ such that $(x,S)\in V$ and $q^{\prime}_{\mathcal{D}}=\partial_{\mathcal{D}}(q_{\mathcal{D}},S)$. Note that $(x,(S,q_{\mathcal{D}}))\in U$ implies $(x,\tau(x,(S,q_{\mathcal{D}})))\in U$, and similarly, that $(x,S)\in V$ implies $(s,\sigma(x,S))\in V$. By definition of $\sigma$, we have $\sigma(x,S)=(S,\gamma_{\oc\mathcal{A}})$ where $\gamma_{\oc\mathcal{A}}$ is the set of all $(\tilde{d},S^{\prime}_{\tilde{d}})$ such that $(\tilde{d},(S^{\prime}_{\tilde{d}},q^{\prime}_{\mathcal{D}}))\in\gamma_{\mathrm{ND}(\mathcal{A})}$, where $((S,q_{\mathcal{D}}),\gamma_{\mathrm{ND}(\mathcal{A})})=\tau(x,(S,q_{\mathcal{D}}))$. But then we are done since we indeed have:

[TABLE]

Now we prove:

Claim 118**.**

[TABLE]

Proof 9.21** (Proof of Claim 118).**

We proceed by $\vartriangleleft$-Induction (Theorem 25). So let $u\in U$ s.t. $\iota_{\mathrm{ND}}\mathchoice{\mathrel{\overset{*}{\underset{\tau}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}_{\tau}}}{\longrightarrow^{*}_{\tau}}{\longrightarrow^{*}_{\tau}}v$ for all $v\vartriangleleft u$ with $v\in U$. The result is trivial if $u=\iota_{\mathrm{ND}}$. Otherwise, by Claim 117, there is $v\in U$ such that $v\mathchoice{\mathrel{\overset{}{\underset{\tau}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\tau}}}{\longrightarrow_{\tau}}{\longrightarrow_{\tau}}u$. But $v\vartriangleleft u$ by Proposition 33, so we have $\iota_{\mathrm{ND}}\mathchoice{\mathrel{\overset{*}{\underset{\tau}{\longrightarrow}}}}{\mathrel{\longrightarrow^{*}_{\tau}}}{\longrightarrow^{*}_{\tau}}{\longrightarrow^{*}_{\tau}}v$ by induction hypothesis and we conclude by Proposition 33, again.

It remains to show

[TABLE]

We first prove:

Claim 119**.**

[TABLE]

Proof 9.22** (Proof of Claim 119).**

The property follows from a case analysis according to the following usual consequence of Induction (see Proposition 6, §3.4.3):

[TABLE]

In the case of $x\mathrel{\doteq}\dot{\varepsilon}$, if $(x,(S,q_{\mathcal{D}}))\in U$ then we must have $S={q}^{\iota}_{\oc\mathcal{A}}$, so that $(x,S)\in V$. Consider now the case of $x\mathrel{\doteq}\mathsf{S}_{d}(y)$. If $u=(x,(S,q_{\mathcal{D}}))\in U$, then it follows from $\varphi(U,u)$ that we have $(x,S)\in V$ and we are done.

We can now prove (24).

Proof 9.23** (Proof of (24)).**

If $u=(x,(S,q_{\mathcal{D}}))$, then $(x,\tau(x,(S,q_{\mathcal{D}})))$ is the unique successor of $u$ in $U$. Assume $u=(x,\tau(x,(S,q_{\mathcal{D}})))$. It then follows from Claim 117 that $(x,(S,q_{\mathcal{D}}))\in U$, and by Claim 119 we also get $(x,S)\in V$. Since $(x,S)\in V$, we have $(\mathsf{S}_{d}(x),S^{\prime}_{d})\in V$ for some unique $d\in\mathscr{D}$ and $S^{\prime}_{d}\in Q_{\oc\mathcal{A}}$. It follows from $\varphi(U,u)$ that $v=(\mathsf{S}_{d}(x),(S^{\prime}_{d},q^{\prime}_{\mathcal{D}}))\in U$, where $q^{\prime}_{\mathcal{D}}=\partial_{\mathcal{D}}(q_{\mathcal{D}},S)$. It remains to show that $v$ is unique such :

[TABLE]

Uniqueness follows from $\varphi(U,u)$ and the fact that $V$ is a play, so it remains to show $u\mathchoice{\mathrel{\overset{}{\underset{\tau}{\longrightarrow}}}}{\mathrel{\longrightarrow_{\tau}}}{\longrightarrow_{\tau}}{\longrightarrow_{\tau}}v$. Note that $(x,(S,q_{\mathcal{D}}))\in U$ implies $(x,\tau(x,(S,q_{\mathcal{D}})))\in U$, and similarly, that $(x,S)\in V$ implies $(s,\sigma(x,S))\in V$. By definition of $\sigma$, we have $\sigma(x,S)=(S,\gamma_{\oc\mathcal{A}})$ where $\gamma_{\oc\mathcal{A}}$ is the set of all $(\tilde{d},S^{\prime}_{\tilde{d}})$ such that $(\tilde{d},(S^{\prime}_{\tilde{d}},q^{\prime}_{\mathcal{D}}))\in\gamma_{\mathrm{ND}(\mathcal{A})}$, where $((S,q_{\mathcal{D}}),\gamma_{\mathrm{ND}(\mathcal{A})})=\tau(x,(S,q_{\mathcal{D}}))$. This finishes the proof since we indeed have:

[TABLE]

Finally, we prove (24), that is:

[TABLE]

Proof 9.24** (Proof of (24)).**

Using the Induction Axiom of $\mathsf{FSO}$ (§3.4.2), we show

[TABLE]

For the base case $x\mathrel{\doteq}\dot{\varepsilon}$, if $(x,S)\in V$ then we must have $S={q}^{\iota}_{\oc\mathcal{A}}$, and we indeed obtain $(x,(S,{q}^{\iota}_{\mathcal{D}}))\in U$. Assume now the property for $x$, and consider some $d\in\mathscr{D}$ and $S^{\prime}_{d}$ such that $(\mathsf{S}_{d}(x),S^{\prime}_{d})\in V$. Since $V$ is a play, it follows from the Predecessor Lemma 37 for Infinite Plays (applied twice) that $(x,S)\in V$ for some $S\in Q_{\oc\mathcal{A}}$. It follows by induction hypothesis that $(x,(S,q_{\mathcal{D}}))\in U$ for some $q\in Q_{\mathcal{D}}$. But now, taking $q^{\prime}_{\mathcal{D}}=\partial_{\mathcal{D}}(q_{\mathcal{D}},S)$, we have $(\mathsf{S}_{d}(x),(S^{\prime}_{d},q^{\prime}_{\mathcal{D}}))\in U$ and we are done.

This concludes the proof of Proposition 116.

10. Conclusion

In this paper, we proposed for each non-empty (hereditarily) finite set $\mathscr{D}$ the theory ${\mathsf{FSO}_{\mathscr{D}}}$ of Functional (Monadic) Second-Order Logic on the full (infinite) $\mathscr{D}$-ary tree. The theory ${\mathsf{FSO}_{\mathscr{D}}}$ (henceforth $\mathsf{FSO}$) is a uniform extension of $\mathsf{MSO}$ on the full $\mathscr{D}$-ary tree with hereditarily finite sets. We formalized in $\mathsf{FSO}$ a basic theory of (alternating) tree automata and (acceptance) games. This allowed us, in the theory of $\mathsf{FSO}$ augmented with an axiom $(\mathit{PosDet})$ of positional determinacy of parity games, to formalize a translation of $\mathsf{MSO}$-formulae to automata adapted from [Wal02]. We then deduced the completeness of $\mathsf{FSO}+(\mathit{PosDet})$ thanks to a variant of the Büchi-Landweber Theorem [BL69], stating that $\mathsf{MSO}$ decides winning for (definable) games of finite graphs (and obtained thanks to the completeness of $\mathsf{MSO}$ over $\omega$-words [Sie70]). By naive proof enumeration, this gives a proof of Rabin’s Tree Theorem [Rab69], the decidability of $\mathsf{MSO}$ on infinite trees. Moreover, since the formal theory $\mathsf{FSO}$ is conservative (w.r.t. the faithful translation $(-)^{\circ}:\mathsf{MSO}\to\mathsf{FSO}$) over a natural set of axioms for $\mathsf{MSO}$, we also get a complete axiomatization of $\mathsf{MSO}$ on infinite trees, namely ${\mathsf{MSO}_{\mathscr{D}}}+\langle\mathit{PosDet}\rangle$ (cf. Definition 5.6.1, §5.6.1).

10.1. Proof theoretic strength of complementation

The present paper does not discuss proof theoretic strength. In the context of second-order arithmetic (in the sense of [Sim10]), it is known that complementation of tree automata is between $\Pi^{1}_{3}$ and $\Delta^{1}_{3}$-comprehension [KM16]. As far as only games are concerned (as opposed to proving the correctness of an internal function for complementation), only $\Pi^{1}_{2}$-comprehension is required for the positional determinacy of (each level of) parity games [KM15, Lemma 4.6].

10.2. Clarifying the status of $\langle\mathit{PosDet}\rangle$

A problem arising from this work is whether the axiom schema $\langle\mathit{PosDet}\rangle$ is indeed independent of $\mathsf{MSO}$. The latter may be seen as the monadic fragment of $\mathsf{PA}2$ (over the appropriate language) and, as we have mentioned, is complete when restricted to infinite words. While it might therefore be natural to suspect that $\mathsf{MSO}$ is already complete without $\langle\mathit{PosDet}\rangle$, we point out that the axiomatization of Weak $\mathsf{MSO}$ over infinite trees given in [Sie78] also augments the natural fragment of Peano arithmetic by an axiom of induction over finite trees. As we mentioned in the Introduction, the completeness of ${\mathsf{MSO}_{\mathscr{D}}}$ was erroneously claimed in the preliminary version of this work [DR15].

10.3. On the notion of proof for $\mathsf{MSO}$

One outcome of this work is that our complete deduction system for $\mathsf{MSO}$ gives a new decision algorithm. Of course, the naive decision algorithm by proof enumeration is not very sophisticated, and it is worth restating that its correctness is itself driven by the usual automata-theoretic argument. Such an algorithm, nonetheless, makes no mention of automata and so can be adapted and improved purely in the setting of proof theory. In this sense, the algorithm is the first of its kind: a decision procedure for $\mathsf{MSO}$ on infinite trees that remains internal to the language, rather than requiring intermediate translations to automata.

A basic motivation for such algorithms is that, even if Rabin’s Tree Theorem proves the existence of decision procedures for $\mathsf{MSO}$ on infinite trees, there is (as far as we know) no working implementation of such procedures.333To our knowledge, the Mona tool (https://www.brics.dk/mona/) only handles Weak $\mathsf{MSO}$. Our axiomatization instead allows the targeting of (semi) automatic approaches, for instance based on proof assistants. As we mentioned in the Introduction, our axiomatization is polynomial-time recognizable and so indeed yields a meaningful notion of ‘proof certificate’: a proof of a theorem may be easily checked, without having to reprove the theorem again.

10.4. Constructive systems and proof interpretations.

A further direction of research is to look for constructive interpretations of $\mathsf{MSO}$. In the case of $\omega$-words, preliminary steps were made in [PR19]. The general idea is to proceed along the following steps:

1. (1)

Determine the relevant computational information one should be able to extract from constructive proofs. 2. (2)

Devise constructive variants of $\mathsf{MSO}$ (together with suitable proof interpretations), which are correct and complete for the chosen class of functions w.r.t. to their provable $\forall\exists$-sentences.

A realizability model for $\mathsf{MSO}$ has been proposed in [Rib20], in which the underlying logic is not only constructive but also linear (in the sense of [Gir87]). Of course, similar approaches may also be considered in more traditional settings for constructive interpretations of proofs [TvD88, Koh08].