Predicative proof theory of PDL and basic applications

TL;DR

This paper develops a proof-theoretic framework for propositional dynamic logic (PDL), establishing cut-elimination, decidability results, and connections to quantified boolean formulas, with implications for complexity and formalization in arithmetic.

Contribution

It introduces a sequent calculus for PDL with cut-elimination and transfinite induction, linking PDL validity to formal proofs in arithmetic and complexity classes.

Findings

Finite cut- and omega-rule-free derivations characterize PDL validity.

PDL-validity of star-free sequents is decidable in polynomial space.

Validity of certain formulas is equivalent to the validity of transparent quantified boolean formulas.

Abstract

Propositional dynamic logic (PDL) is presented in Sch\"{u}tte-style mode as one-sided semiformal tree-like sequent calculus Seq$_\omega^{\text{pdl}}$ with standard cut rule and the omega-rule with principal formulas $\left[ P^{\ast }\right] \!A$. The omega-rule-free derivations in Seq$_{\omega }^{\text{pdl}}$ are finite (trees) and sequents deducible by these finite derivations are valid in PDL. Moreover the cut-elimination theorem for Seq$_{\omega}^{\text{pdl}}$ is provable in Peano Arithmetic (PA)extended by transfinite induction up to Veblen's ordinal $\varphi_\omega\left( 0\right) $. Hence (by the cutfree subformula property) such predicative extension of PA proves that any given $\left[ P^{\ast }\right] $-free sequent is valid in PDL iff it is deducible in Seq$_\omega^{\text{pdl}}$ by a finite cut- and omega-rule-free derivation, while PDL-validity of arbitrary star-free sequents is decidable in polynomial space. The former also implies standard Herbrand-style conclusions, which eventually leads to PSPACE-decidability of PDL-validity of $S$, provided that $P$ is atomic and $A$ is in a suitable \emph{basic conjunctive normal form}. Furthermore we consider star-free formulas $A$ in dual \emph{basic disjunctive normal form}, and corresponding expansions $S=\left\langle P^{\ast }\right\rangle \!A\vee Z$ whose PDL-validity problem is known to be EXPTIME-complete. We show that cutfree-derivability in Seq$_\omega^{\text{pdl}}$ (hence PDL-validity) of such $S$\ is equivalent to plain validity of a suitable "transparent" quantified boolean formula $\widehat{S}$. The whole proof can be formalized in PA extended by transfinite induction along $\varphi_\omega\left( 0\right)$ -- actually in the corresponding primitive recursive weakening, $\mathbf{PRA}_{\varphi_{\omega }\left( 0\right)}$.

Equations

$$\begin{array}[]{c}\begin{array}[]{c}\framebox{$\left(\text{{Ax}}\right)\quad x,\lnot x,\Gamma$}\end{array}\\ \begin{array}[]{c}\framebox{$\left(\vee\right)\quad\dfrac{A,B,\Gamma}{A\vee B,\Gamma}\ $}\framebox{$\left(\wedge\right)\quad\dfrac{A,\Gamma\quad\quad B,\Gamma}{A\wedge B,\Gamma}\ $}\end{array}\\ \ \begin{array}[]{c}\framebox{$\left\langle\cup\right\rangle\quad\dfrac{\left\langle P\right\rangle\!A,\left\langle R\right\rangle\!A,\Gamma}{\left\langle P\cup R\right\rangle\!A,\Gamma}\ $}\framebox{$\left[\cup\right]\quad\dfrac{\left[P\right]\!A,\Gamma\text{\qquad}\left[R\right]\!A,\Gamma}{\left[P\cup R\right]\!A,\Gamma}\ $}\end{array}\\ \begin{array}[]{c}\framebox{$\left\langle;\right\rangle\quad\dfrac{\left\langle P\right\rangle\!\!\left\langle R\right\rangle\!A,\Gamma}{\left\langle P;R\right\rangle\!A,\Gamma}\ $}\framebox{$\left[;\right]\quad\dfrac{\left[P\right]\!\left[R\right]\!A,\Gamma}{\left[P;R\right]\!A,\Gamma}\ $}\end{array}\end{array}$$

$$\begin{array}[]{c}\quad\qquad\begin{array}[]{c}\framebox{$\left\langle\ast\right\rangle\quad\dfrac{\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle P\right\rangle^{m}\!\!A,\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle P^{\ast}\right\rangle\!A,\Gamma}{\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle P^{\ast}\right\rangle\!A,\Gamma}\left(m\geq 0\right)\ $}\\ \framebox{$\left[\ast\right]$\quad$\dfrac{\cdots\quad\left[\overrightarrow{Q}\right]\!\!\left[P\right]^{m}\!\!A,\Gamma\quad\cdots\ \left(\forall m\geq 0\right)}{\left[\overrightarrow{Q}\right]\!\!\left[P^{\ast}\right]\!A,\Gamma\quad\quad\quad\quad\quad}$}\end{array}\\ \text{\ }\qquad\framebox{$\begin{array}[]{c}\left(\text{{Gen}}\right)\quad\dfrac{A_{1},\cdots,A_{n}}{\left(P\right)_{\chi_{1}}\!\!A_{1},\cdots,\left(P\right)\!_{\chi_{n}}\!A_{n},\Gamma}\ \left(n>0\right)\\ \text{if }\overset{n}{\underset{i=1}{\sum}}\chi_{i}=1.\end{array}$}\\ \quad\quad\quad\begin{array}[]{c}\framebox{$\left(\text{{Cut}}\right)\quad\dfrac{C,\Gamma\quad\quad\overline{C},\Pi}{\Gamma\cup\Pi}$}\end{array}\end{array}$$

$$\framebox{$\dfrac{A_{1},A_{2},\cdots,A_{n}\overset{\mathcal{L}}{\equiv}A_{1}\vee A_{2}\vee\cdots\vee A_{n}}{\begin{array}[]{c}\left[P\right]\!\left(A_{1}\vee A_{2}\vee\cdots\vee A_{n}\right)\overset{\mathcal{L}}{\equiv}\left[P\right]\!\left(\lnot\left(A_{2}\vee\cdots\vee A_{n}\right)\!\rightarrow\!A_{1}\right)\!\underset{\left(D2\right)}{\Rightarrow}\\ \left[P\right]\!\left(\lnot\left(A_{2}\!\vee{}\cdots\!\vee{}A_{n}\right)\!\rightarrow\!\left[P\right]\!A_{1}\right)\overset{\mathcal{L}}{\equiv}\left[P\right]\!A_{1}\!\vee\!\left\langle P\right\rangle\!\left(A_{2}\!\vee\!\cdots\!\vee\!A_{n}\right)\\ \underset{\left(D1\right)}{\Rightarrow}\left[P\right]\!A_{1}\vee\left\langle P\right\rangle\!\left(A_{2}\vee\cdots\vee A_{n}\right)\vee\Gamma\underset{\left(D3\right)}{\Rightarrow}\\ \left[P\right]\!A_{1}\!\vee\!\left\langle P\right\rangle\!A_{2}\!\vee\!\cdots\!\vee\!\left\langle P\right\rangle\!A_{n}\!\vee\!\Gamma\overset{\mathcal{L}}{\equiv}\left[P\right]\!A_{1},\!\left\langle P\right\rangle\!A_{2},\!\cdots,\left\langle P\right\rangle\!A_{n},\Gamma\end{array}}\left(\text{{G}}\right)$}$$

$$\begin{array}[]{c}\quad\qquad\,\begin{array}[]{c}\framebox{$\left\langle\overset{+}{\cup}\right\rangle\ \dfrac{\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle P\right\rangle\!A,\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle R\right\rangle\!A,\Gamma}{\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle P\cup R\right\rangle\!A,\Gamma}\ $}\framebox{$\left[\overset{+}{\cup}\right]\ \dfrac{\left[\overrightarrow{Q}\right]\!\!\left[P\right]\!A,\Gamma\text{\qquad}\left[\overrightarrow{Q}\right]\!\!\left[R\right]\!A,\Gamma}{\left[\overrightarrow{Q}\right]\!\!\left[P\cup R\right]\!A,\Gamma}\ $}\end{array}\\ \,\quad\quad\quad\quad\ \quad\begin{array}[]{c}\framebox{$\left\langle\,\overset{+}{;}\right\rangle\quad\dfrac{\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle P\right\rangle\!\!\left\langle R\right\rangle\!A,\Gamma}{\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle P;R\right\rangle\!A,\Gamma}\ $}\framebox{$\left[\,\overset{+}{;}\right]\quad\dfrac{\left[\overrightarrow{Q}\right]\!\!\left[P\right]\!\left[R\right]\!A,\Gamma}{\left[\overrightarrow{Q}\right]\!\!\left[P;R\right]\!A,\Gamma}\ $}\end{array}\end{array}$$

$$\displaystyle\begin{array}[]{c}\framebox{$\left(\text{{W}}\right)\quad\dfrac{\Gamma}{\Gamma,\Pi}\ $\ (weakening)}\end{array}\begin{array}[]{c}\framebox{$\left(\text{{C}}\right)\quad\dfrac{A,A,\Gamma}{A,\Gamma}\ $(contraction)}\end{array}$$

$$\displaystyle\begin{array}[]{c}\framebox{$\left(\vee\right)^{\circlearrowleft}\quad\dfrac{\!A\vee B,\Gamma}{\!A,\!B,\Gamma}\ $}\end{array}\begin{array}[]{c}\framebox{$\left(\wedge\right)_{1}^{\circlearrowleft}\quad\dfrac{\!A\wedge B,\Gamma}{\!A,\Gamma}\ $}\end{array}\begin{array}[]{c}\framebox{$\left(\wedge\right)_{2}^{\circlearrowleft}\quad\dfrac{\!A\wedge B,\Gamma}{\!B,\Gamma}\ $}\end{array}$$

$$\displaystyle\begin{array}[]{c}\framebox{$\left\langle\cup\right\rangle^{\circlearrowleft}\quad\dfrac{\!\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle P\cup R\right\rangle\!A,\Gamma}{\!\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle P\right\rangle\!A,\!\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle R\right\rangle\!A,\Gamma}\ $}\end{array}$$

$$\displaystyle\begin{array}[]{c}\framebox{$\left[\cup\right]_{1}^{\circlearrowleft}\quad\dfrac{\!\left[\overrightarrow{Q}\right]\!\!\left[P\cup R\right]\!A,\Gamma}{\left[\overrightarrow{Q}\right]\!\!\left[P\right]\!A,\Gamma}\ $}\end{array}\begin{array}[]{c}\framebox{$\left[\cup\right]_{2}^{\circlearrowleft}\quad\dfrac{\left[\overrightarrow{Q}\right]\!\!\left[P\cup R\right]\!A,\Gamma}{\!\left[\overrightarrow{Q}\right]\!\!\left[R\right]\!A,\Gamma}\ $}\end{array}$$

$$\displaystyle\begin{array}[]{c}\framebox{$\left\langle\,\cup\right\rangle^{\circlearrowleft}\quad\dfrac{\!\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle P;\!R\right\rangle\!A,\Gamma}{\!\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle P\right\rangle\!\!\left\langle R\right\rangle\!A,\Gamma}\ $}\end{array}\begin{array}[]{c}\framebox{$\,\left[\cup\right]^{\circlearrowleft}\quad\dfrac{\left[\overrightarrow{Q}\right]\!\!\left[P;\!R\right]\!A,\Gamma}{\left[\overrightarrow{Q}\right]\!\!\left[P\right]\!\left[R\right]\!A\!,\Gamma}\ $}\end{array}$$

$$\displaystyle\begin{array}[]{c}\framebox{$\left[\ast\right]^{\circlearrowleft}\quad\dfrac{\left[\overrightarrow{Q}\right]\!\!\left[P^{\ast}\right]\!A,\Gamma}{\left[\overrightarrow{Q}\right]\!\!\left[P\right]^{m}\!A,\Gamma}\ \left(m\geq 0\right)\ $}\end{array}$$

$$\displaystyle\begin{array}[]{c}\framebox{$\left(\overrightarrow{\text{{Gen}}}\right)\quad\dfrac{A_{1},\cdots,A_{n}}{\left(\overrightarrow{P}\right)_{\text{\negthinspace}f_{1}}\!\!\!A_{1},\cdots,\left(\overrightarrow{P}\right)_{\!f_{n}}\!\!\!A_{n},\Gamma}\ \left(n>0\right)$}\end{array}$$

$$\framebox{$\deg\left(\partial\right):=\max\left\{0,\sup\left\{\mathfrak{o\!}\left(C\right)+1:C\ occurs\ as\ cut\ formula\ in\ \partial\right\}\right\}$}$$

$$\displaystyle\framebox{$\left(\partial:\Gamma\right)=\dfrac{\left(\partial_{1}:\Gamma_{1}\right)}{\Gamma}\ \left(\text{{R}}\right)$}\text{ \ , \ }\framebox{$\left(\partial:\Gamma\right)=\dfrac{\left(\partial_{1}:\Gamma_{1}\right)\quad\quad\left(\partial_{2}:\Gamma_{2}\right)}{\Gamma}\ \left(\text{{R}}\right)$}\text{ }$$

$$\displaystyle\text{\ \qquad\qquad\ or \ }\framebox{$\left(\partial:\Gamma\right)=\dfrac{\cdots\quad\left(\partial_{m}:\Gamma_{m}\right)\quad\cdots\left\{m\geq 0\right\}}{\Gamma}\ \left(\text{{R}}\right)=\left[\ast\right]$}$$

$$\displaystyle\framebox{$\left(\mathcal{E\!}\left(\partial\right)\!:\!\Gamma\right)\!=\!\dfrac{\left(\mathcal{E\!}\left(\partial_{1}\right):\Gamma_{1}\right)}{\Gamma}\left(\text{{R}}\right)$}\text{ \ or \ }\framebox{$\left(\mathcal{E\!}\left(\partial\right)\!:\!\Gamma\right)\!=\!\dfrac{\left(\mathcal{E\!}\left(\partial_{1}\right):\Gamma_{1}\right)\ \ \left(\mathcal{E\!}\left(\partial_{2}\right):\Gamma_{2}\right)}{\Gamma}\left(\text{{R}}\right)$}$$

$$\displaystyle\text{\ \qquad\ \qquad or \ }\framebox{$\left(\mathcal{E\!}\left(\partial\right):\Gamma\right)=\dfrac{\cdots\quad\left(\mathcal{E\!}\left(\partial_{m}\right):\Gamma_{m}\right)\quad\cdots\left\{m\geq 0\right\}}{\Gamma}\ \left[\ast\right]$}\text{.}$$

$$\framebox{$\left(\partial:\Gamma\cup\Pi\right)=\dfrac{\left(\partial_{1}:C,\Gamma\right)\quad\quad\left(\partial_{2}:\overline{C},\Pi\right)}{\Gamma\cup\Pi}\ \left(\text{{Cut}}\right)$}$$

$$\framebox{$\left(\mathcal{E\!}\left(\partial\right):\Gamma\cup\Pi\right)=\left(\mathcal{E\!}\left(\!\mathcal{R\!}\left(\dfrac{\left(\mathcal{E\!}\left(\partial_{1}\right):C,\Gamma\right)\quad\left(\mathcal{E\!}\left(\partial_{2}\right):\overline{C},\Pi\right)}{\Gamma\cup\Pi}\left(\text{{Cut}}\right)\!\right)\!\right)\!:\Gamma\cup\Pi\!\right)$}$$

$$\mathrm{deg}\left(\mathcal{R}\left(\partial \right)\right)<\mathrm{deg}\left(\partial \right)$$

$$\framebox{$\left(\partial:\Gamma\cup\Pi\right)=\dfrac{\left(\partial_{1}:C,\Gamma\right)\quad\quad\left(\partial_{2}:\overline{C},\Pi\right)}{\Gamma\cup\Pi}\ \left(\text{{Cut}}\right)$}$$

$$\framebox{$h\left(\mathcal{R}\left(\partial\right)\right)<h\left(\partial_{1}\right)+\!\!\!\!\!+\,\,h\left(\partial_{2}\right)+\omega<h\left(\partial\right)\cdot 2+\omega$}\text{.}$$

$$\framebox{$h\left(\mathcal{E}\left(\partial\right)\right)<\varphi\left(\alpha,h\left(\partial\right)\right)$}\text{,}$$

$$\framebox{$h\left(\mathcal{E}\left(\partial\right)\right)<\underset{n<\omega}{\sup}\,\varphi\left(n,\omega\cdot 2\right)=\varphi\left(\omega,0\right)=\varphi_{\omega}\!\left(0\right)$}$$

$$\framebox{$\left\langle\ast\right\rangle\quad\dfrac{\!\!\!\left\langle P^{\ast}\right\rangle\!A,\left\langle P\right\rangle^{m}\!\!A,\Sigma}{\left\langle P^{\ast}\right\rangle\!A,\Sigma}\ $}$$

$$\begin{array}[]{c}\quad\begin{array}[]{c}\framebox{$\left(\text{{Ax}}\right)\quad x,\lnot x,\Gamma$}\end{array}\\ \begin{array}[]{c}\framebox{$\left(\vee\right)\quad\dfrac{A,B,\Gamma}{A\vee B,\Gamma}\ $}\framebox{$\left(\wedge\right)\quad\dfrac{A,\Gamma\quad\quad B,\Gamma}{A\wedge B,\Gamma}\ $}\end{array}\\ \qquad\qquad\text{\ }\framebox{$\begin{array}[]{c}\left(\text{{Gen}}\right)\quad\dfrac{A_{1},\cdots,A_{n}}{\left(p\right)_{\chi_{1}}\!\!A_{1},\cdots,\left(p\right)\!_{\chi_{n}}\!A_{n},\Gamma}\ \left(n>0\right)\\ \text{if }\overset{n}{\underset{i=1}{\sum}}\chi_{i}=1.\end{array}$}\end{array}$$

$$\framebox{$\ \left(\text{{Gen}}\right)\quad\dfrac{A_{i},\Delta}{\left[p\right]\!A_{1},\cdots,\left[p\right]\!A_{k},\left\langle p\right\rangle\!B_{1},\cdots,\left\langle p\right\rangle\!B_{l}}$}$$

$$⊢{\hat{\Sigma }}_0\iff ⊢A,\Pi \iff \stackrel{m}{\underset{i=1}{\bigwedge }}⊢B_i,⟨p⟩C_i,{\Delta }_i,\Pi \iff$$

$$\framebox{$\left.\begin{array}[]{c}``F\!or\ every\ i\in\!\left[1,m\right]\!,\,either\vdash B_{i},\Pi\ or\\ there\ is\ j\in\!\left[1,n_{i}\right]\ such\ that\vdash C_{i},D_{i,j}"\end{array}\right.$}$$

$$⊬{\hat{\Sigma }}_0\iff ⊬A,\Pi \iff$$

$$\framebox{$\left.\begin{array}[]{c}``T\!here\ is\ i\in\!\left[1,m\right]\!\ such\ that\nvdash B_{i},\Pi\\ and\ f\!or\ every\ j\in\!\left[1,n_{i}\right],\,\nvdash C_{i},D_{i,j}"\,.\end{array}\right.$}$$

$$⊢$$

$$\framebox{$\left.\begin{array}[]{c}``F\!or\ every\ i\in\!\left[1,m\right]\!,\,either\vdash B_{i},\Pi\ or\\ there\ is\ j\in\!\left[1,n_{i}\right]\ such\ that\vdash\widehat{A}_{s},C_{i},D_{i,j}"\end{array}\right.$}$$

$$⊬{\hat{\Sigma }}_{s+1}\iff ⊬{\hat{A}}_{s+1},\Pi \iff$$

$$\framebox{$\left.\begin{array}[]{c}``T\!here\ is\ i\in\!\left[1,m\right]\!\ such\ that\nvdash B_{i},\Pi\\ and\ f\!or\ every\ j\in\!\left[1,n_{i}\right],\,\nvdash\widehat{A}_{s},C_{i},D_{i,j}"\,.\end{array}\right.$}$$

Full text

Predicative proof theory of PDL and basic applications

L. Gordeev

Tübingen University

lew.gordeew@uni-tuebingen.de

1 Extended abstract

Propositional dynamic logic (PDL) was derived by M. J. Fischer and R. Ladner [6], [7] from dynamic logic where it plays the role that classical propositional logic plays in classical predicate logic. Conceptually, it describes the properties of the interaction between programs (as modal operators) and propositions that are independent of the domain of computation. The semantics of PDL is based on Kripke frames and comes from that of modal logic. Corresponding sound and complete Hilbert-style formalism was proposed by K. Segerberg [19] (see also [15], [10]). Gentzen-style treatment is more involved. This is because the syntax of PDL includes starred programs $P^{\ast}$ which make finitary sequential formalism similar to that of (say) Peano Arithmetic with induction (PA) that allows no full cut-elimination. In the case of PA, however, there is a well-known Schütte-style solution in the form of infinitary (also called semiformal) sequent calculus with Carnap-style omega-rule that allows full cut elimination, provably in PA extended by transfinite induction up to Gentzen’s ordinal $\varepsilon_{0}$ (cf. [3], [18]). By the same token, in the case of PDL, we introduce Schütte-style semiformal one-sided sequent calculus Seq${}_{\omega}^{\text{{pdl}}}$ whose inferences include the omega-rule with principal formulas $\left[P^{\ast}\right]\!A$ and prove cut-elimination theorem using transfinite induction up to Veblen’s predicative ordinal $\varphi_{\omega}\left(0\right)$ (that exceeds $\varepsilon_{0}$, see [21], [5]). The ordinal increase in question is caused by higher upper bounds on the complexity of cut formulas that may contain nested occurrences of the starred programs, as compared to plain occurrences of quantifiers in the language of PA. The omega-rule-free derivations in Seq${}_{\omega}^{\text{{pdl}}}$ are finite and sequents deducible by these finite derivations are valid in PDL. Hence by the cutfree subformula property we conclude that any given $\left[P^{\ast}\right]$-free sequent is valid in PDL iff it is deducible in Seq${}_{\omega}^{\text{{pdl}}}$ by a finite cut- and omega-rule free derivation, which by standard methods enables better structural analysis of the validity of $\left[P^{\ast}\right]$-free sequent involved. 111cf. e.g. Gentzen-style conclusion that any given false equation n$$=$$m (in particular $0=1$) is not valid in PA, since obviously it has no cutfree derivation. The latter also refers to the computational complexity of the validity problem in PDL that is known to be EXPTIME-complete (cf. [7], [17]). We show that PDL-validity of $\left[P^{\ast}\right]$-free basic conjunctive normal expansions (: $\mathrm{BCNE}$) is solvable in polynomial space, whereas PDL-validity of dual $\left[P^{\ast}\right]$-free basic disjunctive normal expansions (: $\mathrm{BDNE}$), whose negations express that satisfying Kripke frames encode accepting computations of polynomial-space alternating TM, is EXPTIME-complete. Thus the conjecture EXPTIME = PSPACE holds true iff PDL-validity of $\mathrm{BDNE}$ is decidable in polynomial space. We show that cutfree-derivability in Seq${}_{\omega}^{\text{{pdl}}}$ (and hence PDL-validity) of any given $\mathrm{BDNE}$, $S$, is equivalent to the validity of a suitable “transparent” quantified boolean formula $\widehat{S}$ whose size is exponential in the size of $S$. Hence EXPTIME = PSPACE holds true if $\widehat{S}$ is equivalid with another (hypothetical) quantified boolean formula whose size is polynomial in the size of $S$, for every $S\in$ $\mathrm{BDNE}$. This may reduce the former problem to a complexity problem in quantified boolean logic. The whole proof is formalized in PA extended by transfinite induction along $\varphi_{\omega}\left(0\right)$ (at most) – actually in the corresponding primitive recursive weakening, $\mathbf{PRA}_{\varphi_{\omega}\left(0\right)}$.

2 More detailed exposition

2.1 Hilbert-style proof system PDL

Language $\mathcal{L}$

Programs PRO (abbr.: $P$, $Q$, $R$, $S$, possibly indexed):

1. (a)

include program-variables (PRO0) $\pi_{0}$, $\pi_{1}$, … (abbr.: $p$, $q$, $r$, possibly indexed), 2. (b)

are closed under modal connectives ; and $\cup$ and star operation ∗. 2. 2.

Formulas FOR (abbr.: $A$, $B$, $C$, $D$, $F$, $G$, $H$, $E$, etc., possibly indexed):

1. (a)

include formula-variables $\upsilon_{0}$, $\upsilon_{1}$, … (abbr.: $x$, $y$, $z$, possibly indexed), 2. (b)

are closed under implication $\rightarrow$ , negation $\lnot$ and modal operation

$F\hookrightarrow\left[P\right]F$, where $P\in\mathrm{PRO}$. 222Boolean constants are definable as usual e.g. by* *$1:=v_{0}\rightarrow v_{0}$ and $0:=\lnot 1$.

**Axioms (**cf. e.g. [19], [10])$\mathbb{:}$ 333Standard axiom ($D3$) : $\left[P\right]\left(A\wedge B\right)\leftrightarrow\left(\left[P\right]A\wedge\left[P\right]B\right)$ follows by $\left(\text{{\footnotesize G}}\right)$ from ($D1$)–($D2$), whereas ($D6$) : $\left[A?\right]\!B\leftrightarrow\left(A\rightarrow B\right)$ is obsolete in the chosen ?-free language.

$\left(\text{{D}}1\right)\quad Axioms\ of\ classical\ propositional\ logic.$

$\left(\text{{D}}2\right)\quad\left[P\right]\left(A\rightarrow B\right)\rightarrow\left(\left[P\right]A\rightarrow\left[P\right]B\right)$

$\left(\text{{D}}4\right)\quad\left[P;Q\right]A\leftrightarrow\left[P\right]\!\left[Q\right]A$

$\left(\text{{D}}5\right)\quad\left[P\cup Q\right]A\leftrightarrow\left[P\right]\!A\wedge\left[Q\right]A$

$\left(\text{{D}}7\right)\quad\left[P^{\ast}\right]A\leftrightarrow A\wedge\left[P\right]\left[P^{\ast}\right]A$

$\left(\text{{D}}8\right)\quad\left[P^{\ast}\right]\left(A\rightarrow\left[P\right]A\right)\rightarrow\left(A\rightarrow\left[P^{\ast}\right]A\right)$

**Inference rules:  **

\begin{array}[]{l}\left(\text{{MP}}\right)\quad\dfrac{A\quad A\rightarrow B}{B}\\ \left(\text{{G}}\right)\quad\dfrac{A}{\left[P\right]A}\end{array}

2.2 Semiformal sequent calculus SEQ${}_{\omega}^{\text{PDL}}$

Definition 1

The language of Seq${}_{\omega}^{\text{{pdl}}}$ includes seq-formulas and sequents. Seq-formulas  are built up from literals $x$ and $\lnot x$ by propositional connectives $\vee$ and $\wedge$ and modal operations $\left[P\right]$ and $\left\langle P\right\rangle$ for arbitrary $P\in\mathrm{PRO}$. Seq-negation $\overline{F}$ is defined recursively as follows, for any seq-formula $F$.

$\overline{x}:=\lnot x,\ \vskip 3.0pt plus 1.0pt minus 1.0pt\overline{\lnot x}:=x,$** 2. 2.

$\overline{A\vee B}:=\overline{A}\wedge\overline{B},\ \overline{A\wedge B}:=\overline{A}\vee\overline{B}.$** 3. 3.

$\overline{\left\langle P\right\rangle\!A}:=\left[P\right]\!\overline{A},\overline{\ \left[P\right]\!A}:=\left\langle P\right\rangle\overline{\!A}.$**

In the sequel we use abbreviations $\left\langle P\right\rangle^{m}\!:=\overset{m\ times}{\overbrace{\left\langle P\right\rangle\cdots\left\langle P\right\rangle}}$ and $\left[P\right]^{m}\!:=\overset{m\ times}{\overbrace{\left[P\right]\cdots\left[P\right]}}$. For any $\chi\in\left\{0,1\right\}$, let \!\left(P\right)_{\chi}:=\left\{\begin{array}[]{ccc}\left[P\right]\text{,}&\text{if}&\chi=1\text{,}\\ \left\langle P\right\rangle\text{,}&\text{if}&\chi=0\text{.}\end{array}\right. For any $\overrightarrow{P}=P_{1},\cdots,P_{k}$ ($k\geq 0$) and $f:\left[1,k\right]\rightarrow\left\{0,1\right\}$ let $\left(\overrightarrow{P}\right)_{f}:=\left(P_{1}\right)\!_{f\left(1\right)}\cdots\left(P_{k}\right)\!_{f\left(k\right)}$. By $\left(\overrightarrow{Q}\right)$, $\left\langle\overrightarrow{Q}\right\rangle$ and $\left[\overrightarrow{Q}\right]$ we abbreviate $\left(\overrightarrow{Q}\right)_{f}$ for arbitrary $f$, $f\equiv 0$ and $f\equiv 1$, respectively. Formulas from $\mathrm{FOR}$ are represented as seq-formulas recursively by $\lnot F:=\overline{F}$, $F\rightarrow G:=\overline{F}\vee G$ and, conversely, by $F\vee G:=\lnot F\rightarrow G$, $F\wedge G:=\lnot\left(F\rightarrow\lnot G\right)$, $\left\langle P\right\rangle F:=\lnot\left[P\right]\lnot F$. Sequents (abbr.: $\Gamma$, $\Delta$, $\Pi$, $\Sigma$, possibly indexed) are viewed as multisets (possibly empty) of seq-formulas. A sequent $\Gamma=F_{1},\cdots,F_{n}$ is called valid iff so is the corresponding disjunction $F_{1}\vee\cdots\vee F_{n}$. Plain complexity of a given formula and/or program in $\mathcal{L}$ is its ordinary length (= total number of occurrences of literals and connectives $\vee$, $\wedge\,$, $\cup$ , ; , $\ast$).

Definition 2

Ordinal complexity* $\mathfrak{o\!}\left(-\right)<\omega^{\omega}$ of formulas, programs and sequents in $\mathcal{L}$ is defined recursively as follows, where $\mathfrak{\alpha}+\!\!\!\!+\,\,\mathfrak{\beta}$ is the symmetric sum of ordinals $\alpha$ and $\beta$.*

$\mathfrak{o\!}\left(x\right)=\mathfrak{o\!}\left(\lnot x\right)=\mathfrak{o\!}\left(p\right):=0.$** 2. 2.

$\mathfrak{o\!}\left(A\vee B\right)=\mathfrak{o\!}\left(A\wedge B\right):=\max\left\{\mathfrak{o\!}\left(A\right),\mathfrak{o\!}\left(B\right)\right\}+1.$** 3. 3.

$\mathfrak{o\!}\left(P\cup Q\right):=\max\left\{\mathfrak{o\!}\left(P\right),\mathfrak{o\!}\left(Q\right)\right\}+1,\quad\mathfrak{o\!}\left(P;Q\right):=\mathfrak{o\!}\left(P\right)+\!\!\!\!\!+\,\,\mathfrak{o\!}\left(Q\right)+1.$** 4. 4.

$\mathfrak{o\!}\left(P^{\ast}\right):=\underset{m<\omega}{\sup}\mathfrak{o\!}\left(\left(P\right)^{m}\right)=\mathfrak{o\!}\left(P\right)\cdot\omega,\quad\mathfrak{o\!}\left(\left\langle P\right\rangle\!A\right)=\mathfrak{o\!}\left(\left[P\right]\!A\right):=\mathfrak{o\!}\left(P\right)+\!\!\!\!\!+\,\,\mathfrak{o\!}\left(A\right)+1.$** 5. 5.

$\mathfrak{o\!}\left(\Gamma\right):=\sum\left\{\mathfrak{o\!}\left(A\right):A\in\Gamma\right\}.$**

Definition 3

Seq${}_{\infty}^{\text{{pdl}}}$* includes the following axiom $\left(\text{{Ax}}\right)$ and inference rules $\left(\vee\right)$, $\left(\wedge\right)$, $\left\langle\cup\right\rangle$, $\left[\cup\right]$, $\left\langle;\right\rangle$, $\left[;\right]$, $\left\langle\ast\right\rangle$, $\left[\ast\right]$, $\left(\text{{Gen}}\right)$, $\left(\text{{Cut}}\right)$ in classical one-sided sequent formalism in the language $\mathcal{L}$. In $\left[\ast\right]$ we allow $\overrightarrow{Q}=\left[\overrightarrow{Q}\right]=\emptyset$. 444We assume that all rules exposed have nonempty premises. 555$\left[\ast\right]$ has infinitely many premises. Ii is called the $\omega$-rule.*

[TABLE]

[TABLE]

For the sake of brevity we’ll drop “seq-” when referring to seq-formulas of Seq${}_{\omega}^{\text{{pdl}}}$. $\Gamma$ is called derivable in Seq${}_{\omega}^{\text{{pdl}}}$ if there exists a (tree-like, possibly infinite) Seq${}_{\omega}^{\text{{pdl}}}$ *derivation $\partial$ with the root sequent $\Gamma$ (abbr.: *$\left(\partial:\Gamma\right)$ ). We assume that Seq${}_{\omega}^{\text{{pdl}}}$ derivations are well-founded. The simplest way to implement this assumption is to supply nodes $x$ in $\partial$ with ordinals $ord\left(x\right)$ such that ordinals of premises are always smaller that the ones of the corresponding conclusions. Having this we let $h\left(\partial\right):=ord\left(root\left(\partial\right)\right)$ and call it the height of $\partial$.

In Seq${}_{\omega}^{\text{{pdl}}}$, formulas occurring in $\Gamma$ and/or $\Pi$ are called side formulas, whereas other (distinguished) ones are called principal formulas, of axioms or inference rules exposed. These axioms and inferences, in turn, are called principal with respect to their principal formulas. Principal formulas of $\left(\text{{Cut}}\right)$ are also called the corresponding cut formulas. We’ll sometimes specify $\left(\text{{Gen}}\right)$ as $\left(\text{{Gen}}\right)_{P}$ to indicate principal program $P$ involved.

Theorem 4** (soundness and completeness)**

Seq${}_{\omega}^{\text{{pdl}}}$* is sound and complete with respect to PDL. Moreover any PDL-valid sequent (in particular formula) is derivable in Seq${}_{\omega}^{\text{{pdl}}}$ using ordinals $<\mathfrak{\omega}+\,\!\omega=:\omega\cdot 2$.*

Proof. The soundness says that any sequent $\Gamma$ that is derivable in Seq${}_{\omega}^{\text{{pdl}}}$ is valid in Kripke-style semantics of PDL. It is proved by transfinite induction on $h\left(\partial\right)$ of well-founded $\left(\partial:\Gamma\right)$ involved. 666Plain (finite) induction is sufficient for $\left[\ast\right]$-free derivations. Actually it suffices to verify that every inference rule of Seq${}_{\omega}^{\text{{pdl}}}$ preserves Kripke validity, which is easy (we omit the details; see also Remark 5 below).

The completeness is proved as follows by deducing in Seq${}_{\omega}^{\text{{pdl}}}$ the axioms and inferences $\left(D1\right)$, $\left(D2\right)$, $\left(D4\right)$, $\left(D5\right)$, $\left(D7\right),\left(D8\right)$, $\left(\text{{MP}}\right)$, $\left(\text{{G}}\right)$ of PDL. By $\overset{\mathcal{L}}{\equiv}$ we denote the equivalence in propositional logic.

$\left(D1\right)$ is deducible by standard method via extended axiom $\left(\text{{Ax}}\right)^{+}:$ $F,\overline{F},\Gamma$ whose finite cutfree derivation is constructed by recursion on plain complexity of $F$ (in particular we pass by $\left(\text{{Gen}}\right)$ from $A,\overline{A}$ to $\left[P\right]\!A,\left\langle P\right\rangle\!\overline{A},\Gamma$).

$\left(D4\right)$ and $\left(D5\right)$ are trivial, while $\left(D2\right)$, $\left(D7\right)$, $\left(D8\right)$ are derivable as follows.

$\left(D2\right):\ \left[P\right]\left(A\rightarrow B\right)\rightarrow\left(\left[P\right]A\rightarrow\left[P\right]B\right)\overset{\mathcal{L}}{\equiv}\left\langle P\right\rangle\left(A\wedge\overline{B}\right)\vee\left\langle P\right\rangle\!\overline{A}\vee\left[P\right]\!B$.

\framebox{\dfrac{\quad\dfrac{\quad\dfrac{\overset{\left(\text{{Ax}}\right)^{+}}{A,\overline{A}},B\quad\quad\overset{\left(\text{{Ax}}\right)^{+}}{\overline{B},\overline{A}},B}{A\wedge\overline{B},\overline{A},B}\left(\wedge\right)}{\left\langle P\right\rangle\left(A\wedge\overline{B}\right),\left\langle P\right\rangle!\overline{A},\left[P\right]!B}\left(\text{{Gen}}\right)}{\left\langle P\right\rangle\left(A\wedge\overline{B}\right)\vee\left\langle P\right\rangle!\overline{A}\vee\left[P\right]!B\quad}\left(\vee\right)}

$\left(D7\right):\ \left[P^{\ast}\right]A\leftrightarrow A\wedge\left[P\right]\left[P^{\ast}\right]A$

$\overset{\mathcal{L}}{\equiv}\,\left(\left\langle P^{\ast}\right\rangle\!\overline{A}\vee\left(A\wedge\left[P\right]\!\left[P^{\ast}\right]\!A\right)\right)\wedge\left(\left[P^{\ast}\right]\!A\vee\overline{A}\vee\left\langle P\right\rangle\!\left\langle P^{\ast}\right\rangle\!\overline{A}\right)$.

\dfrac{\dfrac{\dfrac{\overset{\left(\text{{Ax}}\right)^{+}}{\overline{A},\left\langle P^{\ast}\right\rangle\overline{\!A},A}}{\left\langle P^{\ast}\right\rangle\overline{\!A},A}\left\langle\ast\right\rangle}{\left\langle P^{\ast}\right\rangle\overline{\!A}\vee A}\left(\vee\right)\dfrac{\dfrac{\dfrac{\overset{\left(\text{{Ax}}\right)^{+}}{\left\langle P\right\rangle^{m+1}\!\overline{A},\left\langle P^{\ast}\right\rangle\overline{\!A},\left[P\right]^{m+1}\!A}}{\cdots\ \left\langle P^{\ast}\right\rangle\!\overline{A},\left[P\right]^{m+1}A\ \cdots}\left\langle\ast\right\rangle}{\left\langle P^{\ast}\right\rangle\overline{\!A},\left[P\right]\!\left[P^{\ast}\right]\!A}\left[\ast\right]}{\left\langle P^{\ast}\right\rangle\!\overline{A}\vee\left[P\right]\!\left[P^{\ast}\right]\!A}\left(\vee\right)}{\left\langle P^{\ast}\right\rangle\!\overline{A}\vee\left(A\wedge\left[P\right]\!\left[P^{\ast}\right]\!A\right)}\left(\wedge\right)$$\ \&\

$\dfrac{\dfrac{\QDATOP{\left(\text{{Ax}}\right)^{+}}{A,\overline{A},\left\langle P\right\rangle\!\!\left\langle P^{\ast}\right\rangle\overline{\!A}}\ \ \dfrac{\overset{\left(\text{{Ax}}\right)^{+}}{\left[P\right]^{m+1}\!A,\overline{A},\left\langle P\right\rangle^{m+1}\!\overline{A},\left\langle P\right\rangle\!\left\langle P^{\ast}\right\rangle\overline{\!A}}}{\left[P\right]^{m+1}\!\!A,\overline{A},\left\langle P\right\rangle\!\left\langle P^{\ast}\right\rangle\overline{\!A}\ \ \ \cdots}\left\langle\ast\right\rangle}{\left[P^{\ast}\right]\!A,\overline{A},\left\langle P\right\rangle\!\left\langle P^{\ast}\right\rangle\overline{\!A}}\left[\ast\right]}{\left[P^{\ast}\right]\!A\vee\overline{A}\vee\left\langle P\right\rangle\!\left\langle P^{\ast}\right\rangle\overline{\!A}}\left(\vee\right)$

$\left(D8\right):\ \left[P^{\ast}\right]\left(A\rightarrow\left[P\right]A\right)\rightarrow\left(A\rightarrow\left[P^{\ast}\right]A\right)$

$\overset{\mathcal{L}}{\equiv}\left\langle P^{\ast}\right\rangle\left(A\wedge\left\langle P\right\rangle\overline{A}\right)\vee\overline{A}\vee\left[P^{\ast}\right]A$.

$\dfrac{\dfrac{\QDATOP{{}}{\cdots}\ \QDATOP{\overset{\partial_{m}}{\Downarrow}}{\left\langle P^{\ast}\right\rangle\!\left(A\wedge\left\langle P\right\rangle\!\overline{A}\right),\overline{A},\left[P\right]^{m}\!A}\ \QDATOP{{}}{\cdots\ }\quad\quad}{\,\left\langle P^{\ast}\right\rangle\!\left(A\wedge\left\langle P\right\rangle\!\overline{A}\right),\overline{A},\left[P^{\ast}\right]\!A}\left[\ast\right]}{\,\left\langle P^{\ast}\right\rangle\!\left(A\wedge\left\langle P\right\rangle\!\overline{A}\right)\vee\overline{A}\vee\left[P^{\ast}\right]\!A}\left(\vee\right)$ , where:

\partial_{0}=$$\left\langle P^{\ast}\right\rangle\!\left(A\wedge\left\langle P\right\rangle\!\overline{A}\right),\overline{A},A\mathbf{\quad}\left(\text{{Ax}}\right)^{+}

\partial_{1}=$$\dfrac{\dfrac{\overset{\left(\text{{Ax}}\right)^{+}}{A,\left\langle P^{\ast}\right\rangle\!\!\left(A\!\wedge\!\left\langle P\right\rangle\!\overline{A}\right)\!,\overline{A},\left[P\right]\!A}\quad\overset{\left(\text{{Ax}}\right)^{+}}{\left\langle P\right\rangle\!\overline{A},\left\langle P^{\ast}\right\rangle\!\!\left(A\!\wedge\!\left\langle P\right\rangle\!\overline{A}\right)\!,\overline{A},\left[P\right]\!A}}{A\wedge\left\langle P\right\rangle\!\overline{A},\left\langle P^{\ast}\right\rangle\!\left(A\wedge\left\langle P\right\rangle\!\overline{A}\right),\overline{A},\left[P\right]\!A}\left(\wedge\right)}{\left\langle P^{\ast}\right\rangle\!\left(A\wedge\left\langle P\right\rangle\!\overline{A}\right),\overline{A},\left[P\right]\!A}\left\langle\ast\right\rangle

\partial_{2}=$$\dfrac{\dfrac{\QDATOP{\left(\text{{Ax}}\right)^{+}}{\,A,\Pi^{+},\overline{A},\left[P\right]\!A,\left[P\right]\!^{2}A\quad\quad}\dfrac{\dfrac{\overset{\left(\text{{Ax}}\right)^{+}}{\,\overline{A},A,P\!A}\overset{\left(\text{{Ax}}\right)^{+}}{\,\quad\quad\overline{A},\left\langle P\right\rangle\!\overline{A},\left[P\right]\!A}}{\,\overline{A},A\wedge\left\langle P\right\rangle\!\overline{A},\left[P\right]\!A}\left(\wedge\right)}{\left\langle P\right\rangle\!\overline{A},\Pi^{+},\overline{A},\left[P\right]\!A,\left[P\right]\!^{2}A}\left(\text{{Gen}}\right)}{A\wedge\left\langle P\right\rangle\!\overline{A},\left\langle P\right\rangle\!\left(A\wedge\left\langle P\right\rangle\!\overline{A}\right),\Pi,\overline{A},\left[P\right]\!A,\left[P\right]\!^{2}A}\left(\wedge\right)}{\left\langle P^{\ast}\right\rangle\!\left(A\wedge\left\langle P\right\rangle\!\overline{A}\right),\overline{A},\left[P\right]^{2}\!A}\left\langle\ast\right\rangle\left\langle\ast\right\rangle for $\Pi:=\left\langle P^{\ast}\right\rangle\!\left(A\wedge\left\langle P\right\rangle\!\overline{A}\right)$ and $\Pi^{+}:=\left\langle P\right\rangle\!\left(A\wedge\left\langle P\right\rangle\!\overline{A}\right),\Pi$,

etc. via $\left(\wedge\right)$, $\left\langle\ast\right\rangle$ and $\left(\text{{Gen}}\right)$.

Obviously these derivations require ordinal assignments $<\mathfrak{\omega}+\omega$. PDL inferences $\left(\text{{MP}}\right)$ and $\left(\text{{G}}\right)$ are obviously derivable by $\left(\text{{Cut}}\right)$ and $\left(\text{{Gen}}\right)$, respectively. These increase ordinals by one, which makes an arbitrary Hilbert-style PDL deduction interpretable as a Seq${}_{\omega}^{\text{{pdl}}}$ derivation of the height $<\omega\cdot 2$, as required.

Remark 5

The validity of $\left(\text{{Gen}}\right)$ also follows by plain generalzation $\left(\text{{G}}\right)$ from $\left(D1\right)$, $\left(D2\right)$ and derivable (dual) $\left(D3\right):\left\langle P\right\rangle\left(A\vee B\right)\leftrightarrow\left(\left\langle P\right\rangle A\vee\left\langle P\right\rangle B\right)$ (cf. Footnote 3), e.g. like this:

[TABLE]

2.3 Cut elimination procedure

2.3.1 Auxiliary sequent calculus SEQ${}_{\omega\text{{+}}}^{\text{PDL}}$

Definition 6

Seq${}_{\omega\text{{+}}}^{\text{{pdl}}}$* is a modification of Seq${}_{\omega}^{\text{{pdl}}}$ that includes the following upgraded inferences $\left\langle\overset{+}{\cup}\right\rangle$, $\left[\overset{+}{\cup}\right]$, $\left\langle\overset{+}{;}\right\rangle$, $\left[\overset{+}{;}\right]$.*

[TABLE]

Obviously these upgrades are still sound in PDL and cut-free derivable in Seq${}_{\omega}^{\text{{pdl}}}$. Hence Seq${}_{\omega}^{\text{{pdl}}}$ and Seq${}_{\omega\text{{+}}}^{\text{{pdl}}}$ are proof theoretically equivalent. In the sequel for the sake of brevity we use old names $\left\langle\cup\right\rangle$, $\left[\cup\right]$, $\left\langle;\right\rangle$, $\left[;\right]$ also for the corresponding upgrades.

2.3.2 Admissible refinements

Lemma 7

The following inferences are admissible in Seq${}_{\omega\text{{+}}}^{\text{{pdl}}}$ minus$\ \left(\text{{Cut}}\right)$. Moreover, for any inversion $\dfrac{\left(\partial:\Delta\right)}{\!\left(\partial^{\circlearrowleft}:\Gamma\right)}$ involved we have $h\left(\partial^{\circlearrowleft}\right)<h\left(\partial\right)+\,\!\omega$. In $\left(\overrightarrow{\text{{Gen}}}\right)$ we assume that $\overrightarrow{P}=P_{1},\cdots,P_{k}$ $\left(k>0\right)$, $f_{1},\cdots,f_{n}:\left[1,k\right]\rightarrow\left\{0,1\right\}$ and $\left(\forall j\in\left[1,k\right]\right)\overset{n}{\underset{i=1}{\sum}}f_{i}\left(j\right)=1$. Note that $\left(\text{{Gen}}\right)$ is a particular case of $\left(\overrightarrow{\text{{Gen}}}\right)$.

[TABLE]

[TABLE]

Proof. Induction on proof height and/or formula complexity. Cases $\left(\text{{W}}\right)$, $\left(\text{{C}}\right)$ are standard. Note that $\left(\text{{C}}\right)$ with principal $\left(\text{{Gen}}\right)$ is trivial, e.g.

\partial:\$$\dfrac{\left(\partial_{1}:A,A,B\right)}{\left\langle P\right\rangle\!A,\left\langle P\right\rangle\!A,\left[P\right]B,\Gamma}\ \left(\text{{Gen}}\right)$$\quad\hookrightarrow\quad\partial^{\text{{C}}}:\$$\dfrac{\left(\partial_{1}^{\text{{C}}}:A,B\right)}{\left\langle P\right\rangle\!A,\left[P\right]B,\Gamma}\ \left(\text{{Gen}}\right).

Case $\left(\overrightarrow{\text{{Gen}}}\right)$ is an obvious iteration of $\left(\text{{Gen}}\right)$.

Cases $\left(\vee\right)^{\circlearrowleft}$, $\left(\wedge\right)_{1}^{\circlearrowleft}$, $\left(\wedge\right)_{2}^{\circlearrowleft}$ are standard (and trivial) boolean inversions**.**

Case $\left\langle\overset{+}{\cup}\right\rangle^{\circlearrowleft}$ ( $\left[\overset{+}{\cup}\right]^{\circlearrowleft}$ analogous)**. **We omit trivial case of principal inversion of $\left\langle\cup\right\rangle$ and show only the crucial cases of principal $\left(\text{{Gen}}\right)$ (in simple form):

\partial:\$$\dfrac{\left(\partial_{1}:A,B,C\right)}{\left\langle P\cup R\right\rangle\!A,\left\langle P\cup R\right\rangle\!B,\left[P\cup R\right]\!C,\Gamma}\ \left(\text{{Gen}}\right)$$\quad\hookrightarrow\partial^{\left\langle\cup\right\rangle^{\circlearrowleft}}:\vskip 3.0pt plus 1.0pt minus 1.0pt

$\dfrac{\dfrac{\left(\text{{Gen}}\right)\!_{\!P}\dfrac{\left(\partial_{1}:A,B,C\right)}{\left\langle P\right\rangle\!\!A,\!\left\langle R\right\rangle\!\!A,\!\left\langle P\right\rangle\!\!B,\!\left\langle R\right\rangle\!\!B,\!\left[P\right]\!C,\!\Gamma}}{\left\langle P\right\rangle\!\!A,\!\left\langle R\right\rangle\!\!A,\!\left\langle P\!\cup\!\!R\right\rangle\!\!B,\!\left[P\right]\!C,\!\Gamma}\!\left\langle\cup\right\rangle\!\dfrac{\dfrac{\left(\partial_{1}:A,B,C\right)}{\left\langle P\right\rangle\!\!A,\!\left\langle R\right\rangle\!\!A,\!\left\langle P\right\rangle\!\!B,\!\left\langle R\right\rangle\!\!B,\!\left[R\right]\!C,\!\Gamma}\!\left(\text{{Gen}}\right)\!_{\!R}}{\left\langle P\right\rangle\!\!A,\!\left\langle R\right\rangle\!\!A,\!\left\langle P\!\cup\!\!R\right\rangle\!\!B,\!\left[R\right]\!C,\!\Gamma}}{\left\langle P\right\rangle\!\!A,\!\left\langle R\right\rangle\!\!A,\!\left\langle P\!\cup\!\!R\right\rangle\!\!B,\!\left[P\!\cup\!\!R\right]\!C,\!\Gamma}\!\left[\cup\right]$,

\partial:\$$\dfrac{\left(\partial_{1}:\left\langle P\cup R\right\rangle\!A,B\right)}{\left\langle Q\right\rangle\!\!\left\langle P\cup R\right\rangle\!A,\left[Q\right]\!B,\Gamma}\left(\text{{Gen}}\right)$$\quad\hookrightarrow

\partial^{\left\langle\cup\right\rangle^{\circlearrowleft}}:\$$\dfrac{\left(\partial_{1}^{\left\langle\cup\right\rangle^{\circlearrowleft}}:\left\langle P\right\rangle\!A,\left\langle R\right\rangle\!A,B\right)}{\left\langle Q\right\rangle\!\!\left\langle P\right\rangle\!A,\left\langle Q\right\rangle\!\!\left\langle R\right\rangle\!A,\left[Q\right]\!B,\Gamma}\left(\text{{Gen}}\right).

**Case $\left\langle\,;\right\rangle^{\circlearrowleft}$ ($\left[\,;\,\right]^{\circlearrowleft}$ analogous). **As above, we omit trivial case of principal inversion of $\left\langle\,;\right\rangle$ and show the crucial cases of principal $\left(\text{{Gen}}\right)$ (in simple form):

\partial:\$$\dfrac{\left(\partial_{1}:A,B,C\right)}{\left\langle P;\!R\right\rangle\!A,\left\langle P;\!R\right\rangle\!B,\left[P;\!R\right]\!C,\Gamma}\ \left(\text{{Gen}}\right)$$\quad\hookrightarrow

\partial^{\left\langle;\right\rangle^{\circlearrowleft}}:\$$\dfrac{\dfrac{\dfrac{\left(\partial_{1}:A,B,C\right)}{\left\langle P\right\rangle\!\!\left\langle R\right\rangle\!A,\left\langle P\right\rangle\!\!\left\langle R\right\rangle\!B,\left[P\right]\!\left[R\right]\!C,\Gamma}\left(\overrightarrow{\text{{Gen}}}\right)_{R,P}}{\left\langle P\right\rangle\!\!\left\langle R\right\rangle\!A,\left\langle P;\!R\right\rangle\!B,\left[P\right]\!\left[R\right]\!C,\Gamma}\left\langle;\right\rangle}{\left\langle P\right\rangle\!\!\left\langle R\right\rangle\!A,\left\langle P;\!R\right\rangle\!B,\left[P;\!R\right]\!C,\Gamma}\ \left[;\right],

\partial:\$$\dfrac{\left(\partial_{1}:\left\langle P;\!R\right\rangle\!A,B,C\right)}{\left\langle Q\right\rangle\!\!\left\langle P;\!R\right\rangle\!A,\left\langle Q\right\rangle\!B,\left[Q\right]\!C,\Gamma}\ \left(\text{{Gen}}\right)$$\quad\hookrightarrow

\partial^{\left\langle;\right\rangle^{\circlearrowleft}}:\$$\dfrac{\left(\partial_{1}^{\left\langle;\right\rangle^{\circlearrowleft}}:\left\langle P\right\rangle\!\!\left\langle R\right\rangle\!A,B,C\right)}{\left\langle Q\right\rangle\!\!\left\langle P\right\rangle\!\!\left\langle R\right\rangle\!A,\left\langle Q\right\rangle\!B,\left[Q\right]\!C,\Gamma}\left(\text{{Gen}}\right).

Case $\left[\ast\right]^{\circlearrowleft}$ is analogous to $\left(\wedge\right)_{i}^{\circlearrowleft}$, via trivial inversion of $\left[\ast\right]$:

\partial:\ \framebox{\dfrac{\left(\partial_{1}:A,B\right)}{\left[P^{\ast}\right]!A,\left\langle P^{\ast}\right\rangle!B,\Gamma}\ \left(\text{{Gen}}\right)}\quad\hookrightarrow

\partial^{\left[\ast\right]^{\circlearrowleft}}:\ \framebox{\dfrac{\dfrac{\left(\partial_{1}:A,B\right)}{\left[P\right]^{m}!!A,\left\langle P\right\rangle^{m}!!B,\left\langle P^{\ast}\right\rangle!B,\Gamma}\left(\overrightarrow{\text{{Gen}}}\right)\underset{m}{{}_{\underbrace{P\cdots P}}}}{\left[P\right]^{m}!!A,\left\langle P^{\ast}\right\rangle!B,\Gamma}\ \left\langle\ast\right\rangle},

\partial:\ \framebox{\dfrac{\left(\partial_{1}:\left[\overrightarrow{R}\right]!!\left[P^{\ast}\right]!A,B\right)}{\left[Q\right]!!\left[\overrightarrow{R}\right]!!\left[P^{\ast}\right]!A,\left\langle Q\right\rangle!B,\Gamma}\ \left(\text{{Gen}}\right)}\quad\hookrightarrow

\partial^{\left[\ast\right]^{\circlearrowleft}}:\ \framebox{\dfrac{\left(\partial_{1}^{\left[\ast\right]^{\circlearrowleft}}:\left[\overrightarrow{R}\right]!!\left[P\right]^{m}!!A,B,\Gamma\right)}{\left[Q\right]!!\left[\overrightarrow{R}\right]!!\left[P\right]^{m}!!A,\left\langle Q\right\rangle!B,\Gamma}\ \left(\text{{Gen}}\right)}.

Note that $\left(\text{{W}}\right)$, $\left(\text{{C}}\right)$, $\left(\vee\right)^{\circlearrowleft}$, $\left(\wedge\right)_{1}^{\circlearrowleft}$, $\left(\wedge\right)_{2}^{\circlearrowleft}$ don’t increase derivation heights.

2.3.3 Cut elimination proper

We adapt familiar predicative cut elimination techniques ([18], [4], [16], [8], [2]).

Theorem 8** (Predicative cut elimination)**

The following is provable in PA extended by transfinite induction up to Veblen-Feferman ordinal $\varphi_{\omega}\!\left(0\right)>\varepsilon_{0}$. Any sequent derivable in Seq${}_{\omega}^{\text{{pdl}}}$ is derivable in Seq${}_{\omega\text{{+}}}^{\text{{pdl}}}$ minus $\left(\text{{Cut}}\right)$. Hence any PDL-valid sequent (formula) is derivable in the cut-free fraction of Seq${}_{\omega\text{{+}}}^{\text{{pdl}}}$, and hence also in Seq${}_{\omega}^{\text{{pdl}}}$ minus $\left(\text{{Cut}}\right)$.

Proof. Our cut elimination operator $\partial\hookrightarrow\mathcal{E}\left(\partial\right)$ satisfying $\deg\left(\mathcal{E}\left(\partial\right)\right)=0$ is defined for any derivation $\partial$ in Seq${}_{\omega\text{{+}}}^{\text{{pdl}}}$ by simultaneous transfinite recursion on $h\left(\partial\right)$ and ordinal cut-degree $\deg\left(\partial\right)$.

[TABLE]

Namely, for any inference rule $\left(\text{{R}}\right)\neq\left(\text{{Cut}}\right)$ with

[TABLE]

we respectively let

[TABLE]

Otherwise, if $\left(\text{{R}}\right)=\left(\text{{Cut}}\right)$ with

[TABLE]

then we stipulate

[TABLE]

with respect to a suitable cut reduction operation $\partial\hookrightarrow\mathcal{R}\left(\partial\right)$ such that

[TABLE]

which makes $\mathcal{E\!}\left(\partial\right)$, $\deg\left(\mathcal{E}\left(\partial\right)\right)=0$, definable by induction on $\deg\left(\partial\right)$ and $h\left(\partial\right)$.

Now $\mathcal{R}\left(\partial\right)$ is defined for any

[TABLE]

by following double induction on ordinal complexity of $C$ and $\max\left(h\left(\partial_{1}\right),h\left(\partial_{2}\right)\right)$, provided that $\deg\left(\partial_{1}\right)=\deg\left(\partial_{2}\right)=0$.

1. Case $C=L$ and $\overline{C}=\overline{L}$ for $L\in\left\{x,\lnot x\right\}$. This case is standard. Namely, we observe that $L$ is principal left-hand side cut formula only if $\left(\partial_{1}:L,\Gamma\right)$ with $\Gamma=\overline{L},\Gamma^{\prime}$. But then $\left(\partial_{2}:\overline{L},\Pi\right)$ infers $\Gamma\cup\Pi=$ $\overline{L},\Gamma^{\prime},\Pi$ by derivable weakening $\left(\text{{W}}\right)$. Thus following Mints-style graphic presentation, we can construct $\mathcal{R}\left(\partial\right)$ by (1) setting $L:=\Pi$ for every non-principal predecessor of the left-hand cut formula $L$ while ascending $\partial_{1}:L,\Gamma$ either up to its disappearance as a side formula of $\left(\text{{Gen}}\right)$ – then we are done – or else up to its principal occurrence in $\left(\text{{Ax}}\right)$ $\overline{L},L,\Gamma^{\prime\prime}$, which yields sequent $\overline{L},\Pi,\Gamma^{\prime\prime}$ instead, followed in the latter case by (2) setting $\overline{L}:=\overline{L},\Gamma^{\prime\prime}$ for every non-principal predecessor of the right-hand cut formula $\overline{L}$ while ascending $\partial_{2}:$ $\overline{L},\Pi$ up to its disappearance as a side formula of $\left(\text{{Gen}}\right)$ or else up to any occurrence in a leaf. In both cases we eventually arrive at a correct derivation of $\Gamma\cup\Pi$, since in the case of principal occurrence of $\overline{L}$ in a modified axiom $\left(\text{{Ax}}\right)$ $\overline{L},L,\Pi^{\prime}$ we obtain another axiom $\left(\text{{Ax}}\right)$ $\overline{L},\Gamma^{\prime\prime},L,\Pi^{\prime}$.

2. Case $C=A\vee B$ and $\overline{C}=\overline{A}\wedge B$. Use derivable inversions $\left(\vee\right)^{\circlearrowleft},\left(\wedge\right)_{1}^{\circlearrowleft},\left(\wedge\right)_{2}^{\circlearrowleft}$:

\partial:\$$\dfrac{\left(\partial_{1}:A\vee B,\Gamma\right)\quad\left(\partial_{2}:\overline{A}\wedge\overline{B},\Pi\right)}{\Gamma\cup\Pi}\left(\text{{Cut}}\right)$$\quad\hookrightarrow\quad

\mathcal{R}\left(\partial\right):=\$$\dfrac{\dfrac{\left(\partial_{1}^{\left(\vee\right)^{\circlearrowleft}}:A,B,\Gamma\right)\quad\left(\partial_{2}^{\left(\vee\right)_{1}^{\circlearrowleft}}:\overline{A},\Pi\right)}{B,\Gamma\cup\Pi}\left(\text{{Cut}}\right)\QDATOP{{}}{\left(\partial_{2}^{\left(\vee\right)_{2}^{\circlearrowleft}}:\overline{B},\Pi\right)}}{\Gamma\cup\Pi}\left(\text{{Cut}}\right).

3. Case $C=\!\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle P\cup R\right\rangle\!A$ and $\overline{C}=\!\left[\overrightarrow{Q}\right]\!\!\left[P\cup R\right]\overline{A}$. Analogous reduction to $\left(\text{{Cut}}\right)$’s on $\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle P\right\rangle\!A$ and $\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle R\right\rangle\!A$ by derivable inversions $\left\langle\cup\right\rangle^{\circlearrowleft}$, $\left[\cup\right]_{1}^{\circlearrowleft}$, $\left[\cup\right]_{2}^{\circlearrowleft}$.

4. Case $C=\!\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle P;R\right\rangle\!A$ and $\overline{C}=\!\left[\overrightarrow{Q}\right]\!\!\left[P;R\right]\overline{A}$. Immediate reduction to $\left(\text{{Cut}}\right)$ on $\left\langle\overrightarrow{Q}\right\rangle\!\!\left\langle P\right\rangle\!\!\left\langle R\right\rangle\!A$ by derivable inversions $\left\langle\,;\right\rangle^{\circlearrowleft}$, $\left[\,;\right]^{\circlearrowleft}$.

5. Case $C=\left\langle\overrightarrow{Q}\right\rangle\!F$ and $\overline{C}=\left[\overrightarrow{Q}\right]\overline{F}$ where $\overrightarrow{Q}=Q_{1}\cdots Q_{n}$ $\left(n>0\right)$ and $\left(\forall j\in\left[1,n\right]\right)\left(Q_{j}\!=p_{j}\text{ or }Q_{j}=P_{j}^{\ast}\right)$, while $F\neq\left\langle Q\right\rangle F^{\prime}$. The reduction is either trivial, if $\partial_{1}=\left(\text{{Ax}}\right)^{+}$, or else defined hereditarily with respect to left-hand side non-principal subcases like

\partial_{1}:\ \framebox{\dfrac{\left(\partial_{1}^{\prime}:C,\Gamma^{\prime}\right)}{C,\Gamma}\ \left(\text{{R}}\right)} with$\ \partial_{2}:\left[p\right]\!\overline{A},\Pi$, when we let

\mathcal{R}\left(\partial\right):=\framebox{\dfrac{\left(\mathcal{R}\left(\partial^{\prime}\right):\Gamma^{\prime}\cup\Pi\right)}{\qquad\Gamma\cup\Pi\quad\quad}\left(\text{{R}}\right)}\vskip 3.0pt plus 1.0pt minus 1.0pt  for \partial^{\prime}:\$$\dfrac{\left(\partial_{1}^{\prime}:C,\Gamma^{\prime}\right)\quad\quad\left(\partial_{2}:\overline{C},\Pi\right)}{\Gamma^{\prime}\cup\Pi}\left(\text{{Cut}}\right),

or analogous non-principal subcases \partial_{1}:\ \framebox{\dfrac{\left(\partial_{1}^{\prime}:C,\Gamma^{\prime}\right)\quad\left(\partial_{1}^{\prime\prime}:C,\Gamma^{\prime\prime}\right)}{C,\Gamma}\ \left(\text{{R}}\right)}\vskip 3.0pt plus 1.0pt minus 1.0pt,

\partial_{1}:\ \framebox{\dfrac{\cdots\quad\left(\partial^{\left(m\right)}:C,\Gamma^{\left(m\right)}\right)\quad\cdots\left(\forall m\geq 0\right)}{C,\Gamma}\ \left[\ast\right]},

as well as the following principal subcases 5 $\left(\text{a}\right)$, 5 $\left(\text{b}\right)$, 5 $\left(\text{c}\right)$.

5 $\left(\text{{a}}\right)$. $C=\left\langle\overrightarrow{Q}\right\rangle\!F=\left\langle\overrightarrow{Q^{\prime}}\right\rangle\!\!\left\langle P^{\ast}\right\rangle\!\!A$ and

\partial_{1}:\$$\dfrac{\left(\partial_{1}^{\prime}:\left\langle\overrightarrow{Q^{\prime}}\right\rangle\!\!\left\langle P\right\rangle\!^{m}A,\left\langle\overrightarrow{Q^{\prime}}\right\rangle\!\!\left\langle P^{\ast}\right\rangle\!\!A,\Gamma\right)}{\left\langle\overrightarrow{Q^{\prime}}\right\rangle\!\!\left\langle P^{\ast}\right\rangle\!\!A,\Gamma}\left\langle\ast\right\rangle with $\partial_{2}:\left[\overrightarrow{Q^{\prime}}\right]\!\!\left[P^{\ast}\right]\!\overline{A},\Pi$. Let

$\mathcal{R}\left(\partial\right):=$

$\dfrac{\left(\mathcal{R}\left(\partial^{\prime}\right):\left\langle\overrightarrow{Q^{\prime}}\right\rangle\!\!\left\langle P\right\rangle\!^{m}A,\Gamma\cup\Pi\right)\quad\quad\left(\partial_{2}^{\left[\ast\right]^{\circlearrowright}}:\left[\overrightarrow{Q^{\prime}}\right]\!\!\left[P\right]\!^{m}\overline{A},\Pi\right)}{\Gamma\cup\Pi}\left(\text{{Cut}}\right)$  where

\partial^{\prime}:\$$\dfrac{\left(\partial_{1}^{\prime}:\left\langle\overrightarrow{Q^{\prime}}\right\rangle\!\!\left\langle P\right\rangle\!^{m}A,\left\langle\overrightarrow{Q^{\prime}}\right\rangle\!\!\left\langle P^{\ast}\right\rangle\!\!A,\Gamma\right)\quad\left(\partial_{2}:\left[\overrightarrow{Q^{\prime}}\right]\!\!\left[P^{\ast}\right]\!\overline{A},\Pi\right)}{\left\langle\overrightarrow{Q^{\prime}}\right\rangle\!\!\left\langle P\right\rangle\!^{m}A,\Gamma\cup\Pi}\left(\text{{Cut}}\right).

5 $\left(\text{{b}}\right)$. $C=\left\langle\overrightarrow{Q}\right\rangle\!F=\left\langle P^{\ast}\right\rangle\!A$ and

\partial_{1}:\$$\dfrac{\left(\partial_{1}^{\prime}:A,\overrightarrow{B},D\right)}{\begin{array}[]{c}\left\langle P^{\ast}\right\rangle\!A,\left\langle P^{\ast}\right\rangle\!\overrightarrow{B},\left[P^{\ast}\right]\!D,\Gamma^{\prime}\\ =\left\langle P^{\ast}\right\rangle\!A,\Gamma\end{array}}\left(\text{{Gen}}\right) with$\ \partial_{2}:\left[P^{\ast}\right]\!\overline{A},\Pi$. Then let

\mathcal{R}\left(\partial\right):=\$$\dfrac{\cdots\ \left(\partial_{m}^{\prime\prime}:\left\langle P^{\ast}\right\rangle\!\overrightarrow{B},\left[P\right]\!^{m}D,\Gamma^{\prime}\cup\Pi\right)\ \cdots\left(\forall m\geq 0\right)}{\begin{array}[]{c}\left\langle P^{\ast}\right\rangle\!\overrightarrow{B},\left[P^{\ast}\right]\!D,\Gamma^{\prime}\cup\Pi\\ =\Gamma\cup\Pi\end{array}}\left[\ast\right]  where

$\partial_{m}^{\prime\prime}:=\vskip 3.0pt plus 1.0pt minus 1.0pt$

$\dfrac{\dfrac{\dfrac{\left(\partial_{1}^{\prime}:A,\overrightarrow{B},D\right)}{\left\langle P\right\rangle\!^{m}\!A,\left\langle P\right\rangle\!^{m}\!\overrightarrow{B},\left[P\right]\!^{m}\!D,\left\langle P^{\ast}\right\rangle\!\!\overrightarrow{B},\Gamma^{\prime}}\!\left(\overrightarrow{\text{{Gen}}}\right)\ \QDATOP{{}}{\left(\partial_{2}^{\left[\ast\right]^{\circlearrowright}}:\left[P\right]\!^{m}\overline{A},\Pi\right)}}{\left\langle P\right\rangle\!^{m}\!\overrightarrow{B},\left\langle P^{\ast}\right\rangle\!\!\overrightarrow{B},\left[P\right]\!^{m}\!D,\Gamma^{\prime}\cup\Pi}\!\left(\text{{Cut}}\right)}{\left\langle P^{\ast}\right\rangle\!\!\overrightarrow{B},\left[P\right]\!^{m}\!D,\Gamma^{\prime}\cup\Pi}\!\left\langle\ast\right\rangle$.

5 $\left(\text{{c}}\right)$. $C=\left\langle\overrightarrow{Q}\right\rangle\!F=\left\langle p\right\rangle\!A$ and

\partial_{1}:\$$\dfrac{\left(\partial_{1}^{\prime}:A,\overrightarrow{B},D\right)}{\begin{array}[]{c}\left\langle p\right\rangle\!A,\left\langle p\right\rangle\!\overrightarrow{B},\left[p\right]\!D,\Gamma^{\prime}\\ =\left\langle p\right\rangle\!A,\Gamma\end{array}}\left(\text{{Gen}}\right) with$\ \partial_{2}:\left[p\right]\!\overline{A},\Pi$. Then we let

\mathcal{R}\left(\partial\right):=\$$\dfrac{\left(\mathcal{R}^{\prime}\left(\partial_{1}^{\prime},\partial_{2}\right):\left\langle p\right\rangle\!\overrightarrow{B},\left[p\right]\!D,\Pi\right)}{\begin{array}[]{c}\left\langle p\right\rangle\!\overrightarrow{B},\left[p\right]\!D,\Gamma^{\prime}\cup\Pi\\ =\Gamma\cup\Pi\end{array}}\left(\text{{W}}\right),

where $\mathcal{R}^{\prime}\left(\partial_{1}^{\prime},\partial_{2}\right)$ is defined by induction on $h\left(\partial_{2}\right)$ – either trivially, if $\partial_{2}=\left(\text{{Ax}}\right)^{+}$, or hereditarily, in the non-principal subcases, while in the principal subcases

\partial_{2}:\$$\dfrac{\left(\partial_{2}^{\prime}:\overline{A},\overrightarrow{G}\right)}{\begin{array}[]{c}\left[p\right]A,\left\langle p\right\rangle\!\overrightarrow{G},\Pi^{\prime}\\ =\left[p\right]\!A,\Pi\end{array}}\left(\text{{Gen}}\right)  and \partial_{2}:\$$\dfrac{\left(\partial_{2}^{\prime}:\overrightarrow{G},H\right)}{\begin{array}[]{c}\left[p\right]A,\left\langle p\right\rangle\!\overrightarrow{G},\left[p\right]\!H,\Pi^{\prime\prime}\\ =\left[p\right]\!A,\Pi\end{array}}\left(\text{{Gen}}\right)

we respectively let

\mathcal{R}^{\prime}\left(\partial_{1},\partial_{2}\right)\vskip 3.0pt plus 1.0pt minus 1.0pt$$:=\$$\dfrac{\left(\partial_{1}^{\prime}:A,\overrightarrow{B},D\right)\quad\quad\left(\partial_{2}^{\prime}:\overline{A},\overrightarrow{G}\right)}{\dfrac{\overrightarrow{B},D,\overrightarrow{G}}{\begin{array}[]{c}\left\langle p\right\rangle\!\overrightarrow{B},\left[p\right]\!D,\left\langle p\right\rangle\!\overrightarrow{G},\Pi^{\prime}\\ =\left\langle p\right\rangle\!\overrightarrow{B},\left[p\right]\!D,\Pi\end{array}}\left(\text{{Gen}}\right)}\left(\text{{Cut}}\right)  and

\mathcal{R}^{\prime}\left(\partial_{1},\partial_{2}\right)\vskip 3.0pt plus 1.0pt minus 1.0pt\vskip 3.0pt plus 1.0pt minus 1.0pt:=\ \framebox{\dfrac{\left(\partial_{2}^{\prime}:\overrightarrow{G},H\right)}{\begin{array}[]{c}\left\langle p\right\rangle!\overrightarrow{G},\left[p\right]!H,\left\langle p\right\rangle!\overrightarrow{B},\left[p\right]!D,\Pi^{\prime\prime}\ =\left\langle p\right\rangle!\overrightarrow{B},\left[p\right]!D,\Pi\end{array}}\left(\text{{Gen}}\right)}, as desired.

Obviously $\mathcal{R}$ reduces the cut degree of $\partial$. That is, in each case 1–5 we have$\ \deg\left(\mathcal{R}\left(\partial\right)\right)<\deg\left(\partial\right)<\omega^{\omega}$, provided that both $\partial_{1}$ and $\partial_{2}$ involved are cutffree. Moreover it’s readily seen that nodes in $\mathcal{R}\left(\partial\right)$ can be augmented with ordinals such that

[TABLE]

Having this one can define ordinal assignments also for (slightly modified) cutfree derivations $\mathcal{E}\left(\partial\right)$ such that for any $\partial$ with $\deg\left(\partial\right)<\omega^{\alpha}$ it holds

[TABLE]

which for $\deg\left(\partial\right)<\omega^{\omega}$ and $h\left(\partial\right)<\omega\cdot 2$ (cf. Theorem 4) yields

[TABLE]

(see Appendix A for a detailed presentation). It is readily seen that the entire proof is formalizable in $\mathbf{PA}_{\varphi_{\omega}\left(0\right)}$, i.e. $\mathbf{PA}$ extended by schema of transfinite induction along (canonical primitive recursive representation of) ordinal $\varphi_{\omega}\!\left(0\right)$. 777$\varphi_{\omega}\!\left(0\right)=D\!\left(\omega^{\Omega+\omega}\right)$ according to ordinal notations used in [8].

Corollary 9

Let $\Gamma$ be any sequent that does not contain occurrences $\left[P^{\ast}\right]$ and suppose that $\Gamma$ is derivable in Seq${}_{\omega}^{\text{{pdl}}}$. Then $\Gamma$ is derivable in a subsystem of Seq${}_{\omega}^{\text{{pdl}}}$, called Seq${}_{1}^{\text{{pdl}}}$, that does not contain inferences $\left[\ast\right]$ and/or $\left(\text{{Cut}}\right)$. Note that every derivation in Seq${}_{1}^{\text{{pdl}}}$ is finite. Consequently, any given $\left[P^{\ast}\right]$-free seq-formula is valid in PDL iff it is derivable in Seq${}_{1}^{\text{{pdl}}}$.

Proof. This is obvious by the subformula property of cutfree derivations.

Remark 10

Here and below we argue in $\mathbf{PA}_{\varphi_{\omega}\left(0\right)}$ that is a proper extension of $\mathbf{PA}$, as $\varphi_{\omega}\left(0\right)>\varepsilon_{0}$. Actually by standard arguments the whole proof is formalizable in the corresponding primitive recursive weakening, $\mathbf{PRA}_{\varphi_{\omega}\left(0\right)}$.

2.4 Herbrand-style conclusions

Let $\mathcal{L}_{0}$ be the star-free sublanguage of $\mathcal{L}$. Denote by Seq${}_{0}^{\text{{pdl}}}$ the $\mathcal{L}_{0}$-subsystem of Seq${}_{1}^{\text{{pdl}}}$.

Theorem 11

Let $\Sigma=\left\langle P^{\ast}\right\rangle\!A,\Pi$ with $A,\Pi\in\mathcal{L}_{0}$. Suppose that $\Sigma$ is derivable in Seq${}_{\omega}^{\text{{pdl}}}$. Then there exists a $k\geq 0$ such that $\widehat{\Sigma}_{k}:=A,\left\langle p\right\rangle\!A,\cdots,\left\langle p\right\rangle^{k}\!\!A,\Pi$ is derivable in Seq${}_{0}^{\text{{pdl}}}$.

Proof. The nontrivial implication  Seq${}_{\omega}^{\text{{pdl}}}$ $\vdash\Sigma\Rightarrow\,\,\text{{Seq}}_{0}^{\text{{pdl}}}\vdash\widehat{\Sigma}_{k}$ follows by standard arguments from the cut elimination theorem by induction on the height of the corresponding finite cutfree proof $\partial$ of $\Sigma$ in Seq${}_{\omega}^{\text{{pdl}}}$. Since no $\left[P^{\ast}\right]$ occurs in $\Sigma$, no $\left\langle P^{\ast}\right\rangle\!A$ can be principal formula in $\left(\text{{Gen}}\right)$. Thus the only crucial case is when some $\left\langle P^{\ast}\right\rangle\!A$ is principal formula in

[TABLE]

which by the induction hypothesis yields $k$ such that$\ \left\langle P\right\rangle^{m}\!\!A,\widehat{\Sigma}_{k}$ is derivable in Seq${}_{0}^{\text{{pdl}}}$. By $\left(\text{{C}}\right)$ or $\left(\text{{W}}\right)$ this yields the derivability of $\widehat{\Sigma}_{k^{\prime}}$ for $k^{\prime}:=\max\left(k,m\right)$.

Remark 12

By the same token, for any $\left[P^{\ast}\right]$-free seq-formula $F$, one can successively replace all subformulas $\left\langle P^{\ast}\right\rangle\!A$ by appropriate disjunctions $\overset{k}{\underset{i=0}{\bigvee}}\left\langle P\right\rangle^{i}\!\!A$ such that $F$ is $\mathbf{PDL}$-valid iff the resulting expansion $\widehat{F}$ is derivable in Seq${}_{0}^{\text{{pdl}}}$.

2.4.1 PSPACE refinement

Denote by $\mathcal{L}_{00}$ a sublanguage of $\mathcal{L}_{0}$ over atomic programs PRO0 and let $\mathcal{L}_{\emptyset}$ be purely propositional fraction of $\mathcal{L}.$ Note that program operations “;” and “$\cup$” are definable in $\mathcal{L}_{00}$ via $\left(P;Q\right)\!A:=\left(P\right)\!\left(Q\right)\!A$, $\left\langle P\cup Q\right\rangle\!A:=\left\langle P\right\rangle\!A\vee\left\langle Q\right\rangle\!A$ and $\left[P\cup Q\right]\!A:=\left[P\right]\!A\wedge\left[Q\right]\!A$. Let Seq${}_{00}^{\text{{pdl}}}$ be the following $\mathcal{L}_{00}$-restriction of Seq${}_{0}^{\text{{pdl}}}$ (that proves the same $\mathcal{L}_{00}$-sequents as Seq${}_{0}^{\text{{pdl}}}$).

[TABLE]

Remark 13

Theorems 4 and 8 confirm that PDL is a conservative extension of classical propositional logic that is formalized by the $\left(\text{{Gen}}\right)$-free subsystem of Seq${}_{00}^{\text{{pdl}}}$. Thus any $\mathcal{L}_{\emptyset}$-formula $A$ is derivable in Seq${}_{00}^{\text{{pdl}}}-\left(\text{{Gen}}\right)$ iff it is valid in propositional logic, and hence, by contraposition, Seq${}_{00}^{\text{{pdl}}}\nvdash A$ iff $\nvDash A$ (: $\lnot A$ is satisfiable).

Lemma 14** ($p\,$-inversion)**

Suppose that $\left[p\right]\!A_{1},\cdots,\left[p\right]\!A_{j},\left\langle p\right\rangle\!B_{1},\cdots,\left\langle p\right\rangle\!B_{k},\Gamma$, where $\Gamma\!=\!\left(q_{1}\right)\!C_{1},\cdots,\left(q_{l}\right)\!C_{l},\Pi$ for $q_{j}\neq p$, and $\Pi\in\mathcal{L}_{00}$, is derivable in Seq${}_{00}^{\text{{pdl}}}$. Then so is either $\Gamma$ or $A_{i},B_{1},\cdots,B_{k}$, for some $i\in\left[1,j\right]$, without increasing the height of the former derivation.

Proof. By straightforward induction on the derivation height. In the crucial principal case we have

[TABLE]

where $0<i\leq k$ and $\Delta\subseteq B_{1},\cdots,B_{l}$, which by derivable $\left(\text{{W}}\right)$ yields the required derivability of $A_{i},B_{1},\cdots,B_{l}$.

Theorem 15

The derivability in Seq${}_{00}^{\text{{pdl}}}$ is a PSPACE problem. 888Apparently this result is well-known in the context of multimodal version of K.

Proof. For the sake of brevity we consider $\mathcal{L}_{00}$ formulas containing at most one atomic program $p=\pi_{0}$. Furthermore, we refine the notion of Seq${}_{00}^{\text{{pdl}}}$ derivability by asserting that a sequent $\Delta\neq\left(\text{{Ax}}\right)$ is the conclusion of a rule $\left(\text{{R}}\right)$ if one of the following priority conditions 1–3 is satisfied.

$\left(\text{{R}}\right)=\left(\vee\right)$. 2. 2.

$\left(\text{{R}}\right)=\left(\wedge\right)$ and no disjunction $A\vee B$ occurs as formula in $\Delta$; thus $\Delta$ is not a conclusion of any $\left(\vee\right)$. 3. 3.

No disjunction $A\vee B$ or conjunction $A\wedge B$ occurs as formula in $\Delta$. Thus $\Delta$ is not a conclusion of any $\left(\vee\right)$ or $\left(\wedge\right)$, i.e. $\Delta=\left(p\right)_{\xi_{1}}\!\!F_{1},\cdots,\left(p\right)\!_{\xi_{n}}\!F_{n}$ for $\overset{n}{\underset{i=1}{\sum}}\!\,\xi_{i}\geq 1$. In this case we stipulate that $\Delta$ is the conclusion of $\left(\text{{R}}\right)$ if one of the following two conditions holds:

1. (a)

$\overset{n}{\underset{i=1}{\sum}}\!\,\xi_{i}=1$ and $F_{1},\cdots,F_{n}$ is the premise of $\left(\text{{R}}\right)=\left(\text{{Gen}}\right)$. 2. (b)

$\overset{n}{\underset{i=1}{\sum}}\!\,\xi_{i}>1$ and there exists $j\in\left[1,n\right]$ with $\xi_{j}=1$ such that either $\Delta^{\left(j\right)}:=F_{j}\cup\left\{F_{l}\in\Delta:\xi_{l}=0\right\}$ or $\Delta^{\left(-j\right)}:=\Delta\setminus\left\{F_{j}\right\}$ is the premise of $\left(\text{{R}}\right)$. (Note that we have $\left(\text{{R}}\right)=\left(\text{{Gen}}\right)$ and $\left(\text{{R}}\right)=\left(\text{{W}}\right)$ in the former and in the latter case, respectively.)

Having this we consider derivations in the refined Seq${}_{00}^{\text{{pdl}}}$ as at most binary-branching trees $\partial$ whose nodes are labeled with sequents of $\mathcal{L}_{00}$. Actually, for any given $\mathcal{L}_{00}$-sequent $\Sigma$ it will suffice to fix one distinguished proof search tree $\partial_{0}$ with root sequent $\Sigma$ that is defined by bottom-up recursion while applying the conditions 1–3 in a chosen order as long as possible. It is readily seen by inversions in Lemmata 7, 14 that $\Sigma$ is derivable in Seq${}_{00}^{\text{{pdl}}}$ iff $\partial_{0}$ proves $\Sigma$, i.e. every maximal path in $\partial_{0}$ is locally correct with respect to 1–3. Moreover, by the obvious subformula property we conclude that the depth, $d\left(\partial_{0}\right)$, and maximum sequent length, $\max\left\{\left|\Delta\right|:\Delta\in\partial_{0}\right\}$, of $\partial_{0}$ are both proportional to $\left|\Sigma\right|$. Hence every maximal path in $\partial_{0}$ can be encoded by a $\mathcal{L}_{01}$-string of the length proportional to $\left|\Sigma\right|$ whose local correctness is verifiable by TM in $\mathcal{O}\left(\left|\Sigma\right|\right)$ space. The corresponding universal verification runs by counting all maximal paths successively, still in $\mathcal{O}\left(\left|\Sigma\right|\right)$ space, which completes the proof.

Remark 16

Arguing along more familiar lines we can turn $\partial_{0}$ into a Boolean circuit with (binary) AND, OR and (unary) ID gates, where ID$\left(x\right):=x$ for $x\in\left\{0,1\right\}$, such that AND, OR and ID correspond to the above conditions 2, 3 (b) and 1 and/or 3 (a), respectively. The corresponding truth evaluations $val\left(-\right)$ are defined as usual via $val\left(\Delta\right):=1$ $\left(\mathbf{true}\right)$ iff $\Delta=\left(\text{{Ax}}\right)$, for every leaf $\Delta$. Then $val\left(\Sigma\right)=1$ iff $\partial_{0}$ proves $\Sigma$, as required. 999This proof is dual to familiar proof of polynomial space solvability of QSAT (cf. e,g. [14]).

2.4.2 Special cases

Recall that by (a particular case of) Theorem 11, for any $\Sigma=\left\langle p^{\ast}\right\rangle\!A,\Pi$ with $A\in\mathcal{L}_{00},\Pi\in\mathcal{L}_{\emptyset}$ the following holds. Suppose that $\Sigma$ is derivable in Seq${}_{\omega}^{\text{{pdl}}}$. Then there exists a $k\geq 0$ such that $\widehat{\Sigma}_{k}:=A,\left\langle p\right\rangle\!A,\cdots,\left\langle p\right\rangle^{k}\!\!A,\Pi$ is derivable in Seq${}_{00}^{\text{{pdl}}}$. It turns out that in some cases it’s possible to estimate the minimum $k$ and hence the size of $\widehat{\Sigma}_{k}$.

Definition 17

Let $p=\pi_{0}$ be fixed. Call basic conjunctive normal form (abbr.: BCNF) any $\mathcal{L}_{00}$-formula $\overset{m}{\underset{i=1}{\bigwedge}}\left(B_{i}\vee\left\langle p\right\rangle\!C_{i}\vee\overset{n_{i}}{\underset{j=1}{\bigvee}}\left[p\right]\!D_{i,j}\right)$ for $m>0$, $n_{i}\geq 0$ and $B_{i}$, $C_{i}$, $D_{i,j}$ $\in\mathcal{L}_{\emptyset}\cup\left\{\emptyset\right\}$. Formulas $\left\langle p^{\ast}\right\rangle\!A\vee Z$ for $A\in\mathrm{BCNF}$ and $Z\in\!\mathcal{L}_{\emptyset}$ are called basic conjunctive normal expressions (abbr.: BCNE).

Theorem 18

Let $A=\overset{m}{\underset{i=1}{\bigwedge}}\left(\!B_{i}\vee\left\langle p\right\rangle\!C_{i}\vee\overset{n_{i}}{\underset{j=1}{\bigvee}}\left[p\right]\!D_{i,j}\!\right)\!\in$ $\mathrm{BCNF}\!$ and for any $k\geq 0$ and $\Pi\!\in\!\mathcal{L}_{\emptyset}$ let $\widehat{A}_{k}:=$ $A,\left\langle p\right\rangle\!A,\cdots,\left\langle p\right\rangle^{k}\!\!A$, $\widehat{A}_{0}=A$, and $\widehat{\Sigma}_{k}:=\widehat{A}_{k},\Pi$. If any $\widehat{\Sigma}_{k}$ is derivable in Seq${}_{00}^{\text{{pdl}}}$ then so is $\widehat{\Sigma}_{n+1}$ too, where $n=\overset{m}{\underset{i=1}{\sum}}n_{i}$.

Proof. For $i\in\left[1,m\right]$ let $\Delta_{i}:=$ $\left\{\left[p\right]D_{i,j}\!:1\leq j\leq n_{i}\right\}$. So Lemma 14 yields

[TABLE]

[TABLE]

where “$\ \vdash$” stands for “  Seq${}_{00}^{\text{{pdl}}}\vdash$”, and hence

[TABLE]

[TABLE]

By the same token, for any $s\geq 0$ we let $\left\langle p\right\rangle\widehat{\!A}_{s}:=\left\langle p\right\rangle\left(A\vee\left\langle p\right\rangle\!A\vee\cdots\vee\left\langle p\right\rangle^{s}\!\!A\right)$ and arrive at

[TABLE]

[TABLE]

which yields

[TABLE]

[TABLE]

Thus for any $k\geq 0$, the assertion $\nvdash\widehat{\Sigma}_{k}$ is equivalent to the existence of a labeled rooted refutation tree $T_{k}$ of the height $k+1$ such that the following conditions 1–3 hold, where sequents $\ell\left(x\right)$ are the labels of nodes $x\in T_{k}$ ($\rho$ being the root).

$\ell\left(\rho\right)=\Pi$. 2. 2.

$\nvdash\ell\left(x\right)$ holds for every leaf $x\in T_{k}$. 3. 3.

For any inner node $x\in T_{k}$ there exists $i\in\left[1,m\right]$ such that $x$ has $m_{i}+1$ ordered children: $x_{0}$ (the son) with label $\ell\left(x_{0}\right)=B_{i},\ell\left(x\right)$ and $x_{1},\cdots,x_{m_{i}}\!$ (the daughters) labeled $\ell\left(x_{j}\right)=C_{i},D_{i,j}$, respectively; moreover $x_{j}$ ($j\geq 0$) is a leaf iff it is either a son or else a daughter of the depth $k+1$.

Such $T_{k}$ is easily obtained by straightforward geometric interpretation of our translations of the conditions $\nvdash\ \!\!\widehat{\Sigma}_{0}$ and $\nvdash\ \!\widehat{\Sigma}_{s+1}$. Moreover, condition 2 above is equivalent to

2*. $\nvdash\ell\left(x\right)$ holds for every node $x\in T_{k}$,

since daughters are subsequents of their sons. That is, every inner node $x\in T_{k}$ with label $\ell\left(x\right)$ has an upper neighbor (son) $x_{0}$ with label $B_{i},\ell\left(x\right)$ being a leaf in $T_{k}$, which by condition 2 yields $\nvdash B_{i},\ell\left(x\right)$, and hence also $\nvdash\ell\left(x\right)$, as required, by converting the admissible weakness rule of Seq${}_{01}^{\text{{pdl}}}$.

Now if $k\leq n+1$ then $\widehat{\Sigma}_{k}\subseteq\widehat{\Sigma}_{n+1}$, and hence $\vdash\widehat{\Sigma}_{k}$ implies $\vdash\widehat{\Sigma}_{n+1}$. Furthermore, assuming $\nvdash\widehat{\Sigma}_{n+1}$ we’ll infer $\left(\forall s>n\right)\nvdash\widehat{\Sigma}_{s}$ and conclude by contraposition that $\left(\exists k\right)\vdash\widehat{\Sigma}_{k}$ implies (in fact is equivalent to) $\vdash\widehat{\Sigma}_{n+1}$, as required. So assume $\nvdash\widehat{\Sigma}_{n+1}$. We prove by induction on $s>n$ the existence of the refutation trees $T_{s}$, and hence $\nvdash\widehat{\Sigma}_{s}$. Basis case $s=n+1$ holds by the assumption. To pass from $T_{s}$ to $T_{s+1}$ we argue as follows. Let $x\in T_{s}$ be any leaf-daughter and $\theta=\left(\rho,y_{1},\cdots,y_{s}=x\right)$ the corresponding maximal path, in $T_{s}$. Since $\theta$ contains at most $n<s$ different labels $\ell\left(y_{\imath}\right)=C_{i},D_{i,j}$ ($i\in\left[1,m\right]$, $j\in\left[1,n_{i}\right]$), there exist a (say, minimal) pair $0<r<t<s$ such that $\ell\left(y_{r}\right)=\ell\left(y_{t}\right)$. Let $T_{\left(s,x,r,t\right)}$ be a tree that arises from $T_{s}$ by substituting its subtree rooted in $y_{r}$ for that rooted in $y_{t}$. Note that $T_{\left(s,x,r,t\right)}$ is higher than $T_{s}$ – so let $T_{s+1}^{\left(x\right)}$ be a subtree of $T_{\left(s,x,r,t\right)}$ consisting of the nodes of heights $\leq s+1$. Proceeding this way successively with respect to all leaf-daughters $x\in T_{s}$ while keeping in mind condition 2*, we eventually obtain a refutation tree $T_{s+1}$ of the height $s+1$, as required.

By Remark 10 and Theorem 11, the following are provable in $\mathbf{PRA}_{\varphi_{\omega}\left(0\right)}$.

Corollary 19

Let $A\in$ $\mathrm{BCNF}\!$, $n$ and $\Pi$ be as above. Then $\Sigma:=\left\langle p^{\ast}\right\rangle\!A,\Pi$ is derivable in Seq${}_{\omega}^{\text{{pdl}}}$ iff $\widehat{\Sigma}_{n+1}:=\widehat{A}_{n+1},\Pi$ is derivable in Seq${}_{00}^{\text{{pdl}}}$.

Corollary 20

Let $S\in$ $\mathrm{BCNE}$. Problem $\mathbf{PDL}\vdash S$, i.e. $\mathbf{PDL}$-validity of $S$, is solvable by a deterministic TM in $\mathcal{O}\left(\left|S\right|^{2}\right)$ space.

Proof. For $A$ as above we have $n<\left|A\right|$, and hence$\left|\widehat{A}_{n+1}\right|=\mathcal{O}\left(\left|A\right|^{2}\right)$. This yields $\left|\widehat{A}_{n+1},Z\right|=\mathcal{O}\left(\left|A\right|^{2}+\left|Z\right|\right)=\mathcal{O}\left(\left|S\right|^{2}\right)$. Now by Theorem 4 followed by Theorems 11, 18 we have

[TABLE]

while problem Seq${}_{01}^{\text{{pdl}}}\vdash\widehat{A}_{n+1},Z$ is solvable in $\mathcal{O\!}\left(\left|\widehat{A}_{n+1},Z\right|\right)\!=\!\mathcal{O\!}\left(\left|S\right|^{2}\right)$ space.

Now consider (dual) basic disjunctive normal forms.

Definition 21

Call basic disjunctive normal form (abbr.: BDNF) any $\mathcal{L}_{00}$-formula $F\vee\overset{s}{\underset{i=1}{\bigvee}}\left(F_{i}\wedge\left[p\right]G_{i}\right)\vee\overset{t}{\underset{j=1}{\bigvee}}\left(F_{j}\wedge\left\langle p\right\rangle\!H_{j}\right)$ for $s,t>0$ and $F$, $F_{i}$, $G_{i}$, $H_{j}$ $\in\mathcal{L}_{\emptyset}\cup\left\{\emptyset\right\}$. Formulas $\left\langle p^{\ast}\right\rangle\!A\vee Z$ for $A\in\mathrm{BDNF}$ and $Z\in\!\mathcal{L}_{\emptyset}$ are called basic disjunctive normal expressions (abbr.: BDNE).

Problem 22

Let $S\in\mathrm{BDNE}$. Is problem $\mathbf{PDL}\vdash S$ solvable by a TM in $\left|S\right|$-polynomial space?

2.4.3 More on BDNE

PDL-satisfiability problem for certain statements Accepts${}_{M,x}=\left[p^{\ast}\right]V\wedge W$ for $V\in$ $\mathrm{BCNF}\!$, $W\in\mathcal{L}_{\emptyset}$ – expressing that satisfying Kripke frames encode accepting computations of polynomial-space alternating TM – is known to be EXPTIME-complete (cf. [1] and [10]: Theorem 8.5, et al; see also [20]). Hence so is also the PDL-validity problem for the corresponding negations $S:=\overline{\text{{Accepts}}}_{M,x}=\left\langle p^{\ast}\right\rangle\!A\vee Z\in$ $\mathrm{BDNE}$. So the affirmative solution to Problem 22 would infer $\mathbf{EXPTIME=PSPACE}$ (and vice versa, since general PDL-validity is EXPTIME-complete). That is, problem $\mathbf{EXPTIME=PSPACE}$ reduces to a particular case of Problem 22 for $S:=\overline{\text{{Accepts}}}_{M,x}$ (see Appendix B for precise definition).

Now let $S=\left\langle p^{\ast}\right\rangle\!A\vee Z\in$ $\mathrm{BDNE}$ for $A\!=\!F\vee\overset{s}{\underset{i=1}{\bigvee}}\!\left(F_{i}\!\wedge\!\left[p\right]G_{i}\right)\vee\overset{t}{\underset{j=1}{\bigvee}}\!\left(F_{j}\!\wedge\!\left\langle p\right\rangle\!H_{j}\right)\!\in\mathrm{BDNF}$ and $Z\in\!\mathcal{L}_{\emptyset}$. We wish to present the assertion $\mathbf{PDL}\vdash S$ in a suitable “transparent” quantified boolean form. To this end, by DeMorgan laws, we first convert $A\!$ to $R=\underset{\xi\in\Xi}{\bigwedge}R_{\xi}\in\mathrm{BCNF}$, where $R_{\xi}=B_{\xi}\vee\left\langle p\right\rangle\!C_{\xi}\vee\underset{\jmath\in J_{\xi}}{\bigvee}\left[p\right]\!D_{\xi,\jmath}$ for $\Xi:=\left\{\xi=\left(\xi\left(1\right),\cdots,\xi\left(s+t\right)\right)\right\}$ with $\xi\left(k\right)\in\left\{1,2\right\}$, $1\leq k\leq s+t$, while

[TABLE]

Clearly $\mathbf{PDL}\vdash A\leftrightarrow R$ (also by PDL-equivalence $\left\langle p\right\rangle\!H\vee\left\langle p\right\rangle\!H^{\prime}\leftrightarrow$ $\left\langle p\right\rangle\!\left(H\vee\!H^{\prime}\right)\,$). Note that $\left|\Xi\right|=2^{s+t}$ and $\left|R_{\xi}\right|<\left|A\right|$, for every $\xi\in\Xi$.

By the cut-elimination theorem, $\mathbf{PDL}\vdash S$ is equivalent to Seq${}_{\omega}^{\text{{pdl}}}\vdash\left\langle p^{\ast}\right\rangle\!R,Z$, which by Theorem 18 is equivalent to Seq${}_{00}^{\text{{pdl}}}\vdash\widehat{R}_{n+1},Z$, where

[TABLE]

for $n:=\underset{\xi\in\Xi}{\sum}\left|J_{\xi}\right|<s\cdot\left|\Xi\right|=s2^{s+t}$. Arguing as in the proof of Theorem 18 we get

[TABLE]

where $f$ is a boolean-valued binary function that is defined for every $\imath\geq 0$ and propositional formula $X$ by the following recursive clauses 1–2, where “$\,\vdash_{\emptyset}Y$ ” stands for plain boolean validity of propositional formula$\ Y$.

$f\!\left(0,X\right)=1\Leftrightarrow\underset{\xi\in\Xi}{\bigwedge}\left(\vdash_{\emptyset}\left(B_{\xi}\vee X\right)\ or\ \underset{\text{ }j\in_{\xi}}{\bigvee}\vdash_{\emptyset}\left(C_{\xi}\vee D_{\xi,j}\right)\right)$ 2. 2.

$f\!\left(\imath+1,X\right)\!=\!1\Leftrightarrow\underset{\xi\in\Xi}{\bigwedge}\left(\!\vdash_{\emptyset}\left(B_{\xi}\vee X\right)\ or\ \!\underset{\text{ }j\in J_{\xi}}{\bigvee}f\!\left(\imath,C_{\xi}\vee D_{\xi,j}\right)\!={}\!1\right)$

Note that every “$\,\vdash_{\emptyset}Y$ ” involved is expressible in quantified boolean logic as $\forall x_{1}\cdots\forall x_{q}Y$, where $\left\{x_{1},\cdots,x_{q}\right\}$ is the set of propositional variables occurring in $Y$. Having this, by recursion on $\imath$ with respect to clauses 1–2 we obtain a desired “transparent” quantified boolean formula $\widehat{S}$ such that

[TABLE]

(QBL being the canonical proof system for quantified boolean logic).

Remark 23

The size of $\widehat{S}$ is exponential in that of $S$, 101010This is in contrast to analogous polynomial BCNE case, see Corollary 20. whereas quantified boolean validity (and/or satisfiability) is known to be PSPACE-complete (cf. e.g. [14]). Hence $\mathbf{EXPTIME=PSPACE}$ holds if $\widehat{S}$ is equivalid with another quantified boolean formula whose size is polynomial in the size of $S$. Moreover, this holds true of $S:=\overline{\text{{Accepts}}}_{M,x}$ (see above and Appendix B).

2.5 Conclusion

Soundness and completeness together with full cut elimination [Theorems 4, 8] in semiformal (infinite) sequent calculus Seq${}_{\omega}^{\text{{pdl}}}$ shows that Hilbert-Bernays-style proof system PDL is a conservative extension of formal (finite) cutfree sequent calculi Seq${}_{00}^{\text{{pdl}}}\varsubsetneq\,$Seq${}_{0}^{\text{{pdl}}}$ and Seq${}_{10}^{\text{{pdl}}}\varsubsetneq\,$Seq${}_{1}^{\text{{pdl}}}$ with respect to the corresponding classes of formulas FOR${}_{00}\varsubsetneq\,$FOR0 and FOR${}_{10}\varsubsetneq\,$FOR1, respectively. I.e., for any $A\in\,$FOR*†*, $\mathbf{PSP}\vdash A$ implies Seq${}_{\dagger}^{\text{{pdl}}}\vdash A$ ($\dagger\in\left\{0,1,00,10\right\}$). Here we let

FOR1 := subset of FOR whose seq-formulas don’t include occurrences $\left[P^{\ast}\right]$.

FOR10 := subset of FOR1 with atomic programs PRO0.

FOR0 := subset of FOR1 whose seq-formulas don’t include occurrences $\left\langle P^{\ast}\right\rangle$, i.e. FOR0 is just star-free fragment of FOR.

FOR00 := subset of FOR0 with atomic programs PRO0.

Basic program connectives are interpretable in FOR00 and FOR10 by $\left(P;Q\right)\!A:=\left(P\right)\!\left(Q\right)\!A$, $\left\langle P\cup Q\right\rangle\!A:=\left\langle P\right\rangle\!A\vee\left\langle Q\right\rangle\!A$ and $\left[P\cup Q\right]\!A:=\left[P\right]\!A\wedge\left[Q\right]\!A$.

Seq${}_{00}^{\text{{pdl}}}:=$

[TABLE]

Seq${}_{0}^{\text{{pdl}}}:=$

[TABLE]

Seq${}_{10}^{\text{{pdl}}}:=$

[TABLE]

Seq${}_{1}^{\text{{pdl}}}:=$

[TABLE]

It is readily seen that Seq${}_{0}^{\text{{pdl}}}$ and Seq${}_{00}^{\text{{pdl}}}$ (resp. Seq${}_{1}^{\text{{pdl}}}$ and Seq${}_{10}^{\text{{pdl}}}$) have the same provable sequents modulo basic interpretation of FOR0 (FOR1) within FOR00 (FOR10).

As usual in proof theory, our cutfree sequent calculi provide useful help in the verification of PDL-(un)provability of concrete formulas of simple shapes. Concerning general computational complexities the following holds.

1. The derivability (provability) in Seq${}_{00}^{\text{{pdl}}}$ is a PSPACE problem. The case of Seq${}_{0}^{\text{{pdl}}}$ is less clear as our interpretation of program connectives does not preserve polynomial size.

2. The derivability (provability) in Seq${}_{1}^{\text{{pdl}}}$ is EXPTIME-complete and in fact so is the derivability in Seq${}_{10}^{\text{{pdl}}}$, too.

3. The latter is characteristic also for a subclass BDNE (: “basic disjunctive normal expressions”) of FOR10, whereas Seq${}_{10}^{\text{{pdl}}}$-derivability of dual BCNE expressions (: “basic conjunctive normal expressions”) turns out to be decidable in polynomial space. Moreover, Seq${}_{10}^{\text{{pdl}}}$-derivability (and hence PDL-provability) of BDNE is equivalent to QBL-validity of corresponding “transparent” quantified boolean formulas of exponential length.

Proofs of our claims use transfinite induction on predicative ordinal $\varphi_{\omega}\!\left(0\right)$. It is not clear yet whether conservative extension results (see above) are provable in Peano Arithmetic.

Relevant papers

[13] formalized PDL as proof system CSPDL with $\omega$-rule for $\left[\ast\right]$ that is based on hypersequents (more precisely: zoom tree hypersequents), rather than sequents. Sequent calculi proper are not exposed there. Consequently, conservative extension corollaries and complexity connections are not mentioned, either. It might appear, however, that the hypersequents were chosen to allow “deep inferences”, i.e. transformations applied to subformulas of those explicitly shown (along the lines of Herbrand version [13] of Gentzen’s sequent calculus). Cut-elimination theorem is claimed for CSPDL but proof thereof is informal. Apparently it should follow by Schütte-style predicative pattern adapted to the hypersequents, instead of plain sequents. However no ordinal bounds on the height transformation is given and hence proof theoretic strength of the required logic formalism remains unclear. According to well-known predicative cut elimination in the presence of $\omega$-rule(s), one would expect it to be the same as in the present paper, provided that ordinal complexity of cut formulas with nested occurrences of starred programs has the same natural upper bound $\omega^{\omega}$ (which does not explicitly follow from [13]: Definition 3.1).

[11] considered finite sequent calculus GPDL in an extended language with mixed formulas (possibly including atomic programs) and contextual sequents (whose antecedents and/or succedents might include program terms). It is claimed that all but analytic cuts in special form can be eliminated from GPDL derivations. There is no discussion of possible conservative extensions and/or computational complexity connections.

[9] presented a different approach in form of an optimal tableau-based EXPTIME algorithm for deciding satisfiability for PDL with converse (CPDL) without the use of analytic cuts. In order to decide the satisfiability of a given input formula $\phi$ the algorithm builds a suitable directed graph $G$ and checks the applicability of one of the four attached rules $Rule\ 1$, …, $Rule\ 4$. There is no obvious translation into plain sequent calculus formalism.

3 Appendix A: Ordinal assignments

3.1 Ordinal arithmetic

We use basic properties 1–8 of Veblen’s ordinals (abbr.: $\alpha$, $\!\beta$, $\gamma$, $\delta$) ( [21], [5], [16]).

Basic relation $<$ is linearly ordered. 2. 2.

Symmetric sum is associative and commutative. 3. 3.

$0<1=\omega^{0}$, $\omega=\omega^{1}$, $\omega^{\beta}=\varphi\left(0,\beta\right)$. 4. 4.

$\alpha+\!\!\!\!\!+\,0=\alpha$, $\alpha<\beta\rightarrow\alpha+\!\!\!\!\!+\gamma<\beta+\!\!\!\!\!+\gamma+\!\!\!\!\!+\delta$. 5. 5.

$\alpha<\beta\rightarrow\varphi\left(\alpha,\gamma\right)<\varphi\left(\beta,\gamma\right)\wedge\varphi\left(\gamma,\alpha\right)<\varphi\left(\gamma,\beta\right)$. 6. 6.

$\alpha\leq\beta<\varphi\left(\gamma,\delta\right)\rightarrow\alpha+\!\!\!\!\!+\beta<\varphi\left(\gamma,\delta\right)$. 7. 7.

$\alpha<\beta\wedge\gamma<\varphi\left(\beta,\delta\right)\rightarrow\varphi\left(\alpha,\gamma\right)<\varphi\left(\beta,\delta\right)\wedge\varphi\left(\alpha,\varphi\left(\beta,\delta\right)\right)=\varphi\left(\beta,\delta\right)$. 8. 8.

$\alpha\leq\omega^{\alpha}$, $0<\alpha\rightarrow\omega^{\varphi\left(\alpha,\beta\right)}=\varphi\left(\alpha,\beta\right)$.

$\varphi\left(\alpha,\beta\right)$ is also denoted by $\varphi_{\alpha}\!\left(\beta\right)$. Note that $\varepsilon_{0}=\varphi_{1}\left(0\right)<\varphi_{\omega}\!\left(0\right)<\Gamma_{0}$.

In the rest of this chapter we freely use these properties without explicit references.

3.2 Cut elimination $\partial\hookrightarrow\mathcal{E}\left(\partial\right)$

For the sake of brevity we’ll slightly refine our inductive definition of $\mathcal{E\!}\left(\partial\right)$. To this end we upgrade $\mathcal{R}$ to $\mathcal{R}^{+}:\left(\partial\mid\!\!\!\frac{\beta}{\,\rho+\omega^{a}}\,\Delta\right)\hookrightarrow\left(\mathcal{R}^{+}\mathcal{\!}\left(\rho,\alpha,\partial\right)\mid\!\!\!\frac{\,\varphi\left(\alpha,\beta\right)}{\,\rho}\,\Delta\right)$. That is, for any $\rho>0$, $\alpha$ and $\left(\partial:\Delta\right)$ with $\deg\left(\partial\right)<\rho+\omega^{a}$ we define $\left(\mathcal{R}^{+}\mathcal{\!}\left(\rho,\alpha,\partial\right):\Delta\right)$ such that $\deg\left(\mathcal{R}^{+}\mathcal{\!}\left(\rho,\alpha,\partial\right)\right)<\rho$ and $h\left(\mathcal{R}^{+}\mathcal{\!}\left(\rho,\alpha,\partial\right)\right)<$ $\varphi\left(\alpha,h\left(\partial\right)\right)$. Then for any $\partial$ with cuts we let

[TABLE]

and conclude that $\deg\left(\mathcal{E}\left(\partial\right)\right)=0$ and $h\left(\mathcal{E}\left(\partial\right)\right)<\varphi\left(\alpha,h\left(\partial\right)\right)$.

Now $\mathcal{R}^{+}\mathcal{\!}\left(\rho,\alpha,\partial\right)$ is defined for any $\partial$ with $\deg\left(\partial\right)<\rho+\omega^{a}$ as follows by double induction on $\alpha$ and $h\left(\partial\right)$. Let $\left(\text{{R}}\right)$ be the lowermost inference in $\partial$. If $\left(\text{{R}}\right)$ is not a $\left(\text{{Cut}}\right)$ on $C$ with $\mathfrak{o\!}\left(C\right)+1\geq\rho$ then $\mathcal{R}^{+}\mathcal{\!}\left(\rho,\alpha,\partial\right)$ arises from $\partial$ by substituting $\mathcal{R}^{+}\mathcal{\!}\left(\rho,\alpha,\partial_{i}\right)$ for the lowermost subdeductions $\partial_{i}$ (recall that $h\left(\partial_{i}\right)<h\left(\partial\right)$ ). Otherwise, we have

[TABLE]

where $\rho\leq\mathfrak{o\!}\left(C\right)+1\leq\deg\left(\partial\right)<\rho+\omega^{a}$. Let

[TABLE]

and consider two cases.

Case $\alpha=0$.

Let $\mathcal{R}^{+}\mathcal{\!}\left(\rho,\alpha,\partial\right)=\mathcal{R}^{+}\mathcal{\!}\left(\rho,0,\partial\right):=\mathcal{R}\left(\widehat{\partial}\right)$. Recall that

[TABLE]

and hence $\deg\left(\mathcal{R}^{+}\mathcal{\!}\left(\rho,\alpha,\partial\right)\right)=\deg\left(\mathcal{R}\left(\widehat{\partial}\right)\right)<\rho$. On the other hand

[TABLE]

which yields $h\left(\mathcal{R}^{+}\mathcal{\!}\left(\rho,\alpha,\partial\right)\right)\!=\!h\left(\mathcal{R}\left(\widehat{\partial}\right)\right)\!<\!\varphi\left(0,h\left(\widehat{\partial}\right)\right)\!$, as desired.

Case $\alpha>0$.

Thus $\omega^{a}=\omega^{\alpha_{1}}+\cdots+\omega^{\alpha_{n}}$ for $\alpha>\alpha_{1}\geq\cdots\geq\alpha_{n}$ (by Cantor’s normal form). In this case we apply inductive hypotheses successively for $\alpha_{1},\cdots,\alpha_{n}$ and let

[TABLE]

where $\rho_{0}:=\rho$ and $\left(\forall i>0\right)\rho_{i+1}:=\rho_{i}+\omega^{\alpha_{i+1}}$. Then $\deg\left(\mathcal{R}^{+}\mathcal{\!}\left(\rho,\alpha,\partial\right)\right)<\rho$ and $h\left(\mathcal{R}^{+}\mathcal{\!}\left(\rho,\alpha,\partial\right)\right)<\varphi\left(\alpha_{1},\varphi\left(\alpha_{2},\varphi\left(\cdots,\varphi\left(\alpha_{n},h\left(\partial\right)\right)\right)\right)\right)<\varphi\left(\alpha,h\left(\partial\right)\right)$, as desired.

3.3 Formalization

We fix a chosen “canonical” primitive recursive ordinal representation

[TABLE]

(also known as system of ordinal notations) in the language of $\mathbf{PA}$ that is supposed to be well-ordered by $<$ up to $\varphi_{\omega}\left(0\right)$ (at least). To formalize the latter assumption we extend standard formalism of $\mathbf{PA}$ by the transfinite induction axiom (schema) for arbitrary arithmetical formulas, $\mathrm{TI}_{\mathcal{O}}\left(\varphi_{\omega}\!\left(0\right)\right)$. The extended proof system is abbreviated by $\mathbf{PA}_{\varphi_{\omega}\left(0\right)}$. Derivations $\partial$ used in the proofs are interpreted as primitive recursive trees whose nodes $x$ are labeled with sequents and ordinals $ord\left(x\right)<\varphi_{\omega}\!\left(0\right)$. Having this it is easy to formalize in $\mathbf{PA}_{\varphi_{\omega}\left(0\right)}$ the whole cut elimination proof; note that the operators $\mathcal{R}$, $\mathcal{R}^{+}$ and $\mathcal{E}$ involved are constructively defined and $\mathrm{TI}_{\mathcal{O}}\left(\varphi_{\omega}\!\left(0\right)\right)$ is used in the corresponding termination-and-correctness proofs only. Actually we can restrict $\mathrm{TI}_{\mathcal{O}}\left(\varphi_{\omega}\!\left(0\right)\right)$ to primitive recursive induction formulas thus reducing $\mathbf{PA}_{\varphi_{\omega}\left(0\right)}$ to $\mathbf{PRA}_{\varphi_{\omega}\left(0\right)}$.

4 Appendix B: Formula AcceptsM,x 111111This is a recollection of [10]: 8.2.

4.1 Semantics

Consider a given polynomial-space-bounded $k$-tape alternating Turing machine $M$ on a given input $x$ of length $n$ with blanks over $M$’s input alphabet; $\vdash$ and $\dashv$ are the left and right endmarkers, respectively. Formula AcceptsM,x involves the single atomic program Next, atomic propositions Symbol${}_{i}^{a}$ and State${}_{i}^{q}$ for each symbol $a$ in $M$’s tape alphabet, $q$ a state of $M$’s finite control, and $0\leq i\leq n$, and an atomic proposition Accept. Then AcceptsM,x has the property that any satisfying Kripke frame encodes an accepting computation of $M$ on $x$. In any such Kripke frame, states $u$ represent configurations of $M$ occurring in the computation tree of $M$ on input $x=x_{1},\cdots,x_{n}$; the truth values of Symbol${}_{i}^{a}$ and State${}_{i}^{q}$ at state $u$ give the tape contents, current state, and tape head position in the configuration corresponding to $u$. The truth value of Accept will be $\mathbf{1}$ iff the computation beginning in state $u$ is an accepting computation according to the rules of alternating Turing machine acceptance. Then $M$ accepts $x$ iff AcceptsM,x is satisfiable. AcceptsM,x is EXPTIME-complete (cf. [10]: Theorem 8.5) and hence so is the negation $\overline{\text{{Accepts}}}_{M,x}$.

4.2 Formal definition

Let $\Gamma$ be $M$’s tape alphabet and $Q$ the set of states; there is a distinguished start-state $s\in Q$ and left/right annotations $\ell,r\notin Q$. Let $U\subseteq Q$  and $E\subseteq Q$ be the sets of universal and existential states, respectively. Thus $U\cup E=Q$ and $U\cap E=\emptyset$. For each pair $\left(q,a\right)\in Q\times\Gamma$ let $\Delta\left(q,a\right)$ be the set of all triples describing a possible action when scanning $a$ in state $q$. Working in $\mathcal{L}$ we let

[TABLE]

where Acc(ept), State${}_{\left(-\right)}^{\left(-\right)}$, Symbol${}_{\left(-\right)}^{\left(-\right)}\in V\!AR$, Next $\in PRO$ while Start, Config, Move and Acceptance are defined as follows.

1. Start $:=$ State${}_{0}^{s}\wedge\underset{1\leq i\leq n}{\bigwedge}$Symbol${}_{i}^{x_{i}}\wedge\underset{n+1\leq i\leq n^{k}}{\bigwedge}$Symbol${}_{i}^{\Box}$.

2. Config $:=$

$\underset{0\leq i\leq n+1}{\bigwedge}\underset{a\in\Gamma}{\bigvee}\!\left(\!\text{{Symbol}}_{i}^{a}\!\wedge\underset{a\neq b\in\Gamma}{\bigwedge}\overline{\text{{Symbol}}}_{i}^{b}\!\right)\!\wedge$ Symbol${}_{0}^{\vdash}\wedge\overline{\text{{Symbol}}}_{n+1}^{\dashv}\wedge$

$\underset{0\leq i\leq n+1}{\bigvee}\underset{q\in Q}{\bigvee}$State${}_{i}^{q}\wedge\underset{0\leq i\leq n+1}{\bigwedge}\underset{q\in Q\cup\left\{\ell,r\right\}}{\bigvee}\left(\text{{State}}_{i}^{q}\wedge\underset{q\neq p\in Q\cup\left\{l,r\right\}}{\bigwedge}\overline{\text{{State}}}_{i}^{p}\right)\wedge$

$\underset{0\leq i\leq n}{\bigwedge}\underset{q\in Q\cup\left\{\ell\right\}}{\bigwedge}\!\left(\overline{\text{{State}}}_{i}^{q}\vee\text{{State}}_{i+1}^{\ell}\right)\wedge\underset{0\leq i\leq n+1}{\bigwedge}\underset{q\in Q\cup\left\{r\right\}}{\bigwedge}\!\left(\overline{\text{{State}}}_{i}^{q}\vee\text{{State}}_{i-1}^{r}\right)$.

3. Move $:=$

$\underset{0\leq i\leq n+1}{\bigwedge}\left(\overline{\text{{State}}}_{i}^{\ell}\vee\overline{\text{{State}}}_{i}^{r}\vee\underset{a\in\Gamma}{\bigwedge}\left(\overline{\text{{Symbol}}}_{i}^{a}\vee\left[\text{{Next}}\right]\text{{Symbol}}_{i}^{a}\right)\right)\wedge$

\underset{0\leq i\leq n+1}{\bigwedge}\underset{\QATOP{a\in\Gamma}{q\in Q}}{\bigwedge}\left(\begin{array}[]{c}\overline{\text{{Symbol}}}_{i}^{a}\vee\overline{\text{{State}}}_{i}^{q}\vee\\ \left(\begin{array}[]{c}\underset{\left(p,b,d\right)\in\Delta\left(q,a\right)}{\bigwedge}\left\langle\text{{Next}}\right\rangle\left(\text{{Symbol}}_{i}^{b}\wedge\text{{State}}_{i+d}^{p}\right)\wedge\\ \left[\text{{Next}}\right]\left(\underset{\left(p,b,d\right)\in\Delta\left(q,a\right)}{\bigvee}\left(\text{{Symbol}}_{i}^{b}\wedge\text{{State}}_{i+d}^{p}\right)\right)\end{array}\right)\end{array}\right).

4. Acceptance $:=$

$\left(\underset{0\leq i\leq n+1}{\bigwedge}\underset{q\in E}{\bigwedge}\overline{\text{{State}}}_{i}^{q}\vee\left(\left(\text{{Acc}}\vee\left[\text{{Next}}\right]\overline{\text{{Acc}}}\right)\wedge\left(\overline{\text{{Acc}}}\vee\left\langle\text{{Next}}\right\rangle\text{{Acc}}\right)\right)\right)\wedge$

$\left(\underset{0\leq i\leq n+1}{\bigwedge}\underset{q\in U}{\bigwedge}\overline{\text{{State}}}_{i}^{q}\vee\left(\left(\text{{Acc}}\vee\left\langle\text{{Next}}\right\rangle\overline{\text{{Acc}}}\right)\wedge\left(\overline{\text{{Acc}}}\vee\left[\text{{Next}}\right]\text{{Acc}}\right)\right)\right)$.

Hence

[TABLE]

is equivalent to $\left\langle p^{\ast}\right\rangle\!A\vee Z$ for p=\Next, $Z=\overline{\text{{Acc}}}\vee\overline{\text{{Start}}}\in\mathcal{L}_{00}$ and

[TABLE]

where:

$R=\left\{\alpha=\left(i,a,q,\left(p,b,d\right)\right)\in\left[n+1\right]\times\Gamma\times Q\times\Delta\left(q,a\right)\right\}$,

$T=\left\{\beta=\left(i,a\right)\in\left[n+1\right]\times\Gamma\right\}$,

$S=\left\{\gamma=\left(i,a,q\right)\in\left[n+1\right]\times\Gamma\times Q\right\}$,

$F_{0}=\underset{0\leq i\leq n+1}{\bigvee}\underset{a\in\Gamma}{\bigwedge}\!\left(\!\overline{\text{{Symbol}}}_{i}^{a}\!\vee\underset{a\neq b\in\Gamma}{\bigvee}\text{{Symbol}}_{i}^{b}\!\right)\!\vee\overline{\text{{Symbol}}}_{0}^{\vdash}\vee\,$Symbol${}_{n+1}^{\dashv}$

$\vee\underset{0\leq i\leq n+1}{\bigwedge}\underset{q\in Q}{\bigwedge}\overline{\text{{State}}}_{i}^{q}\vee\underset{0\leq i\leq n+1}{\bigvee}\underset{q\in Q\cup\left\{\ell,r\right\}}{\bigwedge}\left(\overline{\text{{State}}}_{i}^{q}\vee\underset{q\neq p\in Q\cup\left\{l,r\right\}}{\bigvee}\text{{State}}_{i}^{p}\right)\vee$

$\underset{0\leq i\leq n}{\bigvee}\underset{q\in Q\cup\left\{\ell\right\}}{\bigvee}\!\left(\overline{\text{{State}}}_{i}^{q}\wedge\text{{State}}_{i+1}^{\ell}\right)\vee\underset{0\leq i\leq n+1}{\bigvee}\underset{q\in Q\cup\left\{r\right\}}{\bigvee}\!\left(\overline{\text{{State}}}_{i}^{q}\wedge\text{{State}}_{i-1}^{r}\right)$,

F_{1}=\underset{0\leq i\leq n+1}{\bigwedge}\underset{q\in E}{\bigwedge}\overline{\text{{State}}}_{i}^{q}\wedge\Acc, $G_{1}=\overline{\text{{Acc}}}$,

$F_{2}=\underset{0\leq i\leq n+1}{\bigwedge}\underset{q\in U}{\bigwedge}\overline{\text{{State}}}_{i}^{q}\wedge\overline{\text{{Acc}}}$, G_{2}=\Acc,

$F_{3}=$ $\underset{0\leq i\leq n+1}{\bigwedge}\underset{q\in E}{\bigwedge}\overline{\text{{State}}}_{i}^{q}\wedge\ \overline{\text{{Acc}}}$, G_{3}=\Acc,

$F_{4}=\underset{0\leq i\leq n+1}{\bigwedge}\underset{q\in U}{\bigwedge}\overline{\text{{State}}}_{i}^{q}\wedge$Acc, $G_{4}=\overline{\text{{Acc}}}$,

F_{\alpha}=\Symbol${}_{i}^{a}\wedge\overline{\text{{State}}}_{i}^{q}$, $G_{\alpha}=\overline{\text{{Symbol}}}_{i}^{b}\vee\overline{\text{{State}}}_{i+d}^{p}$,

F_{\beta}=\State{}_{i}^{\ell}\wedge\State{}_{i}^{r}\wedge\Symbol${}_{i}^{a}$, $G_{\beta}=\overline{\text{{Symbol}}}_{i}^{a}$,

F_{\gamma}=\Symbol${}_{i}^{a}\wedge\overline{\text{{State}}}_{i}^{q}$, $G_{\gamma}=\underset{\left(p,b,d\right)\in\Delta\left(q,a\right)}{\bigvee}\left(\overline{\text{{Symbol}}}_{i}^{b}\vee\overline{\text{{State}}}_{i+d}^{p}\right)$.

Note that $\left|\left\langle p^{\ast}\right\rangle\!A\vee Z\right|$ is at most quadratic in $\left|\text{{Accepts}}_{M,x}\right|$.

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