Optimal Satisfiability Checking for Arithmetic $\mu$-Calculi

TL;DR

This paper extends the complexity bounds and provides practical algorithms for satisfiability checking in coalgebraic $ ext{-}$-calculi, including those with weighted and arithmetic extensions, broadening applicability beyond relational models.

Contribution

It proves exponential-time upper bounds for satisfiability in general coalgebraic  ext{-}$-calculi under broader assumptions and introduces a generic caching algorithm for practical on-the-fly checking.

Findings

New exponential-time bounds for extended graded  ext{-}$-calculi with Presburger arithmetic.

Extended probabilistic  ext{-}$-calculi with polynomial inequalities.

A bound of (((nk)!)^2) on minimum model size for all coalgebraic  ext{-}$-formulas.

Abstract

The coalgebraic $\mu$-calculus provides a generic semantic framework for fixpoint logics with branching types beyond the standard relational setup, e.g. probabilistic, weighted, or game-based. Previous work on the coalgebraic $\mu$-calculus includes an exponential time upper bound on satisfiability checking, which however requires a well-behaved set of tableau rules for the next-step modalities. Such rules are not available in all cases of interest, in particular ones involving either integer weights as in the graded $\mu$-calculus, or real-valued weights in combination with non-linear arithmetic. In the present paper, we prove the same upper complexity bound under more general assumptions, specifically regarding the complexity of the (much simpler) satisfiability problem for the underlying so-called one-step logic, roughly described as the nesting-free next-step fragment of the logic. We also present a generic global caching algorithm that is suitable for practical use and supports on-the-fly satisfiability checking. Example applications include new exponential-time upper bounds for satisfiability checking in an extension of the graded $\mu$-calculus with Presburger arithmetic, as well as an extension of the (two-valued) probabilistic $\mu$-calculus with polynomial inequalities. As a side result, we moreover obtain a new upper bound $\mathcal{O}(((nk)!)^2)$ on minimum model size for satisfiable formulas for all coalgebraic $\mu$-calculi, where $n$ is the size of the formula and $k$ its alternation depth.

Equations

$$\psi ,\varphi ::=\perp \mid \top \mid \psi \wedge \varphi \mid \psi \vee \varphi \mid ♡\varphi \mid X\mid \mu X.\psi \mid \nu X.\psi ♡\in \Lambda ,X\in \mathbf{V}$$

$$[[♡\psi ]{]}_i$$

$$[[X]{]}_i$$

$$[[◊]{]}_U(A)$$

$$\mathcal{B}(U)$$

$$[[⟨k⟩]{]}_U(A)$$

$$\mathcal{D}(U)=\{d:U\to (\mathbb{Q}\cap [0,1])\mid {\sum }_{u\in U}d(u)=1\}.$$

$$[[⟨p⟩]{]}_U(A)$$

$$[[L_{a_1,\dots ,a_n,b}]{]}_U(A_1,\dots ,A_n)$$

$$[[M_{a_1,\dots ,a_n,b}]{]}_U(A_1,\dots ,A_n)$$

$$[[L_{p,b}]{]}_U(A_1,\dots ,A_n)$$

$$[[M_{p,b}]{]}_U(A_1,\dots ,A_n)$$

$$\Sigma =$$

$$\{(\eta X.{\psi }_1,0)\in \mathbf{F}\times \{0\}\}\cup \mathsf{selections},$$

$$\Delta (\psi ,\sigma )$$

$$\Delta (\psi ,({\psi }_0\vee {\psi }_1,b))$$

$$\Delta (\psi ,({\psi }_0\wedge {\psi }_1,0))$$

$$\Delta (\psi ,(\eta X.{\psi }_1,0))$$

$$L({\mathsf{B}}_{\chi })=\overline{L({\mathsf{A}}_{\chi })}=\overline{{\mathsf{BadBranch}}_{\chi }}=:{\mathsf{GoodBranch}}_{\chi },$$

$$[[v]{]}_1=\bigcap _{♡a\in v}[[♡]]\{u\in U\mid a\in u\}.$$

$$f(\mathbf{X})=$$

$$\{v\in \mathsf{states}\mid T(\bigcup _{1\le i\le l}{X}_i(v))\cap [[l(v)]{]}_1\ne \varnothing \}$$

$$g(\mathbf{X})=$$

$$\{v\in \mathsf{states}\mid T(\bigcup _{1\le i\le l}{X}_i(v))\cap [[l(v)]{]}_1=\varnothing \},$$

$$X_i(v)=\{l(u)\in X_i\mid \exists \sigma \in \mathsf{selections}.\delta (v,\sigma )=\{u\},\beta (v,\sigma ,u)=i\}.$$

$${\mathbf{E}}_G$$

$${\mathbf{A}}_G$$

$$v_0\in \mathbf{E}\text{ if and only if χ is satisfiable}.$$

$$\widehat{[[\psi ]]}$$

$$v\in {\widehat{[[♡\psi ]]}}_{\overline{m}}\text{ implies }\xi (v)\in [[♡]]({\widehat{[[\psi ]]}}_{\overline{m}}).$$

$$F=\widehat{[[{\psi }_i]]}\cap \{\lceil w_j\rceil \in V\mid (w_j,{\sigma }_j)\in L_v,{\psi }_i\in \Delta ({♡}_i{\psi }_i,{\sigma }_j)\}.$$

$$f^{\lambda }(U)=\{\begin{array}{ll}U& \text{ if }\lambda =0\\ f(f^{{\lambda }^{\prime }}(U))& \text{ if }\lambda ={\lambda }^{\prime }+1\\ {\bigcup }_{k<\lambda }{f}^k(U)& \text{ if λ is a limit-ordinal}\end{array}$$

$$\widehat{[[\psi ]]}\subseteq [[\psi ]].$$

$$\mu f=f^{\lambda }(\varnothing ).$$

Full text

1 Introduction

Modal fixpoint logics are a well-established tool in the temporal specification, verification, and analysis of concurrent systems. One of the most expressive logics of this type is the modal $\mu$-calculus [19, 2, 3], which features explicit operators for taking least and greatest fixpoints, which roughly speaking serve the specification of liveness properties (least fixpoints) and safety properties (greatest fixpoints), respectively. Like most modal logics, the modal $\mu$-calculus is traditionally interpreted over relational models such as Kripke frames or labelled transition systems. The growing interest in more expressive models where transitions are governed, e.g., by probabilities, weights, or games has sparked a commensurate growth of temporal logics and fixpoint logics interpreted over such systems; prominent examples include probabilistic $\mu$-calculi [5, 16, 23], the alternating-time $\mu$-calculus [1], and the monotone $\mu$-calculus, which contains Parikh’s game logic [26]. The graded $\mu$-calculus [20] features next-step modalities that count successors; it is standardly interpreted over Kripke frames but, as pointed out by D’Agostino and Visser [6], graded modalities are more naturally interpreted over so-called multigraphs, where edges carry integer weights, and in fact we shall see that this modification leads to better bounds on minimum model size for satisfiable formulas.

Coalgebraic logic [27, 31] has emerged as a unifying framework for modal logics interpreted over such more general models; it is based on the principle of casting the transition type of the systems at hand as a set functor, and the systems in question as coalgebras for this type functor, following the paradigm of universal coalgebra [29]; additionally, modalities are interpreted as so-called predicate liftings. The coalgebraic $\mu$-calculus [4] caters for fixpoint logics within this framework, and essentially covers all mentioned (two-valued) examples as instances. It has been shown that satisfiability checking in a coalgebraic $\mu$-calculus is in ExpTime, provided that one exhibits a set of tableau rules for the modalities, so-called one-step rules, that is tractable in a suitable sense. Such rules are known for many important cases, notably including alternating-time logics, the probabilistic $\mu$-calculus even when extended with linear inequalities, and game logic [33, 21, 4]. There are, however, important cases where such rule sets are currently missing, and where there is in fact little perspective for finding suitable rules. One prominent case of this kind is graded modal logic; further cases arise when logics over systems with non-negative real weights, such as probabilistic systems, are taken beyond linear arithmetic to include polynomial inequalities.

The object of the current paper is to fill this gap by proving a generic upper bound ExpTime for coalgebraic $\mu$-calculi in the absence of tractable sets of modal tableau rules. The method we use instead is to analyse the so-called one-step satisfiability problem of the logic on a semantic level – this problem is essentially the satisfiability problem of a very small fragment of the logic, the one-step logic, which excludes not only fixpoints, but also nested next-step modalities, with a correspondingly simplified semantics that no longer involves actual transitions. E.g. the one-step logic of the relational $\mu$-calculus is interpreted over models essentially consisting of a set with a distinguished subset, abstracting the successors of a single state that is not itself part of the model. We have applied this principle to satisfiability checking in coalgebraic (next-step) modal logics [32], coalgebraic hybrid logics [24], and reasoning with global assumptions in coalgebraic modal logics [22]. It also appears implicitly in work on automata for the coalgebraic $\mu$-calculus [8], which however establishes only a doubly exponential upper bound in the case without tractable modal tableau rules.

Our main example applications are on the one hand the graded modal $\mu$-calculus and its extension with Presburger modalities, i.e. with (monotone) linear inequalities, and on the other hand the extension of the (two-valued) probabilistic $\mu$-calculus [4, 23] with (monotone) polynomial inequalities. While the graded $\mu$-calculus as such is known to be in ExpTime [20], the other mentioned instances of our result are, to our best knowledge, new. At the same time, our proofs are fairly simple, even compared to specific ones, e.g. for the graded $\mu$-calculus.

Technically, we base our results on an automata-theoretic treatment by means of standard parity automata with singly-exponential branching degree (in particular on modal steps), thus precisely enabling the singly-exponential upper bound, in contrast to previous work in [8] where the introduced $\Lambda$-automata lead to doubly-exponential branching on modal steps in the resulting satisfiability games. Our new algorithm for satisfiability witnessing the singly-exponential time bound is, in fact, a global caching algorithm [12, 11], and is able to decide the satisfiability of nodes on-the-fly, that is, possibly before the tableau is fully expanded, thus offering a perspective for practically feasible reasoning. A side result of our approach is a singly-exponential bound on minimum model size for satisfiable formulas for all coalgebraic $\mu$-calculi, calculated only in terms of the size of the parse tree of the formula and its alternation depth (again, the best previously known bound for the case without tractable modal tableau rules [8] was doubly exponential). This bound is new even in the case of the graded $\mu$-calculus over multigraphs – over Kripke frames, it is clearly just not true as the model size can depend exponentially on numbers occurring in a formula when these are coded in binary, again illustrating the smoothness of multigraph semantics. Moreover, we identify a criterion for a polynomial bound on branching in models, which holds in all our examples.

The material is organized as follows. In Section 2, we recall the basics of coalgebra and the coalgebraic $\mu$-calculus. We outline our automata-theoretic approach in Section 3, and present the global caching algorithm and its runtime analysis in Section 4. Soundness and completeness of the algorithm are proved in Section 5.

2 The Coalgebraic $\mu$-Calculus

We recall the basics of the framework of coalgebraic logic [27, 31] and the coalgebraic $\mu$-calculus [4]. For ease of notation, we restrict the technical development to unary modalities in this work, noting that all proofs naturally generalize to the $n$-ary setting; in fact, we will liberally use higher arities in examples. We fix a $\mathbf{Set}$-endofunctor $T$, where elements of $TX$ should be regarded as structured collections over $X$ that will later serve as collections of successors of states (in the most basic example, $T$ is powerset $\mathcal{P}$), and a modal similarity type $\Lambda$, that is, a set of unary modal operators. We assume that $\Lambda$ is closed under duals, i.e., that for each modal operator $\heartsuit\in\Lambda$, there is a dual $\overline{\heartsuit}\in\Lambda$ such that $\overline{\overline{\heartsuit}}=\heartsuit$ for all $\heartsuit\in\Lambda$. We interpret modal operators $\heartsuit\in\Lambda$ as $T$-predicate liftings, that is, natural transformations $[\![\heartsuit]\!]:\mathcal{Q}\to\mathcal{Q}\circ T^{\mathit{op}}$ where $\mathcal{Q}:\mathbf{Set}^{\mathit{op}}\to\mathbf{Set}$ denotes the contravariant powerset functor. Predicate liftings thus are just families of functions $[\![\heartsuit]\!]_{X}:\mathcal{Q}(X)\to\mathcal{Q}(TX)$ that satisfy naturality, i.e. $[\![\heartsuit]\!]_{X}(f^{-1}[A])=(Tf)^{-1}[[\![\heartsuit]\!]_{Y}(A)]$ for all $X,Y\in\mathbf{Set}$, all $f:X\to Y$ and all $A\subseteq Y$, where $f^{-1}$ denotes preimage. We assume that $\Lambda$ comes with a predicate lifting $[\![\heartsuit]\!]$ for each $\heartsuit\in\Lambda$; furthermore we require that the duality of modal operators is respected, i.e. that $[\![\heartsuit]\!]_{V}(U)=\overline{[\![\overline{\heartsuit}]\!]_{V}\overline{U}}$ for all sets $V$, $U\subseteq V$, where for all sets $U$ and the according obvious base set $V$, $\overline{U}=\{u\in V\mid u\notin U\}$ denotes the complement of $U$ in $V$. Given a set $U$, a function $f:\mathcal{P}(U)\to\mathcal{P}(U)$ is monotone if for all $A,B\subseteq U$, $A\subseteq B$ implies $f(A)\subseteq f(B)$. To ensure the existence of fixpoints of formulas, we require that all predicate liftings are monotone.

Definition 1 (Coalgebraic $\mu$-calculus [4])

Let $\mathbf{V}$ be an infinite set of fixpoint variables. Formulas of the coalgebraic $\mu$-calculus (over $\Lambda$) are given by the grammar

[TABLE]

Formulas are interpreted over $T$-coalgebras, that is, pairs $(C,\xi)$, consisting of a set $C$ of states and a transition function $\xi:C\to TC$ that assigns a structured collection $\xi(x)\in TC$ of successors (and observations) to $x\in C$; e.g. $\mathcal{P}$-coalgebras are just Kripke frames. The valuation of fixpoint variables requires partial functions i:\mathbf{V}\mathrel{\ooalign{\hfil\mapstochar\mkern 5.0mu$\hfil\cr$\to\cr}}\mathcal{P}(C) that assign sets $i(X)$ of states to fixpoint variables $X$. To interpret formulas over $(C,\xi)$, we define the expected clauses for propositional formulas plus

[TABLE]

where $\mathsf{LFP}$ and $\mathsf{GFP}$ compute the least and greatest fixpoints of their argument functions, respectively, where $[\![\psi]\!]^{X}_{i}(A)=[\![\psi]\!]_{i[X\mapsto A]}$ for $A\subseteq C$ and where $(i[X\mapsto A])(X)=A$ and $(i[X\mapsto A])(Y)=i(Y)$ for $Y\neq X$. Thus we have $x\in[\![\heartsuit\psi]\!]_{i}$ if and only if $\xi(x)\in[\![\heartsuit]\!]_{C}([\![\psi]\!]_{i})$. By the monotonicity of predicate liftings, the extremal fixpoints of the functions $[\![\psi]\!]^{X}_{i}$ are indeed defined. Although the logic does not contain negation as an explicit operator, negation can be defined by taking negation normal forms. Similarly, the framework does not force the inclusion of propositional atoms, which can just be seen as nullary modalities (see Example 1.1.). Fixpoint operators bind their fixpoint variables, so that we have standard notions of bound and free fixpoint variables; a formula is closed if it contains no free fixpoint variables. For closed formulas $\psi$, the valuation of fixpoint variables is irrelevant so that we write $[\![\psi]\!]$ instead of $[\![\psi]\!]_{i}$. A state $x\in C$ satisfies a closed formula $\psi$ (denoted $x\models\psi$) if $x\in[\![\psi]\!]$. Given a set $V$, we put $\Lambda(V)=\{\heartsuit a\mid\heartsuit\in\Lambda,a\in V\}$ and refer to elements $\heartsuit a\in\Lambda(V)$ as modal literals (over $V$). Throughout, we use $\eta\in\{\mu,\nu\}$ to denote extremal fixpoint operators. The size $|\psi|$ of a formula is its length over the alphabet $\{\bot,\top,\wedge,\vee\}\cup\Lambda\cup\mathbf{V}\cup\{\eta X.\mid X\in\mathbf{V}\}$, where we assume that the length of $\heartsuit\in\Lambda$ is the size of its representation. The alternation depth $\mathsf{ad}(\psi)$ of a formula $\psi$ is the depth of dependent nesting of alternating least and greatest fixpoints in $\psi$; we assign even numbers to least fixpoint formulas and odd numbers to greatest fixpoint formulas, and, as usual, assign greater numbers to outermost fixpoints. For a more detailed definition of various flavours of alternation depth, see e.g. [25]. The satisfiability problem of the coalgebraic $\mu$-calculus is to decide, for a given formula $\chi$, whether there is a coalgebra $(C,\xi)$ and a state $x\in C$ that satisfies $\chi$. We restrict our development to formulas in which all fixpoint variables are guarded by modal operators; furthermore, we assume w.l.o.g. that all formulas are clean, i.e that each fixpoint variable is bound by at most one fixpoint operator, and irredundant, i.e each bound variable is used at least once.

As usual in $\mu$-calculi, the unfolding of fixpoints does not affect their semantics, that is, for all $X\in\mathbf{V}$ and all formulas $\psi$, we have $[\![\eta X.\,\psi]\!]=[\![\psi[X\mapsto\eta X.\,\psi]]\!]$.

Example 1

We now detail several instances of the coalgebraic $\mu$-calculus; for further examples, e.g. the alternating-time $\mu$-calculus, see [4].

To obtain the standard modal $\mu$-calculus [18] (which contains CTL as a simple fragment), we use the powerset functor, that is, we put $T=\mathcal{P}$ so that $T$-coalgebras are Kripke frames. To ease readability, we refrain from incorporating propositional atoms into the logic, noting that atoms from a set $P$ can easily be added by switching to the functor $T\times\mathcal{P}(P)$ and then defining their semantics by means of nullary predicate liftings. As modal similarity type, we take $\Lambda=\{\Diamond,\Box\}$ and define the predicate liftings

[TABLE]

for sets $U$ and $A\subseteq U$. Standard examples include the CTL-formula $\mathsf{AF}\,\psi=\mu X.\,(\psi\vee\Box X)$, which states that on all paths, $\psi$ eventually holds, and the fairness formula $\nu X.\,\mu Y.\,((\psi\wedge\Diamond X)\vee\Diamond Y)$, which asserts the existence of a path on which the formula $\psi$ is satisfied infinitely often. 2. 2.

We interpret the graded $\mu$-calculus [20] over multigraphs [6], i.e. $T$-coalgebras for the multiset functor $T=\mathcal{B}$, defined by

[TABLE]

for sets $U,V$ and functions $f:U\to V$, $\theta:U\to\mathbb{N}\cup\{\infty\}$. Thus $\mathcal{B}$-coalgebras $(C,\xi)$ assign multisets $\xi(x)$ to states $x\in C$, with the intuition that $x$ has $y\in C$ as successor with multiplicity $k$ if $(\xi(x))(y)=k$. We use the modal similarity type $\Lambda=\{\langle k\rangle,[k]\mid k\in\mathbb{N}\cup\{\infty\}\}$ and define the predicate liftings

[TABLE]

for sets $U$ and $A\subseteq U$, where $\theta(A)=\sum_{a\in A}\theta(a)$. E.g. the formula $\nu X.\,(\psi\wedge\langle 1\rangle X)$ expresses the existence of an infinite binary tree in which the formula $\psi$ is satisfied globally. 3. 3.

Similarly, the two-valued probabilistic $\mu$-calculus [4, 23] is obtained by using the distribution functor $T=\mathcal{D}$ that maps sets $U$ to probability distributions over $U$ with countable support, defined by

[TABLE]

Then $\mathcal{D}$-coalgebras are just Markov chains. We use the modal similarity type $\Lambda=\{\langle p\rangle,[p]\mid p\in\mathbb{Q}\cap[0,1])\}$ and define the predicate liftings

[TABLE]

for sets $U$ and $A\subseteq U$, where again $d(A)=\sum_{a\in A}d(a)$. 4. 4.

The Presburger $\mu$-calculus is the extension of the graded $\mu$-calculus with Presburger arithmetic; the next step version of the logic was introduced by Demri and Lugiez [7]. Its formulas can be interpreted over the semantic domain from item 2., that is, over multigraphs. We introduce new higher-arity modalities by putting $\Lambda=\{L_{a_{1},\ldots,a_{n},b},M_{a_{1},\ldots,a_{n},b}\mid a_{1},\ldots,a_{n},b,n\in\mathbb{N}\}$ and define the predicate liftings

[TABLE]

for sets $U$ and $A_{1},\ldots,A_{n}\subseteq U$, where $\theta(A)=\textstyle\sum_{a\in A}\theta(a)$. 5. 5.

Similarly, we use the semantic domain from item 3., Markov chains, to obtain the probabilistic $\mu$-calculus with polynomial inequalities [22]. Again, we introduce new higher-arity modalities by putting $\Lambda=\{L_{p,b},M_{p,b}\mid p\in\mathbb{Q}_{>0}[X_{1},\ldots,X_{n}],b\in\mathbb{Q}_{\geq 0}\}$ (i.e. $p$ ranges over polynomials) and

[TABLE]

for sets $U$ and $A_{1},\ldots,A_{n}\subseteq U$, where again $f(A)=\sum_{a\in A}f(a)$.

The logics from the last two items are necessarily less general than the corresponding next-step logics [7, 22], because the definition of $\mu$-calculi requires monotonicity of the involved predicate liftings. To ensure monotonicity, we restrict all coefficients to be positive, and moreover we restrict the relation in item 4. to be $>$ instead of one of the relations $\{>,<,=\}\cup\{\equiv_{k}\mid k\in\mathbb{N}\}$.

3 Tracking Automata

We use parity automata (e.g. [13]) that track single formulas along paths through potential models to decide whether it is possible to construct a model in which all least fixpoint formulas are eventually satisfied. Formally, (nondeterministic) parity automata are tuples $\mathsf{A}=(V,\Sigma,\Delta,q_{0},\alpha)$ where $V$ is a set of nodes; $\Sigma$ is a finite set, the alphabet; $\Delta\subseteq V\times\Sigma\times V$ is the transition relation assigning a set $\Delta(v,a)=\{u\mid(v,a,u)\in\Delta\}$ of nodes to all $v\in V$ and $a\in\Sigma$; $q_{0}\in V$ is the initial node; and $\alpha:V\to\mathbb{N}$ is the priority function, assigning priorities $\alpha(v,a,u)\in\mathbb{N}$ to transitions $(v,a,u)\in\Delta$ (this is the standard in recent work since it yields slightly more succinct automata). If $\Delta$ is a function, then $\mathsf{A}$ is said to be deterministic. The automaton $\mathsf{A}$ accepts an infinite word $w=w_{0}w_{1},\ldots\in\Sigma^{\omega}$ if there is a $w$-path through $\mathsf{A}$ on which the highest priority that is passed infinitely often is even; formally, the language that is accepted by $\mathsf{A}$ is defined by $L(\mathsf{A})=\{w\in\Sigma^{\omega}\mid\exists\rho\in\mathsf{run}(\mathsf{A},w).\,\max(\mathsf{Inf}(\alpha\circ\rho))\text{ is even}\}$, where $\mathsf{run}(\mathsf{A},w)$ denotes the set of infinite sequences $\rho_{0},\rho_{1},\ldots\in V^{\omega}$ such that $\rho_{0}=q_{0}$ and for all $i\geq 0$, $\rho_{i+1}\in\Delta(\rho_{i},w_{i})$ and where, given an infinite sequence $S$, $\mathsf{Inf}(S)$ denotes the elements that occur infinitely often in $S$. Here, we see infinite sequences $\rho\in U^{\omega}$ over some set $U$ as functions $\mathbb{N}\to U$ and write $\rho_{i}$ to denote the $i$-th element of $\rho$.

We now fix a target formula $\chi$ and put $n:=|\chi|$ and $k:=\mathsf{ad}(\chi)$. We let $\mathbf{F}:=\mathsf{FL}(\chi)$ denote the Fischer-Ladner closure [19] of $\chi$; the Fischer-Ladner closure contains all formulas that can arise as subformulas when unfolding each fixpoint in $\chi$ exactly once. We put $\mathsf{selections}:=\mathcal{P}(\mathbf{F}\cap\Lambda(\mathbf{F}))$ where $\mathbf{F}\cap\Lambda(\mathbf{F})$ is the set of formulas from $\mathbf{F}$ that are modal literals. We have $|\mathbf{F}|\leq n$ and hence $|\mathsf{selections}|\leq 2^{n}$.

Definition 2 (Tracking automaton)

The tracking automaton for a formula $\chi$ is a nondeterministic parity automaton $\mathsf{A}_{\chi}=(\mathbf{F},\Sigma,\Delta,q_{0},\alpha)$, where $q_{0}=\chi$,

[TABLE]

for $\psi,\psi_{0},\psi_{1}\in\mathbf{F}$, $\sigma\in\mathsf{selections}$ and $b\in\{0,1\}$,

[TABLE]

E.g. the last clause means that when tracking the unfolding of a fixpoint $\eta X.\,\psi_{1}$ at $\psi$, we track $\psi$ to the unfolding $\psi_{1}[X\mapsto\psi]$ if $\psi$ equals the unfolded fixpoint, and to $\psi$ otherwise; similarly for the other clauses, and in particular a modal literal $\psi=\heartsuit\psi_{0}$ is only tracked to $\psi_{0}$ through a selection $\sigma$ if $\heartsuit\psi_{0}\in\sigma$, i.e. if $\sigma$ selects $\heartsuit\psi_{0}$ to be tracked. The priority function $\alpha$ is derived from the alternation depths of formulas, counting only unfoldings of fixpoints (i.e. all other transitions have priority 1). Formally, $\alpha(\psi,\sigma,\psi^{\prime})=1$ if $\psi=\psi^{\prime}$ or $\psi$ is not a fixpoint literal; if $\psi$ is a fixpoint literal and $\psi\neq\psi^{\prime}$, then we put $\alpha(\psi,\sigma,\psi^{\prime})=\mathsf{ad}(\psi)$.

Intuitively, words from $\Sigma^{\omega}$ encode infinite paths through labelled coalgebras $(C,\xi)$ where letters $\sigma\in\mathsf{selections}$ encode modal steps from states $x\in C$ with label $l(x)$ to states $y\in C$ with label $\{\psi\in\mathbf{F}\mid\exists\heartsuit\in\Lambda.\,\heartsuit\psi\in\sigma\cap l(x)\}$. Letters $(\psi_{0}\vee\psi_{1},b)$ choose disjuncts; the tracking automaton is nondeterministic for letters $(\psi_{0}\wedge\psi_{1},0)$ and accepts exactly the words that encode a path that contains a least fixpoint formula $\psi$ that is unfolded infinitely often without being dominated by any outer fixpoint formula (i.e. one with alternation depth greater than $\mathsf{ad}(\psi)$); denoting these words by $\mathsf{BadBranch}_{\chi}$, we thus have $L(\mathsf{A}_{\chi})=\mathsf{BadBranch}_{\chi}$. The automaton $\mathsf{A}_{\chi}$ has size $n$ and priorities $1$ to $k$. Using the standard construction (e.g. [17]), we transform $\mathsf{\mathsf{A}_{\chi}}$ to an equivalent Büchi automaton of size $nk$. Then we determinize the Büchi automaton using e.g. the Safra/Piterman-construction [30, 28] and obtain an equivalent deterministic parity automaton with priorities [math] to $2nk-1$ and size $\mathcal{O}(((nk)!)^{2})$. Finally we complement this parity automaton by increasing every priority by 1, obtaining a deterministic parity automaton $\mathsf{B}_{\chi}=(D_{\chi},\Sigma,\delta,v_{0},\beta)$ of size $\mathcal{O}(((nk)!)^{2})$, with priorities $1$ to $2nk$ and with

[TABLE]

i.e. $\mathsf{B}_{\chi}$ is a deterministic parity automaton that accepts the words that encode paths that do not contain a least fixpoint whose satisfaction is deferred indefinitely. We use the labelling function $l:D_{\chi}\to\mathcal{P}(\mathbf{F})$ from the determinized automaton.

Remark 1

It has been noted that the standard tracking automata for alternation-free formulas are, in fact, Co-Büchi automata [10, 15] and that the tracking automata for aconjunctive formulas are limit-deterministic parity automata [14]. These considerably simpler automata can be determinized to deterministic Büchi automata of size $3^{n}$ and to deterministic parity automata of size $\mathcal{O}((nk)!)$ and with $2nk$ priorities, respectively. This observation also holds true for the tracking automata in this work so that for formulas of suitable syntactic shape, Lemma 1 below yields accordingly lower bounds on the runtime of our satisfiability checking algorithm.

4 Global Caching for the

Coalgebraic $\mu$-Calculus

We now introduce a generic global caching algorithm that decides the satisfiability problem of the coalgebraic $\mu$-calculus. Given an input formula $\chi$, the algorithm expands the determinized and complemented tracking automaton $\mathsf{B}_{\chi}$ step by step and propagates (un)satisfiability through this graph; the algorithm terminates as soon as the initial node $v_{0}$ is marked as (un)satisfiable. The algorithm bears similarity to standard game-based algorithms for $\mu$-calculi [9, 14, 8]; however, it crucially deviates from these algorithms in the treatment of modal steps: Intuitively, our algorithm decides whether it is possible to remove some of the modal transitions as well as one of the transitions from any reachable pair $((\psi_{1}\vee\psi_{2}),0),((\psi_{1}\vee\psi_{2}),1)$ of disjunction transitions within the automaton $\mathsf{B}_{\chi}$ in such a way that the resulting sub-automaton of $\mathsf{B}_{\chi}$ that no longer contains choices for disjunctions and (possibly) has a reduced set of modal transitions is totally accepting, that is, accepts any word for which there is an infinite run of the automaton. In doing so, it is crucial that the labels of state nodes $v$ in the reduced automaton are one-step satisfied in the set of states that are reachable from $v$ by the remaining modal transitions. This last property is ensured by using instances of the so-called one-step satisfiability problem to propagate (un)satisfiability over modal transitions; these instances can often be solved in time singly-exponential in $|\chi|$, and in fact, this appears to be the case for all currently known examples of decidable coalgebraic $\mu$-calculi. Previous work in [8] casts the modal steps of satisfiability checking for coalgebraic $\mu$-calculi in terms of satisfiability games but leads to a doubly-exponential number of modal moves for one of the players and hence does not yield a singly-exponential upper bound on satisfiability checking (unless a suitable set of tableau rules is provided).

Definition 3 (One-step satisfiability

problem [22, 8])

Let $V$ be a finite set, let $v\subseteq\Lambda(V)$ such that $a\neq b$ whenever $\heartsuit_{1}a,\heartsuit_{2}b\in v$, and let $U\subseteq\mathcal{P}(V)$. The one-step satisfiability problem for inputs $v$ and $U$ is to decide whether $TU\cap[\![v]\!]_{1}\neq\emptyset$, where

[TABLE]

We denote denote the time it takes to solve the problem for input $v$ and $U$ by $t(|v|,|U|)$, where $|v|=\sum_{\heartsuit a\in v}|\heartsuit T|$, the size of $v$, takes the representation of modal operators into account and where $|U|$ is just the number of elements of $U$.

Remark 2

We keep the definition of the actual one-step logic as mentioned in the introduction somewhat implicit in the above definition of the one-step satisfiability problem. One can see that it contains two layers: a purely propositional layer embodied in $U$, which postulates which propositional formulas over $V$ are satisfiable, and a modal layer with nesting depth of modalities uniformly equal to $1$, embodied in the set $v$, which specifies constraints on an element of $TU$.

Example 2

For the one-step fragment of the standard modal $\mu$-calculus (Example 1.1.), the one-step satisfiability problem for given $v\subseteq\Lambda(V)$ and $U\subseteq\mathcal{P}(V)$ consists in deciding whether there is a set $A\in\mathcal{P}(U)\cap[\![v]\!]_{1}$, that is, such that for each $\Diamond a\in v$, there is $u\in A$ such that $\psi\in u$, and for each $\Box a\in v$ and each $u\in A$, $a\in u$. Here we have $t(|v|,|U|)\leq|v|\cdot|U|$. For the one-step fragment of the graded $\mu$-calculus (Example 1.2.), the problem for input $v$ and $U$ consists in deciding whether there is a multiset $\theta\in\mathcal{B}(U)$ with $\theta\in[\![v]\!]_{1}$; the latter is the case if for each $\langle k\rangle a\in v$, we have $\sum_{u\in U_{a}}\theta(u)>k$, where $U_{a}=\{u\in U\mid a\in u\}$ and for each $[k]a\in v$, we have $\sum_{u\in U_{\neg a}}\theta(u)\leq k$, where $U_{\neg a}=\{u\in U\mid a\notin u\}$.

Let $l=2|\chi|\cdot\mathsf{ad}(\chi)$ denote the number of priorities in $\mathsf{B}_{\chi}$. Nodes whose labels consist exclusively of modal literals are referred to as saturated nodes or states. We denote the set of states by $\mathsf{states}\subseteq D_{\chi}$ and the set of pre-states, that is, non-state nodes, by $\mathsf{prestates}\subseteq D_{\chi}$. For each pre-state $v\in\mathsf{prestates}$, we also fix a non-modal formula $\psi_{v}\in l(v)$. We now define $l$-ary set-valued functions $f$ and $g$ that compute one-step (un)satisfiability w.r.t. their argument sets.

Definition 4 (One-step propagation)

For sets $G\subseteq D_{\chi}$ and $\mathbf{X}=X_{1},\ldots,X_{l}\subseteq G^{l}$, we put

[TABLE]

where $\beta(v,(\psi_{v},b))$ abbreviates $\beta(v,(\psi_{v},b),\delta(v,(\psi_{v},b)))$ and where

[TABLE]

Since for states $v$, $l(v)\subseteq\Lambda(\mathbf{F})$ and since $\bigcup_{1\leq i\leq l}X_{i}(v)\subseteq\mathcal{P}(\mathbf{F})$, one-step propagation steps for states are just instances of the one-step satisfiability problem with $V=\mathbf{F}$. Since states have at most $|\chi|$ modal literals in their labels, these instances can be solved in time $t(|\chi|,2^{|\chi|})$.

Definition 5 (Propagation)

Given a set $G$, we put

[TABLE]

where $\mathbf{X}=X_{1},\ldots,X_{l}$ for $X_{i}\subseteq G$, where $\eta_{i}=\mu$ for odd $i$, $\eta_{i}=\nu$ for even $i$ and where $\overline{\eta}=\mu$ if $\eta=\nu$ and $\overline{\eta}=\nu$ if $\eta=\mu$.

The set $\mathbf{E}_{G}$ contains nodes $v\in G$ for which there are choices for all disjunction and modal transitions that are reachable from $v$ within $G$ such that the labels of all reachable states in the chosen sub-automaton of $\mathsf{B}_{\chi}$ are one-step satisfied and such that on all paths through the chosen sub-automaton, the highest priority that is passed infinitely often is even, the intuition being that no least fixpoint is unfolded infinitely often without being dominated. Dually, the set $\mathbf{A}_{G}$ contains nodes for which there exist no such suitable choices.

We recall that $v_{0}\in D_{\chi}$ is the initial state of the determinized and complemented tracking automaton $\mathsf{B}_{\chi}$. The algorithm expands $\mathsf{B}_{\chi}$ step-by-step starting from $v_{0}$; for pre-states $u$, the expansion step adds nodes according to the fixed non-modal formula $\psi_{u}$ that is to be expanded next, and for states, the expansion follows all (matching) selections. The order of expansion can be chosen freely, e.g. by heuristic methods. Optional intermediate propagation steps can be used judiciously to realize on-the-fly solving.

Algorithm 4.1 (Global caching)

To decide the satisfiability of the input formula $\chi$, initialize the sets of unexpanded and expanded nodes, $U=\{v_{0}\}$ and $G=\emptyset$, respectively.

Expansion: Choose some unexpanded node $u\in U$, remove it from $U$ and add it to $G$. If $u$ is a pre-state, then add the set $\{\delta(u,\sigma)\mid\sigma\in\Sigma\cap(\psi_{u}\times\{0,1\})\}$ to $U$. If $u$ is a state, then add the set $\{\delta(u,\sigma)\mid\sigma\in\mathsf{selections}\}$ to $U$. 2. 2.

Optional propagation: Compute $\mathbf{E}_{G}$ and/or $\mathbf{A}_{G}$. If $v_{0}\in\mathbf{E}_{G}$, then return ‘satisfiable‘, if $v_{0}\in\mathbf{A}_{G}$, then return ‘unsatisfiable‘. 3. 3.

If $U\neq\emptyset$, then continue with step 1. 4. 4.

Final propagation: Compute $\mathbf{E}_{G}$. If $v_{0}\in\mathbf{E}_{G}$, then return ‘satisfiable‘, otherwise return ‘unsatisfiable‘.

Lemma 1

Given a target formula $\chi$ with $|\chi|=n$ and $\mathsf{ad}(\chi)=k$, Algorithm 4.1 terminates and runs in time $\mathcal{O}(((nk)!)^{4nk}\cdot t(n,2^{n}))$.

Proof

The loop of the algorithm expands the determinized and complemented tracking automaton node by node and hence is executed at most $|D_{\chi}|\in\mathcal{O}(((nk)!)^{2})\in 2^{\mathcal{O}(nk\log n)}$ times. A single expansion step can be implemented in time $\mathcal{O}(2^{n})$ since propositional expansion is unproblematic and for the modal expansion of a state $u$, all (matching) selections, of which there are (at most) $2^{n}$, have to be considered. A single propagation step consists in computing two fixpoints of nesting depth $l=2nk$ of the functions $f$ and $g$ over $\mathcal{P}(D_{\chi})$ and can hence be implemented in time $2(|D_{\chi}|^{2nk}\cdot t(n,2^{n}))\in\mathcal{O}(((nk!)^{2})^{2nk}\cdot t(n,2^{n}))\in 2^{\mathcal{O}(n^{2}k^{2}\log n+\log(t(n,2^{n})))}$, noting that a single computation of $f(\mathbf{X})$ and $g(\mathbf{X})$ for a tuple $\mathbf{X}\subseteq(D_{\chi})^{k}$ can be implemented in time $\mathcal{O}(t(n,2^{n}))$: for pre-states, the one-step propagation is unproblematic and for states, it consists in solving the one-step satisfiability problem with inputs of size at most $n$ and $2^{n}$, as explained above. Thus the complexity of the whole algorithm is dominated by the complexity of the propagation step.∎

Corollary 1

If the one-step satisfiability problem of a coalgebraic logic for inputs $v$ and $U$ with $|U|\leq 2^{|v|}$ can be solved in time $t(|v|,|U|)\leq 2^{p(|v|)}\cdot p^{\prime}(|U|)$, where $p$ and $p^{\prime}$ are some polynomial functions, then the satisfiability problem of the $\mu$-calculus over the logic is in ExpTime.

The complexity bounds obtained by our current semantic approach thus subsume the earlier bounds obtained by the tableau-based approaches in [4, 15, 14] but also cover new example logics. In particular we have

Lemma 2

The satisfiability problems of the following logics are in ExpTime:

the standard $\mu$-calculus, 2. 2.

the graded $\mu$-calculus, 3. 3.

the (two-valued) probabilistic $\mu$-calculus, 4. 4.

the Presburger $\mu$-calculus, 5. 5.

the (two-valued) probabilistic $\mu$-calculus extended with polynomial inequalities

Proof

It suffices to show that the respective one-step satisfiability problems can be solved in time $t(n,2^{n})\leq 2^{p(n)}\cdot p^{\prime}(2^{n})$, that is, in time singly exponential in $n$, for inputs $v$ and $U$ of sizes $|v|\leq n$ and $|U|\leq 2^{n}$. While this follows by relatively easy arguments (using known bounds on sizes of solutions of systems of real or integer linear inequalities) for all of our examples, we import most of the results from previous work for brevity. For standard Kripke logic, we have $p(x)=\log x$ and $p^{\prime}(x)=x$, see Example 2. For the one-step satisfiability problem of graded modal logic, by Lemma 1 in [20], we have $t(n,2^{n})\leq(2n+2)^{n}\leq 2^{n\log(2n+2)}$ and choose, e.g., $p(x)=x\log(2x+2)$ and $p^{\prime}(x)=x$. The corresponding properties for (two-valued) probabilistic modal logic and the two arithmetic logics (items 4. and 5.) are shown in Example 7 in [22]. ∎

Remark 3

We also obtain a polynomial bound on branching width in models for all our example logics simply by importing Lemma 6 and the observations in Example 7 from [22]. With exception of the standard $\mu$-calculus, this bound appears to be novel for all example logics in this work.

5 Soundness and Completeness

We now prove the central result, that is, the total correctness of Algorithm 4.1. As the sets $\mathbf{E}_{G}$ and $\mathbf{A}_{G}$ grow monotonically with $G$, it suffices to prove equivalence of satisfiability and containment of the initial node $v_{0}$ in $\mathbf{E}:=\mathbf{E}_{D_{\chi}}$.

Theorem 5.1 (Soundness and completeness)

We have

[TABLE]

Proof

By Corollary 2, it suffices to show that there is a pre-semi-tableau (see Definition 6) for $\chi$ with unfolding timeouts (see Definition 8) if and only if $\chi$ is satisfiable. So let there be a pre-semi-tableau for $\chi$ with unfolding timeouts. We use the Existence Lemma (Lemma 5) to obtain a strongly coherent coalgebra (see Definition 10) which by the Truth Lemma (Lemma 6) is a model for $\chi$. For the converse direction, let $\chi$ be satisfiable. We use Lemma 7 to extract a pre-semi-tableau with unfolding timeouts from the model. ∎

Definition 6 (Pre-semi-tableau)

Given two sets $A$ and $B$, a ternary relation $R\subseteq A\times B\times A$ and two elements $a\in A$, $b\in B$, we put $R(a)=\{a^{\prime}\in A\mid\exists b\in B.\,(a,b,a^{\prime})\in R\}$ and $R(a,b)=\{a^{\prime}\in A\mid(a,b,a^{\prime})\in R\}$. Let $W\subseteq D_{\chi}$ be a set of nodes labelled with formulas from $\mathbf{F}$ and put $U=W\cap\mathsf{prestates}$ and $V=W\cap\mathsf{states}$. Given a ternary relation $L\subseteq W\times\Sigma\times W$, the pair $(W,L)$ is a pre-semi-tableau for $\chi$ if $L\subseteq\delta$ and for all $v\in V$, we have $T(L(v))\cap[\![l(v)]\!]_{1}\neq\emptyset$, for all $u\in U$, there is exactly one $b\in\{0,1\}$ such that $L(u,(\psi_{u},b))=\delta(u,(\psi_{u},b))$ and for all other $\sigma\in\Sigma$, $L(u,\sigma)=\emptyset$ and there is no $L$-cycle that contains only elements from $U$. A path through a pre-semi-tableau is an infinite sequence $(v_{0},\sigma_{0}),(v_{1},\sigma_{1}),\ldots\in(W\times\Sigma)^{\omega}$ such that for all $i$, $v_{i+1}\in L(v_{i},\sigma_{i})$. We denote the first state that is reachable by zero or more $L$-steps from a node $v\in W$ by $\lceil v\rceil$ (since there is no $L$-cycle that contains only elements from $U$, such a state always exists).

Given a state $v$, the relation $L$ of a pre-semi-tableau thus picks a set $L(v)$ of nodes over which a coherent observation for $v$ can be built; given a pre-state $u$, $L$ picks a single (pre)state that is obtained from $u$ by transforming the formula $\psi_{u}$.

Definition 7 (Tracking timeouts)

Given a path $\rho=(v_{0},\sigma_{0}),(v_{1},\sigma_{1}),\ldots$ through a pre-semi-tableau, we say that priority $i$ occurs (at position $j$) in $\rho$ if $\beta(v_{j},\sigma_{j},v_{j+1})=i$, recalling that $\beta$ is the priority function of the determinised and complemented tracking automaton $\mathsf{B}_{\chi}$. Then the path $\rho$ has tracking timeouts $\overline{m}=(m_{l},\ldots,m_{1})$, if for each odd $1\leq i<l$, priority $i$ occurs at most $m_{i}$ times in $\rho$ before some priority greater than $i$ occurs in $\rho$. Nothing is said about $m_{i}$ for even $i$, which are in fact irrelevant and serve only to ease notation. An element $w\in W$ has tracking timeouts $\overline{m}$ in some pre-semi-tableau $(W,L)$ if every path through $(W,L)$ that starts at $w$ has tracking timeouts $\overline{m}$. A pre-semi-tableau $(W,L)$ has tracking timeouts if there is, for each $w\in W$, some vector $\overline{m}$ such that $w$ has tracking timeouts $\overline{m}$.

Intuitively, a pre-semi-tableau $(W,L)$ has tracking timeouts if every word that encodes an infinite $L$-path through $W$ is accepted by $\mathsf{B}_{\chi}$. We recall that a run of $\mathsf{B}_{\chi}$ is accepting if the encoded path does not contain a trace that unfolds some least fixpoint formula infinitely often without having it dominated.

Definition 8 (Unfolding timeouts)

Given a path $\rho=(v_{0},\sigma_{0}),(v_{1},\sigma_{1}),\ldots$ through a pre-semi-tableau and a sequence of formulas $\Psi=\psi_{0},\psi_{1},\ldots$, we say that $\Psi$ is a trace of $\psi_{0}$ in $\rho$ (we also say that $\rho$ contains $\Psi$) if $\psi_{0}\in l(v_{0})$ and for all $i>0$, $\psi_{i}\in l(v_{i})\cap\Delta(\psi_{i-1},\sigma_{i-1})$. For $i$ with $\psi_{i}=\eta X.\psi$ for some fixpoint variable $X$ and some formula $\psi$, we say that $\Psi$ unfolds at level $\mathsf{ad}(\psi_{i})$ at position $i$. Then the trace $\Psi$ has unfolding timeouts $\overline{m}=(m_{k},\ldots,m_{1})$ for $\psi_{0}$ if for each odd $1\leq i\leq k$, $\Psi$ unfolds at most $m_{i}$ times at level $i$ before $\Psi$ unfolds at some level greater than $i$. Again the unfolding timeouts for even $i$, that is, for greatest fixpoints, are irrelevant. The path $\rho$ has unfolding timeouts for $\psi_{0}$ if there is, for all its traces $\Psi$ of $\psi_{0}$, some vector $\overline{m}$ such that $\Psi$ has unfolding timeouts $\overline{m}$ for $\psi_{0}$. Given a pre-semi-tableau $(W,L)$, a node $w\in W$ has unfolding timeouts $\overline{m}$ for some formula $\psi$ if every path through $(W,L)$ that starts at $w$ and contains a trace of $\psi$ has unfolding timeouts $\overline{m}$ for $\psi$. A pre-semi-tableau $(W,L)$ has unfolding timeouts if for each element $w\in W$ and each formula $\psi\in l(v)$, there is some vector $\overline{m}$ such that $w$ has unfoldings timeouts $\overline{m}$ for $\psi$. We denote the set of states that have unfolding timeouts $\overline{m}$ for $\psi$ by $\mathsf{uto}(\psi,\overline{m})\subseteq W$.

A pre-semi-tableau $(W,L)$ has unfolding timeouts if for all words that encode an infinite $L$-path through $W$, all runs of the nondeterministic tracking automaton $\mathsf{A}_{\chi}$ on the word are non-accepting. We recall that a run of $\mathsf{A}_{\chi}$ is accepting if it unfolds some least fixpoint infinitely often without having it dominated.

Lemma 3

Let $(W,L)$ be a pre-semi-tableau. Then $(W,L)$ has tracking timeouts if and only if it has unfolding timeouts.

Proof

We recall that $\mathsf{B}_{\chi}$ is obtained from $\mathsf{A}_{\chi}$ by determinization and subsequent complementation so that we have $L(\mathsf{B}_{\chi})=\overline{L(\mathsf{A}_{\chi})}$. The result thus follows directly from the fact that having tracking timeouts ensures that $\mathsf{B}_{\chi}$ accepts all words that encode a path in $(W,L)$ while having unfolding timeouts ensures that $\mathsf{A}_{\chi}$ does not accept any word that encodes a path in $(W,L)$. ∎

Lemma 4

We have $v_{0}\in\mathbf{E}$ if and only if there is a pre-semi-tableau for $\chi$ that has tracking timeouts.

Proof (Sketch)

If $v_{0}\in\mathbf{E}$, then the definition of the function $f$ ensures the existence of suitable transitions in $\mathsf{B}_{\chi}$ that can be used to define a pre-semi-tableau $(W,L)$. The definition of $(W,L)$ has to be executed in a nested inductive-coinductive way, relying on the fact that $v_{0}$ is contained in the nested fixpoint $\mathbf{E}$ to ensure that $(W,L)$ has tracking timeouts. For the converse direction, let there be a pre-semi-tableau for $\chi$ that has tracking timeouts. Then $v_{0}$ has tracking timeouts $\overline{m}$ for some $\overline{m}$. We show $v_{0}\in\mathbf{E}$ by nested induction and coinduction, using $\overline{m}$ as termination measure for the induction parts. The full proof can be found in the appendix. ∎

Combining Lemmas 4 and 3, we obtain

Corollary 2

We have $v_{0}\in\mathbf{E}$ if and only if there is a pre-semi-tableau for $\chi$ that has unfolding timeouts.

Definition 9

Given a pre-semi-tableau $(W,L)$ with set of states $V$, we put

[TABLE]

where $\psi\in\mathbf{F}$, where $\vdash_{\mathsf{PL}}$ denotes propositional entailment and where $\overline{m}$ is a vector of $k$ natural numbers.

Thus we have $v\in[\![\psi]\!]_{\overline{m}}$ if there is a node $u\in W$ such that $\lceil u\rceil=v$ and $u$ has timeouts $\overline{m}$ for $\psi$. This serves to ease the proofs of the upcoming Existence and Truth Lemmas as it anchors the timeout vector $\overline{m}$ at the node $u$ instead of anchoring it at the state $v$ which may not have timeouts $\overline{m}$ for $\psi$ (namely, if a greatest fixpoint is unfolded on the $L$-path from $u$ to $v$).

Definition 10 (Strong coherence)

Let $(W,L)$ be a pre-semi-tableau with unfolding timeouts and set of states $V$. A coalgebra $\mathcal{C}=(V,\xi)$ is strongly coherent if for all states $v\in V$, for all formulas $\heartsuit\psi\in\mathbf{F}$ and for all timeout-vectors $\overline{m}$,

[TABLE]

Lemma 5 (Existence)

Let $(W,L)$ be a pre-semi-tableau with set of states $V$ that has unfolding timeouts. Then there is a strongly coherent coalgebra over $V$.

Proof

Let $v\in V$ be a state with $l(v)=\{\heartsuit_{1}\psi_{1},\ldots,\heartsuit_{n}\psi_{n}\}$ and put $L_{v}=\{(\sigma_{i},v_{i})\in\Sigma\times W\mid(v,\sigma_{i},v_{i})\in L\}$. As $(W,L)$ is a pre-semi-tableau, we have $T(L(v))\cap[\![l(v)]\!]_{1}\neq\emptyset$, so we put $\xi(v)=t$ for some $t\in T(L(v))\cap[\![l(v)]\!]_{1}\neq\emptyset$. Let $i$ be a number for which there is a vector $\overline{m}^{\prime}$ such that $v\in\mathsf{uto}(\heartsuit_{i}\psi_{i},\overline{m}^{\prime})$. Such a vector exists since $(W,L)$ has unfolding timeouts. Now let $\overline{m}$ denote the least such vector (by lexicographic ordering). It suffices to show that $t\in[\![\heartsuit_{i}]\!](\widehat{[\![\psi_{i}]\!]}_{\overline{m}})$. By construction, we have $t\in[\![\heartsuit_{i}]\!](\{u\in W\mid\psi_{i}\in l(v)\}\cap L(v))$ and hence $t\in[\![\heartsuit_{i}]\!](F)$ where

[TABLE]

As $v\in\mathsf{uto}(\heartsuit_{i}\psi_{i},\overline{m})$, every infinite $L$-path $(v,\sigma_{j}),(w_{j},\sigma^{\prime}),\ldots$ such that $(\sigma_{j},w_{j})\in L(v)$ and that contains a trace of $\heartsuit_{i}\psi_{i}$ has unfolding timeouts $\overline{m}$ for $\heartsuit_{i}\psi_{i}$ and $\psi_{i}\in\Delta(\heartsuit_{i}\psi_{i},\sigma_{j})$. Hence all such $w_{j}\in L(v)$ have unfolding timeouts $\overline{m}$ for $\psi_{i}$. Thus $F\subseteq\widehat{[\![\psi_{i}]\!]}\cap\{\lceil w\rceil\in V\mid w\in\mathsf{uto}(\psi_{i},\overline{m})\}=\widehat{[\![\psi_{i}]\!]}_{\overline{m}}$, as required. ∎

Definition 11 (Timed-out satisfaction)

Given sets $W$, $U\subseteq W$, a function $f:\mathcal{P}(W)\to\mathcal{P}(W)$ and an ordinal number $\lambda$, we define

[TABLE]

Given a least fixpoint formula $\mu X.\psi$ with $\mu X.[\![\psi]\!]_{i}^{X}=([\![\psi]\!]_{i}^{X})^{\lambda}(\emptyset)$ where $i:\mathbf{V}\to\mathcal{P}(C)$ valuates fixpoint variables, as usual, we say that $x\in([\![\psi]\!]_{i}^{X})^{\lambda}(\emptyset)$ satisfies $\mu X.\psi$ with timeout $\lambda$ (under $i$) and write $x\in[\![\mu X.\psi]\!]^{\lambda}_{i}$. For all models $(C,\xi)$ and ordinal numbers $\lambda$ with $|C|=\lambda$, all states $x\in C$ and all least fixpoint formulas $\mu X.\psi$ such that $x\models\mu X.\psi$, we have $x\in[\![\mu X.\psi]\!]^{\lambda}_{\epsilon}$ where $\epsilon$ denotes the empty valuation of fixpoint variables. Formulas $\psi$ from nested fixpoints are satisfied (under $i$) with vectors of ordinal numbers $\overline{\lambda}=(\lambda_{k},\ldots,\lambda_{j})$, ordered from outermost fixpoint to innermost fixpoint, as timeouts; here, $i$ is assumed to valuate $k-j$ alternating fixpoint variables. Again the timeouts for greatest fixpoint variables are irrelevant and serve only to ease notation. We write $x\in[\![\psi]\!]_{i}^{\overline{\lambda}}$ to indicate that $x$ satisfies $\psi$ with nested timeouts $\overline{\lambda}$ (under $i$).

We have $[\![\mu X.\psi]\!]_{i}^{0}=\emptyset$, $[\![\mu X.\psi]\!]_{i}^{\lambda+1}=[\![\psi[X\mapsto\mu X.\psi]]\!]_{i}^{\lambda}$ for ordinals $\lambda$ and $[\![\mu X.\psi]\!]_{i}^{\lambda}=\bigcup_{k<\lambda}[\![\mu X.\psi]\!]_{i}^{k}$ for limit-ordinals $\lambda$.

In strongly coherent coalgebras, all least fixpoint literals are satisfied after finitely many unfolding steps:

Lemma 6 (Truth)

In strongly coherent coalgebras, we have that for all $\psi\in\mathbf{F}$,

[TABLE]

Proof (Sketch)

This proof is standard for $\mu$-calculi; we use nested induction and coinduction to show eventual satisfaction of all least fixpoint formulas that occur in some label, using unfolding timeouts as termination measure for the induction parts of the proof. The full proof can be found in the appendix. ∎

Lemma 7 (Soundness)

Let $\chi$ be satisfiable. Then a pre-semi-tableau for $\chi$ with unfolding timeouts can be constructed over a subset of $D_{\chi}$.

Proof (Sketch)

Relying on the information from some fixed model for $\chi$, we construct a pre-semi-tableau for $\chi$ by choosing usable transitions in the determinized and complemented tracking automaton $\mathsf{B}_{\chi}$. The transitions have to be chosen in a way that preserves the timeouts with which formulas are satisfied in the model. Then we use nested transfinite induction and coinduction to show that the constructed pre-semi-tableau has unfolding timeouts, using the timeout vectors $(\lambda_{k},\ldots,\lambda_{j})$ with which formulas are satisfied in the model as termination measure for the transfinite induction parts of the proof. The full proof can be found in the appendix. ∎

Corollary 3

Let $\chi$ be a satisfiable coalgebraic $\mu$-calculus formula. Then $\chi$ has a model of size $\mathcal{O}(((nk)!)^{2})\in 2^{\mathcal{O}(nk\log n)}$.

Proof

Let $\chi$ be satisfiable. By soundness, we have $v_{0}\in\mathbf{E}$. The model for $\chi$ that is constructed during the completeness proof is built over $\mathbf{E}\cap\mathsf{states}\subseteq D_{\chi}$. The stated bound follows since $|D_{\chi}|\in\mathcal{O}(((nk)!)^{2})$. ∎

6 Conclusion

We have shown that the satisfiability problem of the coalgebraic $\mu$-calculus is in ExpTime if the corresponding one-step satisfiability problem can be solved in time singly exponential in $n$ for inputs $v,U$ of sizes $n$ and $2^{n}$. Prominent examples where this is the case include the graded $\mu$-calculus, the (two-valued) probabilistic $\mu$-calculus, the Presburger $\mu$-calculus, and the extension of the two-valued probabilistic $\mu$-calculus with polynomial inequalities; the ExpTime bound appears to be novel for the last two logics. We also have presented a generic satisfiability algorithm that realizes the singly exponential time upper bound under the stated assumption and is suitable for practical use since it supports global caching and on-the-fly solving. Moreover, we obtained a novel singly-exponential bound on minimum model size of satisfiable formulas for all decidable coalgebraic $\mu$-calculi and a polynomial bound on the branching width in models for all example logics mentioned above.

Appendix 0.A Appendix: Omitted Proofs and Lemmas

Full proof of Lemma 4: Let $v_{0}\in\mathbf{E}$. All elements $v$ of the nested fixpoint $\mathbf{E}$ (which has nesting depth $l$) have nested timeouts $\overline{m}=(m_{l},\ldots,m_{1})$ with $m_{i}\leq|D_{\chi}|$ that ensure that $v$ can be shown to be contained in $\mathbf{E}$ while, for all odd $1\leq i\leq l$, unfolding fixpoints $\eta_{i}X_{i}.\eta_{i-1}X_{i-1}.\ldots\eta_{1}X_{1}.f(X_{1},\ldots,X_{i},M(m_{l},\ldots,m_{i+1}),\ldots,M(m_{l}))$ at most $m_{i}$ times before unfolding the fixpoint for some number greater than $i$; here we use $M(\overline{m})$ to denote the elements of $\mathbf{E}$ that have nested timeouts $\overline{m^{\prime}}$ where $\overline{m}^{\prime}$ is just $\overline{m}$ if the length of $\overline{m}$ is even and where $\overline{m}^{\prime}$ is obtained from $\overline{m}$ by decreasing the last element of $\overline{m}$ by one if the length of $\overline{m}$ is odd. Thus there is, for each $v\in\mathbf{E}$, some least (by lexicographic ordering) vector, denoted by $\overline{m}_{v}$, such that $v$ has nested timeouts $\overline{m}_{v}$. Now we define a pre-semi-tableau for $\chi$ over $\mathbf{E}$: Put $V=\mathbf{E}\cap\mathsf{states}$, $U=\mathbf{E}\cap\mathsf{prestates}$ and $W=V\cup U$. Given a pre-state $u\in U$ with nested timeouts $\overline{m}_{u}$, since $u\in\mathbf{E}$, there is a $b$ such that $\delta(u,(\psi_{u},b))\in M(m_{l},\ldots,m_{\beta(u,(\psi_{u},b),\delta(u,(\psi_{u},b)))})$. We put $L(u,(\psi_{u},b))=\delta(u,(\psi_{u},b))$ and $L(u,\sigma)=\emptyset$ for all $\sigma\in\Sigma$ with $\sigma\neq(\psi_{u},b)$. Given a state $v\in V$ with nested timeouts $\overline{m}_{v}$, since $v\in\mathbf{E}$, we have $T(\bigcup_{1\leq i<l}M_{i}(v))\cap[\![l(v)]\!]_{1}\neq\emptyset$ where $M_{i}(v)$ is the set of the labels of nodes $u\in M(m_{l},\ldots,m_{i})$ for which there is a selection $\sigma$ with $\beta(v,\sigma,\delta(v,\sigma))=i$. For each such $\sigma$, we put $L(v,\sigma)=\delta(v,\sigma)$; for all other $\sigma\in\Sigma$, we put $L(v,\sigma)=\emptyset$. As $L\subseteq\delta$, as $T(L(v))\cap[\![l(v)]\!]_{1}\neq\emptyset$ for states $v\in V$, as for pre-states $u\in U$, there is exactly one $b\in\{0,1\}$ such that $L(u,(\psi_{u},b))=\{v\}$ and for all other $\sigma\in\Sigma$, $L(u,\sigma)=\emptyset$, and as there is – by guardedness of fixpoint-variables – no $L$-cycle in $U$, $(W,L)$ is a pre-semi-tableau. Since we constructed $L$ in such a way that nested timeouts are respected and since nested and tracking timeouts both are defined by means of $\beta$, $(W,L)$ has tracking timeouts too.

The proof for the converse direction is analogous: Let $(W,L)$ be a pre-semi-tableau with tracking timeouts, and with set $V$ of states and set $U$ of pre-states. To show that $W$ is contained in the fixpoint $\mathbf{E}$, we proceed by nested induction and coinduction using tracking timeouts as termination measure for the induction parts of the proof. For pre-states $u\in U$ with minimal tracking timeouts $\overline{m}_{u}=(m_{l},\ldots,m_{j})$, there is a single $b$ such that $L(u,(\psi_{u},b))=\delta(u,\psi_{u},b)=:v$ where $v$ has tracking timeouts $\overline{m}_{v}$ that are obtained from $\overline{m}_{u}$ by decreasing the $i$-th element by 1 if $\beta(u,(\psi,b),\delta(u,(\psi,b)))=i$ for $i$ odd; thus $u$ is contained in $f(M(m_{l},\ldots,m_{1}),\ldots,M(m_{l}))$, as required. For states $v\in V$ with minimal tracking timeouts $\overline{m}_{v}=(m_{l},\ldots,m_{j})$, we have $T(L(v))\cap[\![l(v)]\!]_{1}\neq\emptyset$ where, again, the timeouts $\overline{m}_{u}$ for nodes $u\in L(v)$ are determined by $\overline{m}_{v}$ and $\beta(v,\sigma,\delta(v,\sigma))$, where $\sigma$ is such that $\delta(v,\sigma)=u$. Again we have $v\in f(M(m_{l},\ldots,m_{1}),\ldots,M(m,l))$, as required. ∎

Full proof of Lemma 6: Let $(V,\xi)$ be a strongly coherent coalgebra that is built over a pre-semi tableau $(W,L)$ with unfolding timeouts and set of states $V$ and let $v\in\widehat{[\![\psi]\!]}$. We proceed by induction over $\psi$. The interesting cases are the cases with $\psi=\eta X.\psi^{\prime}$ for some fixpoint variable $X\in\mathbf{V}$ and formula $\psi^{\prime}$. In this case we start a second induction over the number $o$ of closed fixpoint operators that are a subformula of $\psi$. If $o=1$, then $\psi^{\prime}$ contains no further closed fixpoint operators. If $o>1$, then we have, for any fixpoint formula $\eta^{\prime}Y.\phi$ that is a subformula of $\psi^{\prime}$, that $\widehat{[\![\eta^{\prime}Y.\phi]\!]}\subseteq[\![\eta^{\prime}Y.\phi]\!]$ by the induction hypothesis. In both cases we are done if we reach a closed formula in the proof by nested induction and coinduction below. Since $(W,L)$ has unfolding timeouts, there is some vector $\overline{m}^{\prime}$ such that $v\in\widehat{[\![\psi]\!]}_{\overline{m}^{\prime}}$. It thus suffices to show that for all timeout vectors $\overline{m}$ and formulas $\psi\in\mathbf{F}$, we have $\widehat{[\![\psi]\!]}_{\overline{m}}\subseteq[\![\psi]\!]_{\overline{m}}$. We show this by nested induction and coinduction, using $(\overline{m},|\psi|)$ as termination measure and distinguishing upon the shape of $\psi$. We consider just the two interesting cases where $\psi=\heartsuit\psi^{\prime}$ and where $\psi=\mu X.\psi^{\prime}$. The former case is directly finished by strong coherence of $(V,\xi)$. In the latter case we use the fact that the unfolding of least fixpoint formulas reduces unfolding timeouts so that we have $v\in\widehat{[\![\psi^{\prime}[X\mapsto\psi]]\!]}_{\overline{m}^{\prime}}$ where $\overline{m}^{\prime}$ is obtained from $\overline{m}$ by reducing $m_{\mathsf{ad}(\psi)}$ by 1 and leaving all other timeouts unchanged so that we have $\overline{m^{\prime}}<\overline{m}$ and the induction hypothesis finishes the case. Thus we have shown that for all timeout vectors $\overline{m}$ and formulas $\psi\in\mathbf{F}$, $\widehat{[\![\psi]\!]}_{\overline{m}}\subseteq[\![\psi]\!]_{\overline{m}}$ which in particular implies $v\in[\![\psi]\!]_{\overline{m}^{\prime}}$, as required. ∎

Lemma 8

Let $G$ be a finite set, let $f:\mathcal{P}(G)\to\mathcal{P}(G)$ be a monotone function and let $n$ be a number such that $\mu f=f^{n}(\emptyset)$. Then we have that for all ordinal numbers $\lambda\geq n$,

[TABLE]

Proof

The proof is by transfinite induction over $\lambda$. If $\lambda=0$, then $n=0$ and $\mu f=f^{0}(\emptyset)=\emptyset$ so that we are done. If $\lambda=\lambda^{\prime}+1$, then we have $f^{\lambda^{\prime}+1}(\emptyset)=f(f^{\lambda^{\prime}}(\emptyset))=f(\mu f)=\mu f$, where the second equality is by the induction hypothesis and the third equality holds since $\mu f$ is a fixpoint. If $\lambda$ is a limit-ordinal, then we have $f^{\lambda}(\emptyset)=\bigcup_{k<\lambda}f^{k}(\emptyset)$. By the induction hypothesis, we have $f^{k}(\emptyset)=\mu f$ for all $k<\lambda$ so that we are done.∎

Full proof of Lemma 7: Let $(C,\xi)$ be a coalgebra and let $x\in C$ be a state with $x\models\chi$. We put $M=\{v\in D_{\chi}\mid\exists y\in C.\,y\models l(v)\}$, $V=M\cap\mathsf{states}$, $U=M\cap\mathsf{prestates}$ and define a pre-semi-tableau over $M$ in a timeout-respecting manner: Let $u\in U$ and $y\in C$ with $y\models u$. Also let $\overline{m}$ be the least (by lexicographic ordering) vector of ordinal numbers such that $y\in[\![\psi_{u}]\!]_{i}^{\overline{m}}$. If $\psi_{u}$ is a disjunction $\psi_{1}\vee\psi_{2}$, then we choose $b$ such that $y\in[\![\psi_{b}]\!]_{i}^{\overline{m}}$. Otherwise, we put $b=0$. Then we put $L(u,(\psi_{u},b))=\delta(u,(\psi_{u},b))$ and $L(u,\sigma)=\emptyset$ for all other $\sigma\in\Sigma$. For $v\in V$ and $y\in C$ with $y\models v$, we have $\xi(x)\in\bigcap_{\heartsuit\psi\in l(v)}[\![\heartsuit]\!][\![\psi]\!]$. For each $\heartsuit\psi\in l(v)$, there even is a least vector $\overline{m}_{\heartsuit\psi}$ of ordinal numbers such that $\xi(x)\in[\![\heartsuit]\!]([\![\psi]\!]_{i}^{\overline{m}_{\heartsuit\psi}})$. For each $\sigma\in\mathsf{selections}$, we define $L(v,\sigma)=\delta(v,\sigma)$. For all other $\sigma\in\Sigma$, we put $L(v,\sigma)=\emptyset$. Then we have $L\subseteq\delta$, $T(L(v))\cap[\![l(v)]\!]_{1}\neq\emptyset$ for states $v\in V$, for pre-states $u\in U$, there is exactly one $b\in\{0,1\}$ such that $L(u,(\psi_{u},b))=\{v\}$ and for all other $\sigma\in\Sigma$, $L(u,\sigma)=\emptyset$, and there is – by guardedness of fixpoint-variables – no $L$-cycle in $U$; thus $(M,L)$ is a pre-semi-tableau.

It remains to show that $(M,L)$ has unfolding timeouts. We let $v\in M$, $\psi\in l(v)$ and $y\in C$ with $y\models l(v)$ where $y\in[\![\psi]\!]_{i}^{\overline{m}}$ and proceed by nested transfinite induction and coinduction over $\overline{m}$. We have to show that if a least fixpoint formula that is a subformula of $\psi$ is satisfied at $y$ in the model with unfolding timeout $\lambda$, then this fixpoint is unfolded in $(M,L)$ at most finitely often before being satisfied when starting from $v$. The latter can be shown by proving containment of $v$ in the least fixpoint of a suitable function $h$, i.e. in $\mu h=h^{n}(\emptyset)$ for some finite number $n$. By Lemma 8, we have $h^{n}(\emptyset)=h^{\lambda}(\emptyset)$. Thus we proceed by transfinite induction over $\overline{m}=(m_{k},\ldots,m_{1})$ and distinguish upon the shape of $\psi$. The interesting case is the case with $\psi=\mu X.\psi^{\prime}$ for some fixpoint variable $X\in\mathbf{V}$ and formula $\psi^{\prime}$ with $j:=\mathsf{ad}(\psi^{\prime})$ even. If $m_{j-1}$ is [math], then we have $[\![\mu X.\psi^{\prime}]\!]_{i}^{0}=\emptyset$ so that there is nothing to show. If $m_{j-1}$ is $\lambda^{\prime}+1$ for some ordinal number $\lambda^{\prime}$, then the induction hypothesis finishes the proof, using the fact that $[\![\mu X.\psi]\!]_{i}^{\lambda^{\prime}+1}=[\![\psi[X\mapsto\mu X.\psi]]\!]_{i}^{\lambda^{\prime}}$ and that unfolding timeouts in the pre-semi-tableau $(M,L)$ count unfoldings in the overlaying model. If $m_{j-1}$ is a limit ordinal, then we have $y\in[\![\mu X.\psi]\!]^{m_{j-1}}(\emptyset)=\bigcup_{k<m_{j-1}}[\![\mu X.\psi]\!]^{k}(\emptyset)$. By the induction hypothesis, $y\in[\![\mu X.\psi]\!]^{k}(\emptyset)$ implies $v\in h^{k}(\emptyset)$ for all $k<m_{j-1}$ so that we have $v\in\bigcup_{k<m_{j-1}}h^{k}(\emptyset)=h^{m_{j-1}}(\emptyset)$, as required.∎