3.1 The \(\mu \)-Calculus
We briefly recall the definition of the \(\mu \)-calculus. We fix a set P of propositions, a set A of actions, and a set \(\mathfrak {V}\) of fixpoint variables. The set \(\mathsf {L}_\mu \) of \(\mu \)-calculus formulas is the set of all formulas \(\phi ,\psi \) that can be constructed by the grammar
$$\begin{aligned} \psi ,\phi :\,\!:= \bot \mid \top \mid p \mid \lnot p \mid X \mid \psi \wedge \phi \mid \psi \vee \phi \mid \langle a \rangle \psi \mid [a]\psi \mid \mu X.\, \psi \mid \nu X.\, \psi \end{aligned}$$
where \(p\in P\), \(a\in A\), and \(X\in \mathfrak {V}\); we write \(|\psi |\) for the size of a formula \(\psi \). Throughout the paper, we use \(\eta \) to denote one of the fixpoint operators \(\mu \) or \(\nu \). We refer to formulas of the form \(\eta X.\,\psi \) as fixpoint literals, to formulas of the form \(\langle a\rangle \psi \) or \([a]\psi \) as modal literals, and to p, \(\lnot p\) as propositional literals. The operators \(\mu \) and \(\nu \) bind their variables, inducing a standard notion of free variables in formulas. We denote the set of free variables of a formula \(\psi \) by \(\mathsf {FV}(\psi )\). A formula \(\psi \) is closed if \(\mathsf {FV}(\psi )=\emptyset \), and open otherwise. We write \(\psi \le \phi \) (\(\psi <\phi \)) to indicate that \(\psi \) is a (proper) subformula of \(\phi \). We say that \(\phi \) occurs free in \(\psi \) if \(\phi \) occurs in \(\psi \) as a subformula that is not in the scope of any fixpoint operator. Throughout, we restrict to formulas that are guarded, i.e. have at least one modal operator between any occurrence of a variable X and an enclosing binder \(\eta X\). (This is standard although possibly not without loss of generality [10].) Moreover we assume w.l.o.g. that input formulas are clean, i.e. all fixpoint variables are mutually distinct and distinct from all free variables, and irredundant, i.e. \(X\in \mathsf {FV}(\psi )\) for all subformulas \(\eta X.\,\psi \). We refer to a variable X that is bound by a least (greatest) fixpoint operator \(\mu X.\chi \) (\(\nu X.\chi \)) in a formula \(\phi \) as a \(\mu \)-variable (\(\nu \)-variable) of \(\phi \), and to the process of substituting such an X with its binding fixpoint literal (\(\mu X.\chi \) or \(\nu X.\chi \), respectively) as unfolding. An occurrence of a subformula \(\psi \) of a formula \(\phi \) contains an active \(\mu \)-variable [15] if \(\psi \) can be converted into a formula containing a free occurrence of a \(\mu \)-variable of \(\phi \) by repeatedly unfolding \(\nu \)-variables of \(\phi \).
Formulas are evaluated over Kripke structures \(\mathcal {K}=(W,(R_a)_{a\in A},\pi )\), consisting of a set W of states, a family \((R_a)_{a\in A}\) of relations \(R_a\subseteq W\times W\), and a valuation \(\pi :P\rightarrow \mathcal {P}(W)\) of the propositions. Given an interpretation \(i:\mathfrak {V}\rightarrow \mathcal {P}(W)\) of the fixpoint variables, define \({[\![\psi ]\!]}_i\subseteq W\) by the obvious clauses for Boolean operators and propositions, \([\![X]\!]_i=i(X)\), \([\![\langle a\rangle \psi ]\!]_i=\{v\in W\mid \exists w\in R_a(v).w\in [\![\psi ]\!]_i\}\), \([\![[a]\psi ]\!]_i=\{v\in W\mid \forall w\in R_a(v).w\in [\![\psi ]\!]_i\}\), \([\![\mu X.\,\psi ]\!]_i =\mu [\![\psi ]\!]^X_i\) and \([\![\nu X.\,\psi ]\!]_i =\nu [\![\psi ]\!]^X_i\), where \(R_a(v)=\{w\in W\mid (v,w)\in R_a\}\), \([\![\psi ]\!]^X_i(G) = [\![\psi ]\!]_{i[X\mapsto G]}\), and \(\mu \), \(\nu \) take least and greatest fixpoints of monotone functions, respectively. If \(\psi \) is closed, then \([\![\psi ]\!]_i\) does not depend on i, so we just write \([\![\psi ]\!]\). We denote the Fischer-Ladner closure [16] of a formula \(\phi \) by \(\mathbf {F}(\phi )\), or just by \(\mathbf {F}\), if no confusion arises; intuitively, \(\mathbf {F}\) is the set of formulas that can arise as subformulas when unfolding each fixpoint operator in \(\phi \) at most once. We note \(\mathbf {F}\le |\phi |\) [16].
The aconjunctive fragment [15] of the \(\mu \)-calculus is obtained by requiring that for all conjunctions that occur as a subformula, at most one of the conjuncts contains an active \(\mu \)-variable. In the weakly aconjunctive fragment [24], this requirement is loosened to the constraint that all conjunctions that occur as a subformula and contain an active \(\mu \)-variable are of the shape \(\psi \wedge \Diamond \psi _1\wedge \ldots \wedge \Diamond \psi _n\wedge \Box (\psi _1\vee \ldots \vee \psi _n)\), where \(\psi \) does not contain active \(\mu \)-variables. For instance, for all n, the formula \(\eta X_n\ldots \mu X_1. \nu X_0. \bigvee _{0\le i\le n} (q_i\wedge \Diamond X_i)\) is aconjunctive (and equivalent to the weakly aconjunctive formula obtained by replacing \(\Diamond X_i\) with \(\Diamond X_i\wedge \Diamond \top \wedge \Box (X_i\vee \top )\)). The permutation satisfiability games that we introduce work for the more expressive weakly aconjunctive fragment.
We will make use of the standard tableau rules [10] (each consisting of one premise and a possibly empty set of conclusions):
$$\begin{aligned} (\bot )\quad&\;\;\;\;\quad \quad \frac{\Gamma ,\bot }{}&(\lightning )\quad&\quad \quad \frac{\Gamma ,p,\lnot p}{}&(\wedge )\quad&\;\quad \quad \frac{\Gamma ,\psi \wedge \phi }{\Gamma ,\psi ,\phi } \\ (\vee ) \quad&\quad \frac{\Gamma ,\psi \vee \phi }{{\Gamma ,\psi }\qquad {\Gamma ,\phi }}&(\langle a\rangle ) \quad&\frac{\Gamma ,[a] \psi _1,\ldots ,[a]\psi _n,\langle a\rangle \phi }{\psi _1,\ldots ,\psi _n,\phi }&(\eta ) \quad&\frac{\Gamma ,\eta X.\, \psi }{\Gamma ,\psi [X\mapsto \eta X.\, \psi ]} \end{aligned}$$
(for \(a\in A\), \(p\in P\)); we refer to the tableau rules by \(\mathcal {R}\) and usually write rule applications with premise \(\Gamma \) and conclusion \(\varSigma =\Gamma _1,\ldots ,\Gamma _n\) sequentially: \((\Gamma /\varSigma )\).
To track fixpoint formulas through pre-tableaux, we will use deferrals, that is, the decomposed form of formulas that are obtained by unfolding fixpoint literals.
Definition 13
(Deferrals). Given fixpoint literals \(\chi _i = \eta X_i.\,\psi _i\), \(i=1,\dots ,n\), we say that a substitution \(\sigma =[X_1\mapsto \chi _1];\ldots ;[X_n\mapsto \chi _n]\) sequentially unfolds \(\chi _n\) if \(\chi _i <_f \chi _{i+1}\) for all \(1\le i<n\), where we write \(\psi <_f \eta X.\,\phi \) if \(\psi \le \phi \) and \(\psi \) is open and occurs free in \(\phi \) (i.e. \(\sigma \) unfolds a nested sequence of fixpoints in \(\chi _n\) innermost-first). We say that a formula \(\chi \) is irreducible if for every substitution \([X_1\mapsto \chi _1];\ldots ;[X_n\mapsto \chi _n]\) that sequentially unfolds \(\chi _n\), we have that \(\chi = \chi _1([X_2\mapsto \chi _2];\ldots ;[X_n\mapsto \chi _n])\) implies \(n=1\) (i.e. \(\chi =\chi _1\)). A formula \(\psi \) belongs to an irreducible closed fixpoint literal \(\theta _n\), or is a \(\theta _n\)-deferral, if \(\psi =\alpha \sigma \) for some substitution \(\sigma = [X_1\mapsto \theta _1];\ldots ;[X_n\mapsto \theta _n]\) that sequentially unfolds \(\theta _n\) and some \(\alpha <_f \theta _1\). We denote the set of \(\theta _n\)-deferrals by \(\mathsf {dfr}(\theta _n)\).
E.g. the substitution \(\sigma =[Y\mapsto \mu Y.\,(\Box X\wedge \Diamond \Diamond Y)];[X\mapsto \theta ]\) sequentially unfolds the irreducible closed formula \(\theta =\nu X.\,\mu Y.\,(\Box X\wedge \Diamond \Diamond Y)\), and \((\Diamond Y)\sigma =\Diamond \mu Y.\,(\Box \theta \wedge \Diamond \Diamond Y)\) is a \(\theta \)-deferral. A fixpoint literal is irreducible if it is not an unfolding \(\psi [X\mapsto \eta X.\,\psi ]\) of a fixpoint literal \(\eta X.\,\psi \); in particular, every clean irredundant fixpoint literal is irreducible.
As a technical tool, we define a measure for the depth of alternation at which a deferral resides inside the fixpoint to which it belongs:
Definition 14
(Alternation level and alternation depth). The alternation level \(\mathsf {al}(\phi \sigma ):=\mathsf {al}(\sigma )\) of a deferral \(\phi \sigma \) is defined inductively over \(|\sigma |\), where \(\mathsf {al}(\epsilon )=\mathsf {al}(\epsilon )_\mu =\mathsf {al}(\epsilon )_\nu =0\), for the empty substitution \(\epsilon \), \(\mathsf {al}(\sigma ;[X\mapsto \eta X.\,\psi ])=\mathsf {al}(\sigma )_\mu +1\) if \(\eta =\mu \) and \(\mathsf {al}(\sigma ;[X\mapsto \eta X.\,\psi ])=\mathsf {al}(\sigma )_\nu \) otherwise, and
$$\begin{aligned} \mathsf {al}(\sigma ;[X\mapsto \eta X.\,\psi ])_\mu = {\left\{ \begin{array}{ll} \mathsf {al}(\sigma )_\mu &{}\text {if }\eta =\mu \\ \mathsf {al}(\sigma )_\nu +1&{}\text {otherwise} \end{array}\right. } \\ \mathsf {al}(\sigma ;[X\mapsto \eta X.\,\psi ])_\nu = {\left\{ \begin{array}{ll} \mathsf {al}(\sigma )_\nu &{}\text {if }\eta =\nu \\ \mathsf {al}(\sigma )_\mu +1&{}\text {otherwise} \end{array}\right. } \end{aligned}$$
This definition assigns greater numbers to inner fixpoint literals, i.e. to deferrals which occur at higher nesting depth, i.e. with more alternation inside their sequence \(\sigma \). Given a formula \(\psi \), its alternation depth \(\mathsf {ad}(\phi )\) is defined as \(\mathsf {ad}(\phi )= \max \{\mathsf {al}(\delta )\mid \delta \in \mathbf F,\exists \theta .\delta \in \mathsf {dfr}(\theta )\}\).
3.2 Limit-Deterministic Tracking Automata
As a first step towards deciding the satisfiability of a weakly aconjunctive \(\mu \)-calculus formula \(\phi \), we now construct a tracking automaton that takes branches of (that is, infinite paths through) standard pre-tableaux for \(\phi \) as input and accepts a branch if and only if it contains a least fixpoint formula whose satisfaction is deferred indefinitely on that branch. To this end, we import the following notions of threads and tableaux from [10]:
Definition 15
A pre-tableau for a formula \(\phi \) is a graph the nodes of which are labelled with subsets of the Fischer-Ladner closure \(\mathbf F\); the graph structure L of a pre-tableau is constructed by applying tableau rules from \(\mathcal {R}\) to the labels of nodes with the requirement that for each rule application \((\Gamma /\varSigma )\) to the label \(\Gamma \) of a node v, there is a w with \((v,w)\in L\) such that the label of w is contained in \(\varSigma \). Nodes whose labels are saturated (i.e. do not contain propositional or fixpoint operators) are called states. Formulas are tracked through rule applications by the connectedness relation \(\leadsto \subseteq (\mathcal {P}(\mathbf F)\times \mathbf F)^2\) that is defined by putting \(\Phi ,\phi \leadsto \Psi ,\psi \) if and only if \(\Psi \) is a conclusion of an application of a rule from \(\mathcal {R}\) to \(\Phi \) such that \(\phi \in \Phi \), \(\psi \in \Psi \), and the rule application transforms \(\phi \) to \(\psi \); if the rule application does not change \(\phi \), then \(\phi =\psi \). E.g. we have \(\Phi ,\psi _1\wedge \psi _2\leadsto \Psi ,\psi _i\), where \(i\in \{1,2\}\) and \(\Psi \) is obtained from \(\Phi \) by applying the rule \((\wedge )\) to \(\psi _1\wedge \psi _2\). A branch \(\Psi _0,\Psi _1\ldots \) in a pre-tableau is a sequence of labels such that for all \(i>0\), \(\Psi _{i+1}\) is an L-successor of \(\Psi _i\). A thread on an infinite branch \(\Psi _0,\Psi _1,\ldots \) is an infinite sequence \(t=\psi _0,\psi _1\ldots \) of formulas with \(\Psi _0,\psi _0\leadsto \Psi _1,\psi _1\leadsto \ldots \). A \(\mu \)-thread is a thread t such that \(\min (\mathsf {Inf}(\mathsf {al}\circ t))\) is odd, i.e. the outermost fixpoint literal that is unfolded infinitely often in t is a least fixpoint literal. A bad branch is an infinite branch that contains a \(\mu \)-thread. A tableau for \(\phi \) is a pre-tableau for \(\phi \) that does not contain bad branches.
We import from [10] the well-known fact that the existence of tableaux in the sense defined above characterizes satisfiability. In [10], the result is shown for the more general unguarded \(\mu \)-calculus; we note that the restriction to guarded formulas does not invalidate the theorem.
Theorem 16
([10]). A \(\mu \)-calculus formula \(\psi \) is satisfiable if and only if there is a tableau for \(\psi \).
Given a formula \(\phi \), we define the alphabet \(\varSigma _{\phi }\) to consist of letters that each identify a rule \(R\in \mathcal {R}\), a principal formula from \(\mathbf {F}\) and one of the conclusions of R. E.g. the letter \(((\vee ),0,p\vee \Diamond q)\) identifies the application of the disjunction rule to a principal formula \(p\vee \Diamond q\) and the choice of the left conclusion; thus this letter identifies the transition from \(p\vee \Diamond q\) to p by use of rule \((\vee )\). We note \(|\varSigma _{\phi }|\in \mathcal {O}(|\phi |)\). Further, we denote the set of all words that encode some branch and some bad branch in some pre-tableau for \(\phi \) by \(\mathsf {Branch}(\phi )\) and \(\mathsf {BadBranch}(\phi )\), respectively.
As a crucial result, we now show that limit-deterministic automata are expressive enough to exactly recognize the bad branches in pre-tableaux for weakly aconjunctive formulas.
Lemma 17
Let \(\phi \) be a weakly aconjunctive formula. Then there is a limit-deterministic PA \(\mathcal {A}=(V,\varSigma _{\phi },\delta ,\phi ,\alpha )\) with \(|V|\le |\phi |\) and \(\mathsf {idx}(\mathcal {A})\le \mathsf {ad}(\phi )+1\) such that \(L(\mathcal {A})\cap \mathsf {Branch}(\phi )=\mathsf {BadBranch}(\phi )\).
Proof
(Sketch). The automaton nondeterministically guesses formulas to be tracked, one at a time; the set of states of the automaton is the Fischer-Ladner closure of \(\phi \). The priorities of the transitions in the automaton are derived from the alternation level of the target formula of the respective transition; then every word \(w\in L(\mathcal {A})\) that encodes some branch encodes a bad branch. Once a deferral is tracked, weak aconjunctivity implies that all compartments to which the tracked formula belongs are internally deterministic; this is the case since for conjunctions \(\psi =\psi _0\wedge \Diamond \psi _1\wedge \ldots \wedge \Diamond \psi _n\wedge \Box (\psi _1\vee \ldots \vee \psi _n)\) – the only case that can introduce nondeterminism – each next modal step determines just one of the formulas \(\psi _i\) that has to be tracked; the conjunct \(\psi _0\) does not contain active \(\mu \)-variables, so tracking it causes the automaton to leave all compartments to which \(\psi \) belongs. Thus the automaton is limit-deterministic. \(\square \)
Example 18
We consider the aconjunctive formula
$$\begin{aligned} \phi =\mu X.(\;p\wedge \nu Y.\;(\Diamond (Y\wedge p)\vee \Diamond X)) \end{aligned}$$
which expresses the existence of a finite or infinite path on which p holds everywhere. We have the \(\phi \)-deferrals \(\phi \epsilon \), \(\psi :=(p\wedge \nu Y.\;(\Diamond (Y\wedge p)\vee \Diamond X))\sigma _1\), \(\theta :=(\nu Y.\;(\Diamond (Y\wedge p)\vee \Diamond X))\sigma _1\), \(\chi :=(\Diamond (Y\wedge p)\vee \Diamond X)\sigma _2\), \((\Diamond (Y\wedge p))\sigma _2\), \(\tau :=(Y\wedge p)\sigma _2\), \(Y\sigma _2\), \(\Diamond X\sigma _2\) and \(X\sigma _2\), where \(\sigma _1=[X\mapsto \phi ]\) and \(\sigma _2=[Y\mapsto \psi ];\sigma _1\). We consider a pre-tableau \(P_\phi \) for \(\phi \) and like in the proof of Lemma 17, we construct the limit-deterministic tracking automaton \(\mathcal {A}_\phi \), depicted below:
The priorities in \(\mathcal {A}_\phi \) are derived as follows: As \(\mathsf {ad}(\phi )=2\) is even, we put \(k=\mathsf {ad}(\phi )+1=3\); since \(\mathsf {al}(\phi )=\mathsf {al}(\psi )=1\), \(\alpha (\phi ,(\mu ),\psi )=\alpha (\Diamond \phi ,(\Diamond ),\phi )=k-\mathsf {al}(\phi )=2\) and since \(\mathsf {al}(p)=0\), \(\alpha (\psi ,(\wedge ),p)=\alpha (\varsigma ,(\wedge ),p)=k-\mathsf {al}(\phi )=3\). All other formulas have alternation level 2 and transitions to them obtain priority 1. The tracking automaton accepts exactly those branches in \(P_\phi \) that start at node 1 and take the loop through node 9 infinitely often; in these branches, \(\phi \) can be tracked forever and evolves to \(\phi \) infinitely often, i.e. their dominating formula is the least fixpoint formula \(\phi \). All other branches loop through node 7 without passing node 9 from some point on; their dominating fixpoint formula is \(\theta \), a greatest fixpoint formula. We observe that due to the aconjunctivity of \(\phi \), \(\mathcal {A}_\phi \) is limit-deterministic since the only two nondeterministic states \(\psi \) and \(\varsigma \) each have only one outgoing \((\wedge )\)-transition with priority less than \(k=3\).
Given a weakly aconjunctive formula \(\phi \), we use Lemma 17 to construct a limit-deterministic tracking automaton \(\mathcal {A}_\phi \) with \(L(\mathcal {A}_\phi )\cap \mathsf {Branch}(\phi )=\mathsf {BadBranch}(\phi )\). Then we put Lemma 11 to use to obtain an equivalent BA in which all states from \(Q=\mathsf {reach}(\pi _3[F])\) are levelled deferrals, i.e. pairs \((\psi ,q)\) consisting of a deferral \(\psi \) and a number \(q\le \lceil \frac{k}{2}\rceil \), the level of the pair \((\psi ,q)\); the level q encodes the odd alternation level \(2q-1\). A levelled deferral \((\psi ,q)\) is active if \(\mathsf {al}(\psi )=2q-1\) and the automaton accepts branches which contain a levelled deferral that is active infinitely often without being finished. The set \(\overline{Q}\) is just a subset of \(\mathbf {F}\). Next we use Theorem 7 to transform this BA to a DPA \(\mathcal {B}_\phi \) with \(L(\mathcal {A}_\phi )=L(\mathcal {B}_\phi )\). We complement \(\mathcal {B}_\phi \) to a DPA \(\mathcal {C}_\phi =(W,\varSigma _{\phi },\delta ,\phi ,\alpha )\) by decreasing the priority of each state in \(\mathcal {B}_\phi \) by one; we have \(L(\mathcal {C}_\phi )=\overline{L(\mathcal {B}_\phi )}\), that is, \(\mathcal {C}_\phi \) accepts exactly those words that encode only ‘good’ branches, if they encode some branch in some pre-tableau for \(\phi \). By construction, \(|W|\in \mathcal {O}((nk)!)\) and \(\mathcal {C}_\phi \) has at most \(nk+1\) priorities, and (recalling Definitions 6 and 10) the states in the carrier W of \(\mathcal {C}_\phi \) are of the shape (U, l), where U is a subset of \(\mathbf {F}\) and l is a partial permutation of levelled deferrals. For a transition \(t=((U,l),r,(V,l'))\) with \((U,l),(V,l')\in W\), \(r\in \varSigma _{\phi }\), if \(\alpha (t)=2(n-a)+1\), then a is the lowest number such that \(\mathsf {al}(\phi )=2q-1\), where \(l'(a)=(\phi ,q)\) and the a-th element of l is not removed by the transition t (i.e. \(\alpha (t)\) references the oldest levelled deferral in \(l'\) that is active but not removed by the transition t) and if \(\alpha (t)=2(n-r)+2\), then \(\alpha (t)\) is the index of the oldest levelled deferral \((\phi ,2q-1)\) that is finished (i.e. removed from l) in the transition t of the automaton \(\mathcal {C}_\phi \), which means that the according r-transition in \(\mathcal {A}_\phi \) makes \(\phi \) leave its \(2q-1\)-compartment. For a state \(v=(U,l)\), we define the label \(\Gamma (v)\) of v as \(\Gamma (v)=U\).
3.3 Permutation Games
The deterministic parity automaton \(\mathcal {C}_\phi \) can now be combined with applications of tableau rules from \(\mathcal {R}\) to form a satisfiability game for \(\phi \). We proceed to recall the definition of parity games and some ensuing basic notions. A parity game is a graph \(\mathcal {G}=(V,E,\alpha )\) that consists of a set of nodes V, a set of edges \(E\subseteq V\times V\) and a priority function \(\alpha :E\rightarrow \mathbb {N}\), assigning priorities to edges. We assume
, that is, every node in V either belongs to player \(\mathsf {Eloise}\) (\(V_\exists \)) or to player \(\mathsf {Abelard}\) (\(V_\forall \)). A play \(\rho \) of \(\mathcal {G}\) is a (possibly infinite) sequence \(v_0v_1\ldots \) such that for all \(i\ge 0\), \(v_i\in V\) and \((v_i,v_{i+1})\in E\). A play \(\rho \) of \(\mathcal {G}\) is won by \(\mathsf {Eloise}\) if and only if \(\rho \) is finite and ends in a node that belongs to \(\mathsf {Abelard}\) or \(\rho \) is infinite and \(\max (\mathsf {Inf}(\alpha \circ \mathsf {trans}(\rho )))\) is even (where \(\mathsf {trans}(\rho )\) is defined by \(\mathsf {trans}(\rho )(i)=(\rho (i),\rho (i+1))\)); \(\mathsf {Abelard}\) wins a play \(\rho \) if and only if \(\mathsf {Eloise}\) does not win \(\rho \). A (memoryless) strategy
assigns moves to states. A play \(\rho \) conforms to a strategy s if for all \(\rho (i)\in \mathsf {dom}(s)\), \(\rho (i+1)=s(\rho (i))\). \(\mathsf {Eloise}\) has a winning strategy for a node v if there is a strategy \(s:V_\exists \rightarrow V\) such that every play of \(\mathcal {G}\) that starts at v and conforms to s is won by \(\mathsf {Eloise}\); we have a dual notion of winning strategies for \(\mathsf {Abelard}\). The winning regions \(\mathsf {win}_\exists (\mathcal {G})\) and \(\mathsf {win}_\forall (\mathcal {G})\) are the sets of those nodes for which \(\mathsf {Eloise}\) and \(\mathsf {Abelard}\) have winning strategies, respectively. Solving a parity game \(\mathcal {G}\) (locally) for a particular node \(v\in V\) amounts to computing the winner of v.
Now we are ready to define permutation games for weakly aconjunctive formulas \(\phi \), using the DPA \(\mathcal {C}_\phi =(W,\varSigma _{\phi },\delta ,\phi ,\alpha )\) from the previous section.
Definition 19
(Permutation games). Let \(\phi \) be a weakly aconjunctive formula. We define the permutation game \(\mathcal {G}(\phi )=(W,E,\beta )\) to be a parity game that has the carrier of \(\mathcal {C}_\phi \) as set of nodes. For every node \(v\in W\) for which \(\Gamma (v)\) is not a state, we fix a single rule that is to be applied to \(\Gamma (v)\) and a single principal formula \(\psi _v\in \Gamma (v)\) to which the rule is to be applied. If \((\vee )\) is to be applied to \(\Gamma (v)\), then we put \(v\in W_\exists \); otherwise, \(v\in W_\forall \). In particular, all state nodes are contained in \(W_\forall \). For \(v\in W\), we put \(E(v)= \bigcup \{\delta (v,a)\mid a\in \varSigma _v\}\), where \(\varSigma _v\subseteq \varSigma _{\phi }\) consists of all letters a that encode the application of some rule to \(\Gamma (v)\) with the condition that the principal formula of the rule application must be \(\psi _v\) if v is not a state node. Finally, we put \(\beta (v,w)=\alpha (v,a,w)\) for \((v,w)\in E\), where \(a\in \varSigma _v\) encodes the rule application that leads from v to w.
Theorem 20
Let \(\phi \) be a closed, irreducible and weakly aconjunctive formula. Then we have \((\{\phi \},[\,])\in \mathsf {win}_\exists (\mathcal {G}(\phi ))\) if and only if \(\phi \) is satisfiable.
Proof
By construction, \(\mathsf {Eloise}\) wins \((\{\phi \},[\,])\) if and only if there is a tableau for \(\phi \) (labelled by the labelling function \(\Gamma \)); we are done by Theorem 16. \(\square \)
Due to the relatively simple structure and the asymptotically smaller size of the determinized automata \(\mathcal {C}_\phi \), the resulting permutation games are somewhat easier to construct and can be solved asymptotically faster than the structures created by standard satisfiability decision procedures for the full \(\mu \)-calculus (e.g. [5, 10]) which employ the full Safra/Piterman-construction; note however, that our method is restricted to the weakly aconjunctive fragment.
Corollary 21
The satisfiability of weakly aconjunctive \(\mu \)-calculus formulas can be decided by solving parity games of size \(\mathcal {O}((nk)!)\) and \(\mathcal {O}(nk)\) priorities.
The winning strategies for \(\mathsf {Eloise}\) or \(\mathsf {Abelard}\) in these games define models for or refutations of the respective formulas, so that we have
Corollary 22
Satisfiable weakly aconjunctive \(\mu \)-calculus formulas have models of size \(\mathcal {O}((nk)!)\).