Global Caching for the Flat Coalgebraic µ-Calculus

Abstract

We present a sound, complete, and optimal single-pass tableau algorithm for the alternation-free µ-calculus. The algorithm supports global caching with intermediate propagation and runs in time 2 O(n) . In game-theoretic terms, our algorithm integrates the steps for constructing and solving the Büchi game arising from the input tableau into a single procedure; this is done onthe-fly, i.e. may terminate before the game has been fully constructed. This suggests a slogan to the effect that global caching = game solving on-the-fly. A prototypical implementation shows promising initial results.Parikh's game logic [31], or probabilistic fixpoint logic. To aid readability, we phrase our results in terms of the relational µ-calculus, and discuss the coalgebraic generalization only at the end of Section 4. The model construction in the completeness proof yields models of size 2 O(n) .We have implemented of our algorithm as an extension of the Coalgebraic Ontology Logic Reasoner COOL, a generic reasoner for coalgebraic modal logics [21]; given the current state of the implementation of instance logics in COOL, this means that we effectively support alternation-free fragments of relational, monotone, and alternating-time [1] µ-calculi, thus in particular covering CTL and ATL. We have evaluated the tool in comparison with existing reasoners on benchmark formulas for CTL [18] (which appears to be the only candidate logic for which well-developed benchmarks are currently available) and on random formulas for ATL and the alternation-free relational µ-calculus, with promising results; details are discussed in Section 5. Related WorkThe theoretical upper bound ExpTime has been established for the full coalgebraic µ-calculus [5] (and earlier for instances such as the alternating-time µ-calculus AMC [35]), using a multi-pass algorithm that combines games and automata in a similar way as for the standard relational case, in particular involving the Safra construction. Global caching has been employed successfully for a variety of description logics [17, 20], and lifted to the level of generality of coalgebraic logics with global assumptions [15] and nominals [16].A tableaux-based non-optimal (NExpTime) decision procedure for the full µ-calculus has been proposed in [23]. Friedmann and Lange [12] describe an optimal tableau method for the full µ-calculus that, unlike most other methods including the one we present here, makes do without requiring guardedness. Like earlier algorithms for the full µ-calculus, the algorithm constructs and solves a parity game, and in principle allows for an on-thefly implementation. The models constructed in the completeness proof are asymptotically larger than ours, but presumably the proof can be adapted for the alternation-free case by using determinization of co-Büchi automata [28] instead of Safra's determinization of Büchi automata [34] to yield models of size 2 O(n) , like ours. For non-relational instances of the coalgebraic µ-calculus, including the alternation-free fragment of the alternatin...

Quasipolynomial Computation of Nested Fixpoints

“…It has been shown in previous work [30] that model checking for coalgebraic µ-calculi against coalgebras with state space U reduces to solving a canonical fixpoint equation system over the powerset lattice P(U ), where the involved function interprets modal operators using predicate liftings, as described in [14,30]. This canonical equation system can alternatively be seen as the winning region of Eloise in coalgebraic parity games, a highly general variant of parity games where the game structure is a coalgebra and nodes are annotated with modalities.…”

“…This level of generality is achieved by abstracting system types as set functors and systems as coalgebras for the given functor following the paradigm of universal coalgebra [49]. It was previously shown [30] that the model checking problem for coalgebraic µ-calculi reduces to the computation of a nested fixpoint. This fixpoint may be seen as a coalgebraic generalization of a parity game winning region but can be literally phrased in terms of small standard parity games (implying quasipolynomial run time) only in restricted cases.…”

Permutation Games for the Weakly Aconjunctive $$\mu $$ μ -Calculus

“…The use of games in μ-calculus satisfiability checking goes back to Niwiński and Walukiewicz [18] and has since been extended to the unguarded μcalculus [10] and the coalgebraic μ-calculus [2]. Game-based procedures for the relational μ-calculus have been implemented in MLSolver [9], and for the alternation-free coalgebraic μ-calculus in COOL [13].…”

“…Similarly, if Eloise wins a parity game with n priorities, then she can ensure that in each play, each odd priority 1 ≤ i ≤ n is visited only finitely often, unless a priority greater than i is visited infinitely often (the converse does not hold in general [4]): Additionally, we devise two series of unsatisfiable formulas that exhibit the advantages of COOL's global caching and on-the-fly-solving capabilities. These formulas are inspired by the CTL-formula series early(n, j, k) and early gc (n, j, k) from [13] but contain fixpoint-alternation of depth 2 k inside the subformula θ:…”

“…The resulting decision procedure generalizes to the weakly aconjunctive coalgebraic μ-calculus, thus covering also, e.g., probabilistic and alternatingtime versions of the μ-calculus. The generic algorithm has been implemented as an extension of the Coalgebraic Ontology Logic Reasoner (COOL) [11,13]. Our implementation constructs and solves the presented permutation games onthe-fly, possibly finishing satisfiability proofs early, and shows promising initial results.…”