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...
Abstract. Coalgebra has in recent years been recognized as the framework of choice for the treatment of reactive systems at an appropriate level of generality. Proofs about the reactive behavior of a coalgebraic system typically rely on the method of coinduction. In comparison to 'traditional' coinduction, which has the disadvantage of requiring the invention of a bisimulation relation, the method of circular coinduction allows a higher degree of automation. As part of an effort to provide proof support for the algebraic-coalgebraic specification language CoCasl, we develop a new coinductive proof strategy which iteratively constructs a bisimulation relation, thus arriving at a new variant of circular coinduction. Based on this result, we design and implement tactics for the theorem prover Isabelle which allow for both automatic and semiautomatic coinductive proofs. The flexibility of this approach is demonstrated by means of examples of (semi-)automatic proofs of consequences of CoCasl specifications, automatically translated into Isabelle theories by means of the Bremen heterogeneous Casl tool set Hets.
The coalgebraic µ-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 µ-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 µ-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 µcalculus with Presburger arithmetic, as well as an extension of the (twovalued) probabilistic µ-calculus with polynomial inequalities. As a side result, we moreover obtain a new upper bound O(((nk)!) 2 ) on minimum model size for satisfiable formulas for all coalgebraic µ-calculi, where n is the size of the formula and k its alternation depth.
We introduce a natural notion of limit-deterministic parity automata and present a method that uses such automata to construct satisfiability games for the weakly aconjunctive fragment of the μ-calculus. To this end we devise a method that determinizes limit-deterministic parity automata of size n with k priorities through limit-deterministic Büchi automata to deterministic parity automata of size O((nk)!) and with O(nk) priorities. The construction relies on limit-determinism to avoid the full complexity of the Safra/Pitermanconstruction by using partial permutations of states in place of Safra-Trees. By showing that limit-deterministic parity automata can be used to recognize unsuccessful branches in pre-tableaux for the weakly aconjunctive μ-calculus, we obtain satisfiability games of size O((nk)!) with O(nk) priorities for weakly aconjunctive input formulas of size n and alternation-depth k. A prototypical implementation that employs a tableau-based global caching algorithm to solve these games on-the-fly shows promising initial results.
This paper presents work in the context of the certification of a safety component for autonomous service robots, and investigates the potential advantages offered by formally modelling the domain knowledge, specification and implementation in a theorem prover in higher-order logic. This allows safety properties to be stated in an abstract manner close to textbook mathematics. The automatic proof checking alleviates correctness concerns, and provides a seamless development process from high-level safety requirements down to concrete implementation. Moreover, the formalisation can be checked for correctness automatically, and the certification review process can focus on the correctness of the specification and safety cases.
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