Reasoning with Global Assumptions in Arithmetic Modal Logics

Kupke, Clemens; Pattinson, Dirk; Schröder, Lutz

doi:10.1007/978-3-319-22177-9_28

Cited by 5 publications

(9 citation statements)

References 38 publications

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“…This example (as well as the previous one) can be extended to admit (monotone) polynomial inequalities among probabilities (or multiplicities, respectively) instead of only comparison with constants, allowing, e.g., for expressing probabilistic independence [23,31,39]. In more detail, we can introduce n-ary modalities says roughly that if we independently sample two successors of the current state, then with probability at least 0.8, the first successor state will satisfy p, and then X again (continuing indefinitely), and the second successor state will remain on a path where it satisfies Y again until it eventually reaches q. as the monotone µ-calculus [22].…”

Section: A2 Details On Applications To Coalgebraic µ-Calculimentioning

confidence: 99%

Quasipolynomial Computation of Nested Fixpoints

Hausmann¹,

Schröder²

2021

Preprint

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It is well-known that the winning region of a parity game with n nodes and k priorities can be computed as a k-nested fixpoint of a suitable function; straightforward computation of this nested fixpoint requires O(n k 2 ) iterations of the function. Calude et al.'s recent quasipolynomial-time parity game solving algorithm essentially shows how to compute the same fixpoint in only quasipolynomially many iterations by reducing parity games to quasipolynomially sized safety games. Universal graphs have been used to modularize this transformation of parity games to equivalent safety games that are obtained by combining the original game with a universal graph. We show that this approach naturally generalizes to the computation of solutions of systems of any fixpoint equations over finite lattices; hence, the solution of fixpoint equation systems can be computed by quasipolynomially many iterations of the equations. We present applications to modal fixpoint logics and games beyond relational semantics. For instance, the model checking problems for the energy µ-calculus, finite latticed µ-calculi, and the graded and the (two-valued) probabilistic µ-calculus -with numbers coded in binary -can be solved via nested fixpoints of functions that differ substantially from the function for parity games but still can be computed in quasipolynomial time; our result hence implies that model checking for these µ-calculi is in QP. Moreover, we improve the exponent in known exponential bounds on satisfiability checking.

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Section: A2 Details On Applications To Coalgebraic µ-Calculimentioning

confidence: 99%

Quasipolynomial Computation of Nested Fixpoints

Hausmann¹,

Schröder²

2021

Preprint

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“…On the other hand, some decidable fragments of GF 2 extended with Presburger constraints are known. We already know that the complexity of the modal logic K or the description logic ALC do not differ from their Presburger versions, see [2,5,11]. We believe that to obtain tight complexity bounds for Presburger GF 2 one should start with a more modest goal, i.e., to establish the exact complexity of Presburger ALCI, namely an extension of ALC with inverse relations.…”

Section: Future Workmentioning

confidence: 99%

One-variable logic meets Presburger arithmetic

Bednarczyk

2020

Theoretical Computer Science

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We consider the one-variable fragment of first-order logic extended with Presburger constraints. The logic is designed in such a way that it subsumes the previously-known fragments extended with counting, modulo counting or cardinality comparison and combines their expressive powers. We prove NP-completeness of the logic by presenting an optimal algorithm for solving its finite satisfiability problem.

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“…5. Similarly, we use the semantic domain from item 3., Markov chains, to obtain the probabilistic µ-calculus with polynomial inequalities [22]. Again, we introduce new higher-arity modalities by putting…”

Section: The Presburger µ-Calculus Is the Extension Of The Graded µ-C...mentioning

confidence: 99%

“…The logics from the last two items are necessarily less general than the corresponding next-step logics [7,22], because the definition of µ-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 {>,…”

Section: The Presburger µ-Calculus Is the Extension Of The Graded µ-C...mentioning

confidence: 99%

“…the one-step logic of the relational µ-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 µ-calculus [8], which however establishes only a doubly exponential upper bound in the case without tractable modal tableau rules.…”

Section: Introductionmentioning

confidence: 99%

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Optimal Satisfiability Checking for Arithmetic $$\mu $$-Calculi

Hausmann

Schröder

2019

Lecture Notes in Computer Science

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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.

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Reasoning with Global Assumptions in Arithmetic Modal Logics

Cited by 5 publications

References 38 publications

Quasipolynomial Computation of Nested Fixpoints

Quasipolynomial Computation of Nested Fixpoints

One-variable logic meets Presburger arithmetic

Optimal Satisfiability Checking for Arithmetic $$\mu $$-Calculi

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