Coalgebraic Satisfiability Checking for Arithmetic $μ$-Calculi
Daniel Hausmann, Lutz Schröder
TL;DR
The paper addresses satisfiability for the coalgebraic μ-calculus over general branching types by reducing to a tractable one-step satisfiability problem and solving a doubly-exponential satisfiability game in singly exponential time via lazy, on-the-fly techniques. It introduces tracking automata and model-checking games to manage fixpoint unfoldings, then derives tableaux and satisfiability games whose correctness is established through a sequence of truth and existence lemmas. The main technical contribution is a generic ExpTime bound that does not require guardedness or complete modal tableau rule sets, applicable to logics such as the graded μ-calculus and probabilistic μ-calculus with polynomial inequalities. The approach yields a small-model property with polynomial branching under OSPMP and offers a practical reasoning method, implemented in related tooling, for a broad class of coalgebraic logics with potential extensions to fusion of logics. Overall, the work significantly broadens the scope of ExpTime Satisfiability for fixpoint logics beyond traditional relational settings.
Abstract
The coalgebraic $μ$-calculus provides a generic semantic framework for fixpoint logics over systems whose branching type goes 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 relies on the availability of tableau rules for the next-step modalities that are sufficiently well-behaved in a formally defined sense; in particular, rule matches need to be representable by polynomial-sized codes, and the sequent duals of the rules need to absorb cut. While such rule sets have been identified for some important cases, they are not known to exist 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 work, we prove the same upper complexity bound under more general assumptions, specifically regarding the complexity of the (much simpler) satisfiability problem for the underlying one-step logic, roughly described as the nesting-free next-step fragment of the logic. The bound is realized by a generic global caching algorithm that supports on-the-fly satisfiability checking. Notably, our approach directly accommodates unguarded formulae, and thus avoids use of the guardedness transformation. Example applications include new exponential-time upper bounds for satisfiability checking in an extension of the graded $μ$-calculus with polynomial inequalities (including positive Presburger arithmetic), as well as an extension of the (two-valued) probabilistic $μ$-calculus with polynomial inequalities.