Optimal Satisfiability Checking for Arithmetic $μ$-Calculi

Abstract: The coalgebraic $\mu$-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 $\mu$-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 $\mu$-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 $\mu$-calculus with Presburger arithmetic, as well as an extension of the (two-valued) probabilistic $\mu$-calculus with polynomial inequalities. As a side result, we moreover obtain a new upper bound $\mathcal{O}(((nk)!)^2)$ on minimum model size for satisfiable formulas for all coalgebraic $\mu$-calculi, where $n$ is the size of the formula and $k$ its alternation depth.