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

Abstract

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.

Quasipolynomial Computation of Nested Fixpoints

“…Similarly, the satisfiability problem of the coalgebraic μ-calculus has been reduced to a computation of a nested fixpoint [25], and our present results imply a marked improvement in the exponent of the associated exponential time bound. Specifically, the nesting depth of the relevant fixpoint is exponentially smaller than the basis of the lattice.…”

“…(3) Under mild assumptions on the modalities (see [25]), we obtain a novel upper bound 2 O(nd log n) for the satisfiability problems of coalgebraic μ-calculi, in particular including the monotone μ-calculus, the alternating-time μ-calculus, the graded μ-calculus and the (two-valued) probabilistic μ-calculus, even when the latter two are extended with (monotone) polynomial inequalities. This improves on the best previous bounds in all cases.…”

“…Furthermore, the satisfiability problem of the coalgebraic μ-calculus has been reduced to solving canonical fixpoint equations systems over lattices P(U ), where U is the state set of a determinized parity automaton and where the innermost equation checks for joint one-step satisfiability of sets of coalgebraic modalities [25]. By interpreting coalgebraic formulae over finite lattices d U rather than over powerset lattices, one obtains the finite-valued coalgebraic μ-calculus (with values {0, .…”

Quasipolynomial Computation of Nested Fixpoints

“…Similarly, the satisfiability problem of the coalgebraic µ-calculus has been reduced to a computation of a nested fixpoint [31], and our present results imply a marked improvement in the exponent of the associated exponential time bound. Specifically, the nesting depth of the relevant fixpoint is exponentially smaller than the basis of the lattice.…”

“…Furthermore, the satisfiability problem of the coalgebraic µ-calculus has been reduced to solving canonical fixpoint equations systems over lattices P(U ), where U is the state set of a determinized parity automaton and where the innermost equation checks for joint one-step satisfiability of sets of coalgebraic modalities [31]. By interpreting coalgebraic formulae over finite lattices d U rather than over powerset lattices, one obtains the finite-valued coalgebraic µ-calculus (with values {0, .…”