Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Optimal Satisfiability Checking for Arithmetic $\mu$-Calculi   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.

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The categories of this paper are: math-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: ao-sci
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The categories of this paper are: astro-ph
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The categories of this paper are: math
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The categories of this paper are: nucl-th
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The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: patt-sol
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The categories of this paper are: physics
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The categories of this paper are: stat
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The categories of this paper are: econ
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The categories of this paper are: chao-dyn
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The categories of this paper are: bayes-an