Bisimulation-invariant PTIME and higher-dimensional µ-calculus

Abstract Consider the class of all those properties of worlds in finite Kripke structures (or of states in finite transition systems), that are • • recognizable in polynomial time, and • • closed under bisimulation equivalence. It is shown that the class of these bisimulation-invariant Ptime queries has a natural logical characterization. It is captured by the straightforward extension of propositional μ-calculus to arbitrary finite dimension. Bisimulation-invariant Ptime , or the modal fragment of Ptime , thus proves to be one of the very rare cases in which a logical characterization is known in a setting of unordered structures. It is also shown that higher-dimensional μ-calculus is undecidable for satisfiability in finite structures, and even ∑ 1 1 -hard over general structures.

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