The Complexity of Regularity in Grammar Logics and Related Modal Logics

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Abstract
A modal reduction principle of the form [i1] … [in]p ⇒ [j1] … [jn′]p can be viewed as a production rule i1 · … ·in → j1 · … · jn′ in a formal grammar. We study the extensions of the multimodal logic Km with m independent K modal connectives by finite addition of axiom schemes of the above form such that the associated finite set of production rules forms a regular grammar. We show that given a regular grammar G and a modal formula Ø, deciding whether the formula is satisfiable in the extension of Km with axiom schemes from G can be done in deterministic exponential‐time in the size of G and Ø, and this problem is complete for this complexity class. Such an extension of Km is called a regular grammar logic. The proof of the exponential‐time upper bound is extended to PDL‐like extensions of Km and to global logical consequence and global satisfiability problems. Using an equational characterization of context‐free languages, we show that by replacing the regular grammars by linear ones, the above problem becomes undecidable. The last part of the paper presents non‐trivial classes of exponential time complete regular grammar logics.

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