The Complexity of Regularity in Grammar Logics and Related Modal Logics
Résumé
A modal reduction principle of the form [i 1 ]. .. [i n ]p ⇒ [j 1 ]. .. [j n ]p can be viewed as a production rule i 1 •. .. • i n → j 1 •. .. • j n in a formal grammar. We study the extensions of the multimodal logic K m 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 φ, deciding whether the formula is satisfiable in the extension of K m with axiom schemes from G can be done in deterministic exponential-time in the size of G and φ, and this problem is complete for this complexity class. Such an extension of K m is called a regular grammar logic. The proof of the exponential-time upper bound is extended to PDL-like extensions of K m and to global logical consequence and global satisfiability problems. Using an equational characterization of contextfree 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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