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Linear temporal logic with until and next, logical consecutions. (English) Zbl 1147.03008
Summary: While specifications and verifications of concurrent systems employ Linear Temporal Logic (\(\mathcal{LTL}\)), it is increasingly likely that logical consequence in \(\mathcal{LTL}\) will be used in the description of computations and parallel reasoning. Our paper considers logical consequence in the standard \(\mathcal{LTL}\) with temporal operations \(\mathbf U\) (until) and \(\mathbf N\) (next). The prime result is an algorithm recognizing consecutions admissible in \(\mathcal{LTL}\), so we prove that \(\mathcal{LTL}\) is decidable w.r.t. admissible inference rules. As a consequence we obtain algorithms verifying the validity of consecutions in \(\mathcal{LTL}\) and solving the satisfiability problem. We start by a simple reduction of logical consecutions (inference rules) of \(\mathcal{LTL}\) to equivalent ones in the reduced normal form (which have uniform structure and consist of formulas of temporal degree 1). Then we apply a semantic technique based on \(\mathcal{LTL}\)-Kripke structures with formula definable subsets. This yields necessary and sufficient conditions for a consecution to be not admissible in \(\mathcal{LTL}\). These conditions lead to an algorithm which recognizes consecutions (rules) admissible in \(\mathcal{LTL}\) by verifying the validity of consecutions in special finite Kripke structures of size square polynomial in reduced normal forms of the consecutions. As a consequence, this also solves the satisfiability problem for \(\mathcal{LTL}\).

03B44 Temporal logic
03B25 Decidability of theories and sets of sentences
03B70 Logic in computer science
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