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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 $$\mathcal{LTL}$$.

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