an:05348354
Zbl 1147.03008
Rybakov, V.
Linear temporal logic with until and next, logical consecutions
EN
Ann. Pure Appl. Logic 155, No. 1, 32-45 (2008).
00232087
2008
j
03B44 03B25 03B70
algorithms; linear temporal logic; admissible consecutions; logical consequence; admissible inference rules; Kripke structures
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 \(\mathcal{LTL}\).