The complexity of linear-time temporal logic over the class of ordinals

Stéphane Demri (1, 2) , Alexander Rabinovich (3)

1 LSV - Laboratoire Spécification et Vérification
2 CNRS - Centre National de la Recherche Scientifique
3 TAU - Tel Aviv University

We consider the temporal logic with since and until modalities. This temporal logic is expressively equivalent over the class of ordinals to first-order logic by Kamp's theorem. We show that it has a PSPACE-complete satisfiability problem over the class of ordinals. Among the consequences of our proof, we show that given the code of some countable ordinal α and a formula, we can decide in PSPACE whether the formula has a model over α. In order to show these results, we introduce a class of simple ordinal automata, as expressive as Büchi ordinal automata. The PSPACE upper bound for the satisfiability problem of the temporal logic is obtained through a reduction to the nonemptiness problem for the simple ordinal automata.

Mots clés: complexity; ordinal; linear-time temporal logic LTL; Buchi ordinal automaton; PSPACE; temporal logic; Kamp's theorem; satisfiability problem

Domaines: Informatique [cs]; Logique en informatique [cs.LO]; Théorie et langage formel [cs.FL]

Stéphane Demri, Alexander Rabinovich. The complexity of linear-time temporal logic over the class of ordinals. Logical Methods in Computer Science, 2010, 6 (4), ⟨10.2168/LMCS-6(4:9)2010⟩. ⟨hal-03190247⟩