The Complexity of Temporal Logic with Until and Since over Ordinals

Abstract

Abstract. 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 thanks to 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.

Completeness and Decidability in Sequence Logic

“…This complexity theoretic result is extended to all countable ordinals in [4]. In the next section we give a simple argument for decidability of |= ω based on a form of finite model property, Lemma 11, which may be of independent interest.…”

Automata and temporal logic over arbitrary linear time

“…Over ordinals, LTL with Until and Since has been shown to have a PSPACE-complete satisfiability problem [7]. Over the ordering of the real numbers, satisfiability of LTL with until and since is PSPACE-complete, but satisfiability of MSO is undecidable.…”

“…Reynolds shows in [13] that the satisfiability problem of temporal logic with only the Until connective is also PSPACE-complete, and conjectures that this might stay true when adding the Since connective. The upper bound in [7] is obtained by reducing the satisfiability of LTL formulae to the accessibility problem in an appropriate automata model, accepting words indexed by ordinals. In this paper, we focus on the general case of arbitrary linear orderings, using the full logic with Until, Since and both Stavi connectives.…”