Temporal logics on strings with prefix relation

Abstract

We show that linear-time temporal logic over concrete domains made of finite strings and the prefix relation admits a PSpace-complete satisfiability problem. Actually, we extend a known result with the concrete domain made of the set of natural numbers and the greater than relation (corresponding to the singleton alphabet case) and we solve an open problem mentioned in several publications. Since the prefix relation is not a total ordering, it is not possible to take advantage of existing techniques dedicated to temporal logics with concrete domains that are essentially linearly ordered structures. Instead, we introduce an adequate encoding of string constraints into length constraints that allows us to reduce the problem on strings to the problem on natural numbers. To do so, we also propose an extended version of the logic on strings that is able to compare lengths of longest common prefixes and for which the satisfiability problem is shown in PSpace. Finally, we show how to lift the result for the branching-time case in order to get decidability when the underlying temporal logic is CTL .

First Steps Towards Taming Description Logics with Strings

“…To reduce the nonemptiness problem for TCA on D Σ to the nonemptiness problem for TCA on the concrete domain pN, ă, ", p" n q nPN q, we show how to take advantage of [17,Lemma 6] dedicated to the transformation of prefix constraints into Boolean combinations of atomic constraints on N. For the sake of being selfcontained, we recall below a few definitions useful in Section 4.2.…”

“…Our motivation in this work is to investigate the decidability/complexity status of ALCF P pD Σ q, that is, for the nontrivial string domain with the prefix order. To do so, we take advantage of recent results on tree constraint automata on Z from [19] combined with an encoding of string constraints by numerical constraints from [17]. These are only some first steps to tame reasoning tasks for description logics with string domains, and of course, other string domains are possibly interesting, see e.g.…”

“…In Section 4, the nonemptiness problem for TCA is shown ExpTime-complete. Though ExpTime-hardness is a consequence of [19, Section 3.1], the ExpTime-membership is by reduction to the nonemptiness problem for TCA on N by lifting arguments from [17,Section 3] to the automata-based setting. In Section 5, we show how to reduce the concept satisfiability problem w.r.t.…”

Model Checking Constraint LTL over Trees

“…Remark 5. Demri and Deter [8] conjectured that if the arity k of the tree is part of the input to the satisfiability problem, it is still in PSPACE. Our proof confirms that this branching degree uniform satisfiability problem is PSPACE-complete.…”

“…Upon finishing our paper, we have become aware that Demri and Deters (abbreviated DD in the following) have submitted a paper [8] that shows above mentioned results on satisfiability using a reduction of constraints over trees to constraints over the integers. Even though the main results of both papers coincide, there are major differences.…”