One-Pass and Tree-Shaped Tableau Systems for TPTL and TPTLb+Past

Abstract

In this paper, we propose a novel one-pass and tree-shaped tableau method for Timed Propositional Temporal Logic and for a bounded variant of its extension with past operators. Timed Propositional Temporal Logic (TPTL) is a real-time temporal logic, with an EXPSPACE-complete satisfiability problem, which has been successfully applied to the verification of real-time systems. In contrast to LTL, adding past operators to TPTL makes the satisfiability problem for the resulting logic (TPTL+P) non-elementary. In this paper, we devise a one-pass and tree-shaped tableau for both TPTL and bounded TPTL+P (TPTL b +P), a syntactic restriction introduced to encode timeline-based planning problems, which recovers the EXPSPACE-complete complexity. The tableau systems for TPTL and TPTL b +P are presented in a unified way, being very similar to each other, providing a common skeleton that is then specialised to each logic. In doing that, we characterise the semantics of TPTL b +P in terms of a purely syntactic fragment of TPTL+P, giving a translation that embeds the former into the latter. Soundness and completeness of the system are proved fully. In particular, we give a greatly simplified model-theoretic completeness proof, which sidesteps the complex combinatorial argument used by known proofs for the one-pass and tree-shaped tableau systems for LTL and LTL+P. reflected in the computational complexity of its satisfiability problem, which is EXPSPACE-complete. Originally proposed as a formal tool for the verification of real-time systems, it recently found interesting applications in the area of artificial intelligence, to encode a meaningful class of timeline-based planning problems [5]. This and other application scenarios benefit from/require the use of past operators, which allow the logic to compactly predicate about events in the past of the current time point. However, in contrast to the case of LTL, where past operators can be supported without harm, adding them to TPTL greatly increases the complexity of its satisfiability problem, which becomes non-elementary [1]. For this reason, bounded TPTL with Past (TPTL b +P) has been introduced [5], which supports past operators, but suitably restricts their use in order to recover an EXPSPACE-complete satisfiability problem. While initially introduced as a specific tool to encode planning problems, TPTL b +P is interesting by itself, since it enables the use of past operators in a fairly natural way.In this paper, we exploit the extensibility of the aforementioned tableau system to provide a one-pass and tree-shaped tableau method for TPTL and TPTL b +P. We present both tableau systems, which are very similar, in a unified way by first (i) factoring out the common structure, and then (ii) showing how to specialise it in the case of TPTL (future-only) and TPTL b +P (bounded) formulae, thus obtaining a one-pass and tree-shaped tableau system for both logics. To show how the tableau for TPTL b +P formulae works, (iii) we characterise the semantics of the logic in terms of a guar...

A SAT-Based Encoding of the One-Pass and Tree-Shaped Tableau System for LTL

“…In particular, the extension to past operators [14] appears to be easy to be encoded, without resorting to the virtual unrollings technique used in other bounded approaches [15]. Reynolds' tableau system has also been extended to timed logics [13], in particular TPTL [2] and a TPTL b +P [11]. It is natural to ask whether the approach used here to encode the LTL tableau to SAT can be adapted to encode the timed extensions of the tableau to SMT.…”

“…We now describe Reynolds' tableau system for LTL. After its original formulation in [22], the system was extended to support past operators [14] and more expressive real-time logics [13]. Here, we briefly recall its original future-only version, which is the one considered for the SAT encoding described in the next section.…”

“…If a branch is rejected, it happens either because of a logical contradiction, that triggers the CONTRADICTION rule, or because of the PRUNE rule, which avoids the tableau to infinitely postpone a request that is impossible to fulfil. From a model-theoretic point of view [13], the PRUNE rule allows one not to consider models that contain redundant segments, i.e., segments that just repeat some previously done piece of work without contributing further to the satisfaction of all the pending requests. Recent work [13] provided a model-theoretic interpretation of this mechanism, showing a characterization of the discarded models.…”

“…In our system, termination is guaranteed by a suitable encoding of the tableau's PRUNE rule. This rule was the main novelty of Reynolds' one-pass and tree-shaped system when it was originally proposed [22], has a clean model-theoretic interpretation [13], and the important role it plays in our encoding adds up to its interesting properties. The result is a simple and complete bounded satisfiability checking procedure based on a small and much simpler SAT encoding.…”

“…Recently, a tree-shaped tableau for LTL has been proposed by Reynolds [22], which only requires a single pass to decide whether a given branch has to be accepted or not. The smaller size of the tree with regards to the full graph structure of previous methods, and its simple rule-based tree search mechanism, led to an efficient implementation [3], a simple and fruitful parallelization [21], and modular extensions to more expressive logics [13,14].…”