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

Abstract

Recently, a new one-pass and tree-shaped tableau system for LTL satisfiability checking has been proposed, with the distinguishing features of (i) being really tree-shaped, in contrast to previous graphshaped tableau methods, and (ii) requiring only one pass to either accept or reject a branch of the tableau. Despite its simplicity, it proved itself to be effective in practice. In this paper, we provide a SAT-based encoding of such a tableau system, based on the technique of bounded satisfiability checking. Starting with a single-node tableau, i.e., depth k of the treeshaped tableau equal to zero, we proceed in an incremental fashion. At each iteration, the tableau rules are encoded in a Boolean formula, representing all branches of the tableau up to the current depth k. A typical downside of such bounded techniques is the effort needed to understand when to stop incrementing the bound, to guarantee the completeness of the procedure. In contrast, termination and completeness of the proposed algorithm is guaranteed without computing any upper bound to the length of candidate models, thanks to the Boolean encoding of the PRUNE rule of the original tableau system. We conclude the paper by describing a tool that implements our procedure, and comparing its performance with other state-of-the-art LTL solvers.

Linear Temporal Logic Modulo Theories over Finite Traces

“…Theorem 8 might seem a rather weak result in practice, but it is worth to note that it is in fact a more general statement of what is already implemented in the BLACK temporal reasoning framework (Geatti, Gigante, and Montanari 2019; Geatti, Gigante, and Montanari 2021) when dealing with LTL f modulo theories (LTL MT f ), a restricted fragment of FOLTL with rigid predicates and non-rigid constants (Geatti, Gianola, and Gigante 2022;Geatti et al 2023).…”

Succinctness of Cosafety Fragments of LTL via Combinatorial Proof Systems

“…Related and future work are discussed in Section 8. An extended version of the paper, complete of all proofs, can be found in [13].…”

Decidable Fragments of LTLf Modulo Theories

“…Several directions for future work can be considered. Following the path taken by [24], an SMT encoding of our PRUNE rule would allow for its implementation in the BLACK temporal reasoning framework [27]. Moreover, whether these results can be extended to a version of LTL MT f supporting time-varying relations is still open.…”

“…In general, LTL MT f is undecidable, and it has been shown to be semi-decidable if applied to decidable first-order fragments and/or theories [24,25]: Crucially, the semi-decidability result has been shown by providing an effective SMT-based encoding of a tree-shaped tableau that, once implemented in the BLACK temporal reasoning system [27,28], has proved to work well in practice. Moreover, being theory-agnostic, the technique works in many different scenarios, leveraging the many expressive theories, and combinations thereof, supported by modern SMT solvers [2].…”