A Parallel Linear Temporal Logic Tableau

McCabe-Dansted, John Christopher; Reynolds, Mark

doi:10.4204/eptcs.256.12

Cited by 3 publications

(2 citation statements)

References 15 publications

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“…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].…”

Section: Introductionmentioning

confidence: 99%

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

Geatti

Gigante

Montanari

2019

Lecture Notes in Computer Science

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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.

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Section: Introductionmentioning

confidence: 99%

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

Geatti

Gigante

Montanari

2019

Lecture Notes in Computer Science

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“…Various ways of overcoming such a limitation have been proposed in the literature, including incremental [7] and single pass techniques [12]. Recently, a one-pass and tree-shaped tableau system for LTL has been devised [11], which does not build any huge preliminary structure and, thanks to its pure rule-based tree-search structure, proved to be amenable to efficient implementation and easy parallelisation [3,10]. Recent work also suggested that its modular structure makes it possible to easily extend it to other linear time logics (the extension to LTL with past operators is described in [6]).…”

Section: Introductionmentioning

confidence: 99%

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

Geatti

Gigante

Montanari

et al. 2018

Electron. Proc. Theor. Comput. Sci.

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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...

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SAT Meets Tableaux for Linear Temporal Logic Satisfiability

Geatti,

Gigante,

Montanari

et al. 2024

J Autom Reasoning

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Linear temporal logic ($$\textsf{LTL}\,$$ LTL ) and its variant interpreted on finite traces ($$\textsf{LTL}_{\textsf{f}\,}$$ LTL f ) are among the most popular specification languages in the fields of formal verification, artificial intelligence, and others. In this paper, we focus on the satisfiability problem for $$\textsf{LTL}\,$$ LTL and $$\textsf{LTL}_{\textsf{f}\,}$$ LTL f formulas, for which many techniques have been devised during the last decades. Among these are tableau systems, of which the most recent is Reynolds’ tree-shaped tableau. We provide a SAT-based algorithm for $$\textsf{LTL}\,$$ LTL and $$\textsf{LTL}_{\textsf{f}\,}$$ LTL f satisfiability checking based on Reynolds’ tableau, proving its correctness and discussing experimental results obtained through its implementation in the BLACK satisfiability checker.

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A Parallel Linear Temporal Logic Tableau

Cited by 3 publications

References 15 publications

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

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

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

SAT Meets Tableaux for Linear Temporal Logic Satisfiability

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