Journal of Automated Reasoning

, Volume 50, Issue 1, pp 1–49

# Invariant-Free Clausal Temporal Resolution

• Jose Gaintzarain
• Montserrat Hermo
• Paqui Lucio
• Marisa Navarro
• Fernando Orejas
Article

## Abstract

Resolution is a well-known proof method for classical logics that is well suited for mechanization. The most fruitful approach in the literature on temporal logic, which was started with the seminal paper of M. Fisher, deals with Propositional Linear-time Temporal Logic (PLTL) and requires to generate invariants for performing resolution on eventualities. The methods and techniques developed in that approach have also been successfully adapted in order to obtain a clausal resolution method for Computation Tree Logic (CTL), but invariant handling seems to be a handicap for further extension to more general branching temporal logics. In this paper, we present a new approach to applying resolution to PLTL. The main novelty of our approach is that we do not generate invariants for performing resolution on eventualities. Hence, we say that the approach presented in this paper is invariant-free. Our method is based on the dual methods of tableaux and sequents for PLTL that we presented in a previous paper. Our resolution method involves translation into a clausal normal form that is a direct extension of classical CNF. We first show that any PLTL-formula can be transformed into this clausal normal form. Then, we present our temporal resolution method, called trs-resolution, that extends classical propositional resolution. Finally, we prove that trs-resolution is sound and complete. In fact, it finishes for any input formula deciding its satisfiability, hence it gives rise to a new decision procedure for PLTL.

### Keywords

Propositional linear-time temporal logic Resolution Invariant-free Clausal normal form

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## Authors and Affiliations

• Jose Gaintzarain
• 1
• Montserrat Hermo
• 2
• Paqui Lucio
• 2
• Marisa Navarro
• 2
• Fernando Orejas
• 3
1. 1.The University of the Basque CountryBilbaoSpain
2. 2.The University of the Basque CountrySan SebastiánSpain
3. 3.Technical University of CataloniaBarcelonaSpain