Dual Systems of Tableaux and Sequents for PLTL

Abstract

On one hand, traditional tableau systems for temporal logic (TL) generate an auxiliary graph that must be checked and (possibly) pruned in a second phase of the refutation procedure. On the other hand, traditional sequent calculi for TL make use of a kind of inference rules (mainly, invariant-based rules or infinitary rules) that complicates their automatization. A remarkable consequence of using auxiliary graphs in the tableaux framework and invariants or infinitary rules in the sequents framework is that TL fails to carry out the classical correspondence between tableaux and sequents. In this paper, we first provide a tableau method TTM that does not require auxiliary graphs to decide whether a set of PLTL-formulas is satisfiable. This tableau method TTM is directly associated to a one-sided sequent calculus called TTC. Since TTM is free from all the structural rules that hinder the mechanization of deduction, e.g. weakening and contraction, then the resulting sequent calculus TTC is also free from this kind of structural rules. In particular, TTC is free of any kind of cut, including invariant-based cut. From the deduction system TTC, we obtain a two-sided sequent calculus GTC that preserves all these good freeness properties and is finitary, sound and complete for PTL. Therefore, we show that the classical correspondence between tableaux and sequent calculi can be extended to TL. Every deduction system is proved to be complete. In addition, we provide illustrative examples of deductions in the different systems.