Cut-free sequent systems for temporal logic

Abstract

Currently known sequent systems for temporal logics such as linear time temporal logic and computation tree logic either rely on a cut rule, an invariant rule, or an infinitary rule. The first and second violate the subformula property and the third has infinitely many premises. We present finitary cut-free invariant-free weakeningfree and contraction-free sequent systems for both logics mentioned. In the case of linear time all rules are invertible. The systems are based on annotating fixpoint formulas with a history, an approach which has also been used in game-theoretic characterisations of these logics.

Cyclic Proofs for Linear Temporal Logic

“…The first observation, see [2], is that system LT naive almost works: the only thing that goes wrong is that we cannot derive induction principles as is shown in the following example.…”

“…It is not even clear how to design a finitary deductive system for linear time temporal logic LTL with nice proof-theoretic properties. In this context, deductive systems featuring infinite long proof branches (together with a global soundness condition) and their cyclic variants have recently obtained much attention, see, for instance, [1,2,3,6,7,8].…”

“…Brünnler and Lange [2] proposed an elegant formalism with cyclic proofs for LTL using focus games from Lange and Stirling [5] as an inspiration. The main technical feature of their system are annotated sequents, which are employed to derive greatest fixed points.…”

“…However, some very basic prooftheoretic problems turn out to be surprisingly hard in this setting. For instance, despite the admissibility of several structural rules, including cut, being proved semantically in [2], it remains to prove the same facts by proper proof-theoretic methods. Even the admissibility of weakening, which is quite trivial for most types of sequent calculi, is far from being simple for annotated sequents due to the presence of sequent contexts in the annotations.…”

“…As Brünnler and Lange point out [2], the problem with weakening is to be expected since the fact that a certain statement is provable by induction does not imply that a weaker statement is also provable by induction.…”