Proof theory and proof search of bi-intuitionistic and tense logic

TLDR: This thesis develops a new extended sequent calculus for bi-intuitionistic logic using a framework of derivations and refutations and solves an open problem about taming proof search in display calculi.

Abstract: In this thesis, we consider bi-intuitionistic logic and tense logic, as well as the combined bi-intuitionistic tense logic. Each of these logics contains a pair of dual connectives, for example, Rauszer’s bi-intuitionistic logic [100] contains intuitionistic implication and dual intuitionistic exclusion. The interaction between these dual connectives makes it non-trivial to develop a cut-free sequent calculus for these logics. In the first part of this thesis we develop a new extended sequent calculus for biintuitionistic logic using a framework of derivations and refutations. This is the first purely syntactic cut-free sequent calculus for bi-intuitionistic logic and thus solves an open problem. Our calculus is sound, semantically complete and allows terminating backward proof search, hence giving rise to a decision procedure for bi-intuitionistic logic. In the second part of this thesis we consider the broader problem of taming proof search in display calculi [12], using bi-intuitionistic logic and tense logic as case studies. While the generality of display calculi makes it an excellent framework for designing sequent calculi for logics where traditional sequent calculi fail, this generality also leads to a large degree of non-determinism, which is problematic for backward proof-search. We control this non-determinism in two ways: 1. First, we limit the structural connectives used in the calculi and consequently, the number of display postulates. Specifically, we work with nested structures which can be viewed as a tree of traditional Gentzen’s sequents, called nested sequents, which have been used previously by Kashima [73] and, independently, by Brunnler and Strasburger [17; 21; 20] and Poggiolesi [97] to present several modal and tense logics. 2. Second, since residuation rules are largely responsible for the difficulty in finding a proof search procedure for display-like calculi, we show how to eliminate these residuation rules using deep inference in nested sequents. Finally, we study the combined bi-intuitionistic tense logic, which contains the well-known intuitionistic modal logic as a sublogic. We give a nested sequent calculus for bi-intuitionistic tense logic that has cut-elimination, and a derived deep inference nested sequent calculus that is complete with respect to the first calculus and where contraction and residuation rules are admissible. We also show how our calculi can capture Simpson’s intuitionistic modal logic [104] and Ewald’s intuitionistic tense logic [39].