On the correspondence between display postulates and deep inference in nested sequent calculi for tense logics

We consider two styles of proof calculi for a family of tense logics, presented in a formalism based on nested sequents. A nested sequent can be seen as a tree of traditional single-sided sequents. Our first style of calculi is what we call "shallow calculi", where inference rules are only applied at the root node in a nested sequent. Our shallow calculi are extensions of Kashima's calculus for tense logic and share an essential characteristic with display calculi, namely, the presence of structural rules called "display postulates". Shallow calculi enjoy a simple cut elimination procedure, but are unsuitable for proof search due to the presence of display postulates and other structural rules. The second style of calculi uses deep-inference, whereby inference rules can be applied at any node in a nested sequent. We show that, for a range of extensions of tense logic, the two styles of calculi are equivalent, and there is a natural proof theoretic correspondence between display postulates and deep inference. The deep inference calculi enjoy the subformula property and have no display postulates or other structural rules, making them a better framework for proof search

On the Correspondence between Display Postulates and Deep Inference in Nested Sequent Calculi for Tense Logics

We consider two styles of proof calculi for a family of tense logics, presented in a formalism based on nested sequents. A nested sequent can be seen as a tree of traditional single-sided sequents. Our first style of calculi is what we call "shallow calculi", where inference rules are only applied at the root node in a nested sequent. Our shallow calculi are extensions of Kashima's calculus for tense logic and share an essential characteristic with display calculi, namely, the presence of structural rules called "display postulates". Shallow calculi enjoy a simple cut elimination procedure, but are unsuitable for proof search due to the presence of display postulates and other structural rules. The second style of calculi uses deep-inference, whereby inference rules can be applied at any node in a nested sequent. We show that, for a range of extensions of tense logic, the two styles of calculi are equivalent, and there is a natural proof theoretic correspondence between display postulates and deep inference. The deep inference calculi enjoy the subformula property and have no display postulates or other structural rules, making them a better framework for proof search

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

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 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 bi-intuitionistic 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, 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 and, independently, by Brunnler and Strafsburger and Poggiolesi 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

Deep Inference in Bi-intuitionistic Logic

Bi-intuitionistic logic is the extension of intuitionistic logic with exclusion, a connective dual to implication. Cut-elimination in bi-intuitionistic logic is complicated due to the interaction between these two connectives, and various extended sequent calculi, including a display calculus, have been proposed to address this problem. In this paper, we present a new extended sequent calculus DBiInt for bi-intuitionistic logic which uses nested sequents and "deep inference", i.e., inference rules can be applied at any level in the nested sequent. We show that DBiInt can simulate our previous "shallow" sequent calculus LBiInt. In particular, we show that deep inference can simulate the residuation rules in the display-like shallow calculus LBiInt. We also consider proof search and give a simple restriction of DBiInt which allows terminating proof search. Thus our work is another step towards addressing the broader problem of proof search in display logic

A cut-free sequent calculus for bi-intuitionistic logic

Bi-intuitionistic logic is the extension of intuitionistic logic with a connective dual to implication. Bi-intuitionistic logic was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent "cut-free" sequent calculus for BiInt has recently been shown by Uustalu to fail cut-elimination. We present a new cut-free sequent calculus for BiInt, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between implication and its dual, similarly to future and past modalities in tense logic. Our calculus handles this interaction using extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of failed derivation trees. Our simple termination argument allows our calculus to be used for automated deduction, although this is not its main purpose

An experimental evaluation of global caching for ALC (system description)

Goré and Nguyen have recently given the first optimal and sound method for global caching for the description (modal) logic ALC, and various extensions. We report on an experimental evaluation for ALC plus its reflexive and reflexive-transitive extensio

Combining Derivations and Refutations for Cut-free Completeness in Bi-intuitionistic Logic

Bi-intuitionistic logic is the union of intuitionistic and dual intuitionistic logic, and was introduced by Rauszer as a Hilbert calculus with algebraic and Kripke semantics. But her subsequent 'cut-free' sequent calculus has recently been shown to fail cut-elimination. We present a new cut-free sequent calculus for bi-intuitionistic logic, and prove it sound and complete with respect to its Kripke semantics. Ensuring completeness is complicated by the interaction between intuitionistic implication and dual intuitionistic exclusion, similarly to future and past modalities in tense logic. Our calculus handles this interaction using derivations and refutations as first class citizens. We employ extended sequents which pass information from premises to conclusions using variables instantiated at the leaves of refutations, and rules which compose certain refutations and derivations to form derivations. Automated deduction using terminating backward search is also possible, although this is not our main purpose