On Combinatorial Proofs for Modal Logic

Abstract

In this paper we extend Hughes' combinatorial proofs to modal logics. The crucial ingredient for modeling the modalities is the use of a self-dual non-commutative operator that has first been observed by Retoré through pomset logic. Consequently, we had to generalize the notion of skew fibration from cographs to Guglielmi's relation webs. Our main result is a sound and complete system of combinatorial proofs for all normal and non-normal modal logics in the S4-tesseract. The proof of soundness and completeness is based on the sequent calculus with some added features from deep inference.

Towards a Combinatorial Proof Theory

“…No Weakening = Surjectivity No Contraction = Injectivity Atomic Contraction = Fullness Shallow Inference = Fibrations Deep Inference = Skew Fibrations An important line of future research is the extension of these results to modal logics [3] and first-order logic [19].…”

“…Let S be a structural proof system. We define G(S) = {G(φ) | φ ∈ D S }, the set of cograph homomorphisms generated from derivations in S. 3 We now need to introduce some terminology to better characterise classes of homomorphisms.…”

Exponentially Handsome Proof Nets and Their Normalization

“…As discussed in Section 2.1, RB-cograph are an alternative syntax for MLL proof nets. In this paper we consider RGB-cographs from [6], which are an extension of the RB-cograph syntax. We provide a further extension of RGB-cographs and we establish a correspondence between these graphs and MELL -proofs.…”

“…These conditions allow us to restraint the correspondence between contractions-weakening derivations and skew fibrations [23,46,6,5,34] on the formulas on the form ?A. However, since in relation web we consider formulas modulo associativity and commutativity of ⊗ and `(see Theorem 11), we need to also consider the additional (deep) rules in Figure 13 taking care of this equivalence.…”

“…We then extend the results in [6] in order to define combinatorial proofs for MELL j . This allows us to represent equivalent cut-free derivations in MELL j .…”