Towards a Combinatorial Proof Theory

International audienceThe main part of a classical combinatorial proof is a skew fi-bration, which precisely captures the behavior of weakening and contraction. Relaxing the presence of these two rules leads to certain substruc-tural logics and substructural proof theory. In this paper we investigate what happens if we replace the skew fibration by other kinds of graph homomorphism. This leads us to new logics and proof systems that we call combinatorial

Combinatorial Proofs for Constructive Modal Logic

International audienceCombinatorial proofs form a syntax-independent presentation of proofs, originally proposed by Hughes for classical propositional logic. In this paper we present a notion of combinatorial proofs for the constructive modal logics CK and CD, we show soundness and completeness of combinatorial proofs by translation from and to sequent calculus proofs, and we discuss the notion of proof equivalence enforced by these translations

On Combinatorial Proofs for Logics of Relevance and Entailment

International audienceHughes' combinatorial proofs give canonical representations for classical logic proofs. In this paper we characterize classical combi-natorial proofs which also represent valid proofs for relevant logic with and without the mingle axiom. Moreover, we extend our syntax in order to represent combinatorial proofs for the more restrictive framework of entailment logic