An Analytic Propositional Proof System On Graphs

Matteo Acclavio (1) , Ross Horne (2) , Lutz Strassburger (3)

1 Dipartimento di Matematica e Fisica [Roma]
2 University of Luxembourg [Luxembourg]
3 PARTOUT - Automatisation et ReprésenTation: fOndation du calcUl et de la déducTion

In this paper we present a proof system that operates on graphs instead of formulas. Starting from the well-known relationship between formulas and cographs, we drop the cograph-conditions and look at arbitrary (undirected) graphs. This means that we lose the tree structure of the formulas corresponding to the cographs, and we can no longer use standard proof theoretical methods that depend on that tree structure. In order to overcome this difficulty, we use a modular decomposition of graphs and some techniques from deep inference where inference rules do not rely on the main connective of a formula. For our proof system we show the admissibility of cut and a generalization of the splitting property. Finally, we show that our system is a conservative extension of multiplicative linear logic with mix, and we argue that our graphs form a notion of generalized connective.

Mots clés: Proof theory; prime graphs; cut elimination; deep inference; splitting; analyticity

Domaines: Logique en informatique [cs.LO]

Matteo Acclavio, Ross Horne, Lutz Strassburger. An Analytic Propositional Proof System On Graphs. Logical Methods in Computer Science, 2022, 18 (4), ⟨10.46298/LMCS-18(4:1)2022⟩. ⟨hal-03087392v2⟩