The three dimensions of proofs

In this document, we study a 3-polygraphic translation for the proofs of SKS, a formal system for classical propositional logic. We prove that the free 3-category generated by this 3-polygraph describes the proofs of classical propositional logic modulo structural bureaucracy. We give a 3-dimensional generalization of Penrose diagrams and use it to provide several pictures of a proof. We sketch how local transformations of proofs yield a non contrived example of 4-dimensional rewriting. Outline In the first section of this paper, we give a 2-dimensional translation of the formulas of system SKS, a formal system for propositional classical logic [Brünnler 2004] expressed in the style of the calculus of structures [Guglielmi 2004]. The idea consists in the replacement of formulas by circuit-like objects organized in a 2-polygraph [Burroni 1993]. This construction is formalized in theorem 1.4.16. We proceed to section 2, whose purpose is to translate the proofs of SKS into 3-dimensional objects that form a 3-polygraph. There we note that every inference rule can be interpreted as a directed 3-cell between two circuits. We prove theorem 2.4.3 stating that the 3-polygraph we have built can be equipped with a proof theory which is the same as the SKS one. Section 3 is where the 3-dimensional nature of proofs happens to be useful: theorem 3.3.1 states that the structural bureaucracy of SKS [Guglielmi 2004] corresponds to topological moves of 3-cells, called exchange relations. In section 4 we draw several 3-dimensional representations of a given proof. Section 5 is an informal discussion about the 4-dimensional nature of local transformations of 3-dimensional proofs. The final section 6 describes how to adapt the work done here to SLLS, the calculus of structures-style formalism for linear logic [Straßburger 2003]. 1 The two dimensions of formulas This section gives a 2-dimensional translation of SKS formulas, heavily inspired by the one already known for terms, studied in [Burroni 1993], [Lafont 2003] and [Guiraud 2004]. After having described the SKS formulas (1.1), we give the intuition behind their translation into circuit-like objects (1.2): this works by replacing variables with explicit local resources management operators. This construction requires some theoretical material which is recalled at this moment (1.3). Then we formalize the translation and study its properties (1.4): the main purpose of this technical part, that can be skipped on a first approach, is to prove that we can compute a canonical representative for circuits corresponding to the same SKS formula (theorem 1.4.16). Finally we translate the structural congruence on SKS formulas into a congruence on the corresponding circuits (1.5). Institut de mathématiques de Luminy, Marseille, France http://iml.univ-mrs.fr/∼guiraud