Ribbon tensorial logic A functorial bridge between proofs and knots

Tensorial logic is a primitive logic of tensor and negation which refines linear logic by relaxing the hypothesis that tensorial negation A 7→ ¬A is involutive. The resulting logic of linear continuations provides a proof-theoretic account of game semantics, where the formulas and proofs of the logic reflect univoquely dialogue games and innocent strategies. In the present paper, we introduce a topologically-aware version of tensorial logic, called ribbon tensorial logic. We associate to every proof of the logic a ribbon tangle which tracks the flow of tensorial negations inside the proof. The translation is functorial: it is performed by exhibiting a correspondence between the notion of dialogue category in proof theory and the notion of ribbon category in knot theory. Our main theorem is that the translation is faithful: two tensorial proofs are equal modulo commuting conversions if and only if the associated ribbon tangles are equal up to topological deformation. The “proof-as-tangle” theorem may be also understood as a coherence theorem for ribbon dialogue categories. By connecting in this functorial way tensorial logic and knot theory, we hope to investigate further the unexpected topological nature of proofs and programs, and of their dialogical interpretation in game semantics.