Interaction and Depth against Nondeterminism in Proof Search

Deep inference is a proof theoretic methodology that generalizes the standard notion of inference of the sequent calculus, whereby inference rules become applicable at any depth inside logical expressions. Deep inference provides more freedom in the design of deductive systems for different logics and a rich combinatoric analysis of proofs. In particular, construction of exponentially shorter analytic proofs becomes possible, however with the cost of a greater nondeterminism than in the sequent calculus. In this paper, we show that the nondeterminism in proof search can be reduced without losing the shorter proofs and without sacrificing proof theoretic cleanliness. For this, we exploit an interaction and depth scheme in the logical expressions. We demonstrate our method on deep inference systems for multiplicative linear logic and classical logic, discuss its proof complexity and its relation to focusing, and present implementations.

💡 Research Summary

The paper “Interaction and Depth against Nondeterminism in Proof Search” addresses a central challenge in deep inference systems: while deep inference allows inference rules to be applied at any depth inside logical expressions, thereby enabling dramatically shorter proofs (often polynomial‑size where sequent‑calculus proofs are exponential), it also introduces a severe explosion of nondeterminism in proof search. The authors propose a principled method to tame this nondeterminism without sacrificing the proof‑size advantages or the clean proof‑theoretic properties of deep inference.

The key observation is that the main source of nondeterminism lies in the switch rule, which is responsible for context management in deep inference. The switch rule rearranges a structure of the form (