Interaction and Depth against Nondeterminism in Deep Inference Proof Search

Deep inference is a proof theoretical methodology that generalises the traditional notion of inference of the sequent calculus. Deep inference provides more freedom in design of deductive systems for different logics and a rich combinatoric analysis of proofs. In particular, construction of exponentially shorter analytic proofs becomes possible, but with the cost of a greater nondeterminism than in the sequent calculus. In this paper, we extend our previous work on proof search with deep inference deductive systems. We argue that, by exploiting an interaction and depth scheme in the logical expressions, the nondeterminism in proof search can be reduced without losing the shorter proofs and without sacrificing from proof theoretical cleanliness.