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Deep inference is a proof theoretic methodology that generalizes the standardnotion of inference of the sequent calculus, whereby inference rules becomeapplicable at any depth inside logical expressions. Deep inference providesmore freedom in the design of deductive systems for different logics and a richcombinatoric analysis of proofs. In particular, construction of exponentiallyshorter analytic proofs becomes possible, however with the cost of a greaternondeterminism than in the sequent calculus. In this paper, we show that thenondeterminism in proof search can be reduced without losing the shorter proofsand without sacrificing proof theoretic cleanliness. For this, we exploit aninteraction and depth scheme in the logical expressions. We demonstrate ourmethod on deep inference systems for multiplicative linear logic and classicallogic, discuss its proof complexity and its relation to focusing, and presentimplementations.

Interaction and Depth against Nondeterminism in Proof Search