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. Introduction Proof search is of central importance in automated reasoning, especially for a broad range of applications in the fields of automated theorem proving, software verification and artificial intelligence. The development of formalisms, techniques and principles that allow the construction of shorter proofs in different logics is a requirement for the applications in these fields. This requirement gains an even greater emphasis for logics, where highly optimized techniques, such as classical resolution, are not applicable. Deep inference [18] is a proof theoretic methodology that generalizes the notion of inference in formalisms such as the sequent calculus, natural deduction and analytic tableaux by introducing a top-down symmetry of the inference rules. This symmetry brings about a combinatoric wealth that gives rise to properties that are otherwise not observable, and thereby results in a broad spectrum of theoretical and practical consequences. For example, at the theoretical front, a manifestation of the top-down symmetry of the deep inference rules is observed as the cut rule and the axiom become dual deep inference rules [18, 3, 37].1 In the standard sequent calculus, this duality remains implicit in the inference rules, and can only be revealed explicitly, for example, by constructing a proof net or an atomic flow graph [19, 21], that is, by removing the deductive information in the proofs that displays their 2012 ACM CCS: [Theory of computation]: Logic.