Deep Inference in Proof Search: The Need for Shallow Inference

Deep inference is a proof theoretical formalism that generalises the “shallow inference” of sequent calculus by permitting the application of inference rules on subformulae like term rewriting rules. Deep inference makes it possible to build shorter proofs than sequent calculus proofs. However, deep inference in proof search introduces higher nondeterminism, an obstacle in front of applications. Deep inference is essential for designing system BV, an extension of multiplicative linear logic (MLL) with a self-dual non-commutative operator. MLL has shallow inference systems, whereas BV is impossible with a shallow-only system. As Tiu showed, any restriction on rule depth makes a system incomplete for BV. This paper shows that any restriction that rules out shallow rules makes the system incomplete, too. Our results indicate that for system BV, shallow and deep rules must coexist for completeness. We provide extensive empirical evidence that deep inference can still be faster than shallow inference when used strategically with a proof theoretical technique for reducing nondeterminism. We show that prioritising deeper rule instances, in general, reduces the cost of proof search by reducing the size of the managed contexts, consequently providing more immediate access to shorter proofs. Moreover, we identify a class of MLL formulae with deep inference proof search times that grow linearly in the number of atoms in contrast to an exponential growth pattern with shallow inference. We introduce a large and exhaustive benchmark for MLL, with and without mix, and a proof search framework to apply various search strategies, which should be of independent interest.