Section: New Results

Recovering Proof Structures in the Sequent Calculus

Participants : Kaustuv Chaudhuri, Stefan Hetzl, Dale Miller.

The sequent calculus is often criticized as a proof syntax because it contains a lot of noise. It records the precise minute sequence of operations that was used to construct a proof, even when the order of some proof steps in the sequence is irrelevant and when some of the steps are unnecessary or involve detours. These features lead to several technical problems: for example, cut-elimination in the classical sequent calculus LK, as originally developed by Gentzen, is not confluent, and hence proof composition in LK is not associative. Many people choose to discard the sequent calculus when attempting to design a better proof syntax with the desired properties.

In recent years, there has been a project at Parsifal to recover some of these alternative proof syntaxes by imposing a certain abstraction over sequent proofs. The earliest example of this was in  [37] , where we showed a class of sequent proofs that were isomorphic to proof nets for multiplicative linear logic. In 2012, we were able to obtain a similar result for first-order classical logic, wherein we defined a class of sequent proofs that are isomorphic to expansion trees, a generalization of Herbrand disjunctions that is in some sense a minimalistic notion of proof for classical logic. This result was published at the CSL 2012 conference [22] and a journal version is in preparation.

Our technique for recovering these dramatically different proof structures directly in the sequent calculus involves the use of maximal multi-focusing which gives a syntactic characterization of those sequent proofs that: (1) have a “don't care” ordering of proof steps where the order does not matter, and (2) groups larger logical steps, called actions, into a maximally parallel form where only important orderings of actions are recorded. This technique was pioneered at Parsifal, and we have barely scratched the surface of its applications.