## 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.