## Section: New Results

### A multi-focused proof system isomorphic to expansion proofs

Participants : Kaustuv Chaudhuri, Stefan Hetzl [Vienna University of Technology, Vienna, Austria] , Dale Miller.

The sequent calculus is often criticized for requiring proofs to contain large
amounts of low-level syntactic details that can obscure the essence of a given
proof.
Because each inference rule introduces only a single connective, sequent
proofs can separate closely related steps—such as instantiating a block of
quantifiers—by irrelevant noise.
Moreover, the sequential nature of sequent proofs forces proof steps that are
syntactically non-interfering and permutable to nevertheless be written in
some arbitrary order.
The sequent calculus thus lacks a notion of *canonicity*: proofs that
should be considered essentially the same may not have a common syntactic
form.
To fix this problem, many researchers have proposed replacing the sequent
calculus with proof structures that are more parallel or geometric.
Proof-nets, matings, and atomic flows are examples of such
*revolutionary* formalisms.
In [13] , we propose, instead, an
*evolutionary* approach to recover canonicity within the
sequent calculus, which we illustrate for classical first-order
logic.
The essential element of our approach is the use of a *multi-focused*
sequent calculus as the means for abstracting away low-level details
from classical cut-free sequent proofs.
We show that, among the multi-focused proofs, the *maximally
multi-focused* proofs that collect together all possible
parallel foci are canonical.
Moreover, if we start with a certain focused sequent proof system,
such proofs are isomorphic to *expansion proofs*—a well
known, minimalistic, and parallel generalization of Herbrand
disjunctions—for classical first-order logic.
This technique appears to be a systematic way to recover the “essence of
proof” from within sequent calculus proofs.