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Journal Articles Journal of Logic and Computation Year : 2014

## A Multi-Focused Proof System Isomorphic to Expansion Proofs

Kaustuv Chaudhuri
Stefan Hetzl
• Function : Author
Dale Miller

#### Abstract

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. 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.

#### Domains

Logic in Computer Science [cs.LO]

### Dates and versions

hal-00937056 , version 1 (27-01-2014)

### Identifiers

• HAL Id : hal-00937056 , version 1

### Cite

Kaustuv Chaudhuri, Stefan Hetzl, Dale Miller. A Multi-Focused Proof System Isomorphic to Expansion Proofs. Journal of Logic and Computation, 2014. ⟨hal-00937056⟩

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