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Conference Papers Year : 2012

## A Systematic Approach to Canonicity in the Classical Sequent Calculus

Kaustuv Chaudhuri
Stefan Hetzl
• Function : Author
• PersonId : 933430
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 of abstracting away the details from classical cut-free sequent proofs. We show that, among the multi-focused proofs, the maximally multi-focused proofs that make the foci as parallel as possible are canonical. Moreover, such proofs are isomorphic to expansion proofs---a well known, minimalistic, and parallel generalization of Herbrand disjunctions---for classical first-order logic. This technique is a systematic way to recover the desired essence of any sequent proof without abandoning the sequent calculus.

#### Domains

Logic in Computer Science [cs.LO]

### Dates and versions

hal-00772396 , version 1 (10-01-2013)

### Identifiers

• HAL Id : hal-00772396 , version 1
• DOI :

### Cite

Kaustuv Chaudhuri, Stefan Hetzl, Dale Miller. A Systematic Approach to Canonicity in the Classical Sequent Calculus. 21st EACSL Annual Conference on Computer Science Logic, Sep 2012, Fontainebleau, France. pp.183-197, ⟨10.4230/LIPIcs.CSL.2012.183⟩. ⟨hal-00772396⟩

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