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# A Systematic Approach to Canonicity in the Classical Sequent Calculus

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LIPIcs.CSL.2012.183.pdf
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• 15 pages

## Cite As

Kaustuv Chaudhuri, Stefan Hetzl, and Dale Miller. A Systematic Approach to Canonicity in the Classical Sequent Calculus. In Computer Science Logic (CSL'12) - 26th International Workshop/21st Annual Conference of the EACSL. Leibniz International Proceedings in Informatics (LIPIcs), Volume 16, pp. 183-197, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)
https://doi.org/10.4230/LIPIcs.CSL.2012.183

## 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.
##### Keywords
• Sequent Calculus
• Canonicity
• Classical Logic
• Expansion Trees

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