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A Systematic Approach to Canonicity in the Classical Sequent Calculus
"... The sequent calculus is often criticized for requiring proofs to be laden with large volumes of lowlevel syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, cutfree sequent proofs can separate closely related steps—such ..."
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The sequent calculus is often criticized for requiring proofs to be laden with large volumes of lowlevel syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, cutfree 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 noninterfering 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 revolt against the sequent calculus and replace it with proof structures that are more parallel or geometric. Proofnets, matings, and atomic flows are examples of such revolutionary formalisms. In this paper, we propose taking, instead, an evolutionary approach to recover canonicity within the sequent calculus, an approach we illustrate for classical firstorder logic. We use a multifocused sequent system as our means of abstracting away the details from classical sequent proofs. We then show that, among the focused sequent proofs, the maximally multifocused proofs, which make the foci as parallel as possible, are canonical. Moreover, such proofs are isomorphic to expansion tree proofs—a well known, simple, and parallel generalization of Herbrand disjunctions—for classical firstorder logic. We thus provide a systematic method of recovering the essence of any sequent proof without abandoning the sequent calculus. 1
A MultiFocused Proof System Isomorphic to Expansion Proofs
, 2013
"... The sequent calculus is often criticized for requiring proofs to contain large amounts of lowlevel 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 instantiatin ..."
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The sequent calculus is often criticized for requiring proofs to contain large amounts of lowlevel 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 noninterfering 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. Proofnets, 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 firstorder logic. The essential element of our approach is the use of a multifocused sequent calculus as the means for abstracting away lowlevel details from classical cutfree sequent proofs. We show that, among the multifocused proofs, the maximally multifocused 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 firstorder logic. This technique appears to be a systematic way to recover the “essence of proof ” from within sequent calculus proofs. 1