On the non-confluence of cut-elimination

Miller

2014

J Logic Computation

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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.

Section: Related Workmentioning

confidence: 99%

A multi-focused proof system isomorphic to expansion proofs

Chaudhuri

Miller

2014

J Logic Computation

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“…The large normal forms of acyclic proofs are therefore only due to repetitions of the same formulas and thus mathematically meaningless. The analogous question for cyclic proofs is open: it has been shown in [1] that a (cyclic) proof can have a non-elementary number of reachable Herbranddisjunctions. It is unclear however whether there exists a (cyclic) proof having infinitely many reachable Herbrand-disjunctions.…”

Section: Lemma 10 Every Directed Proof Is Acyclicmentioning

confidence: 99%

“…Each restriction of the full set of proof rewrite rules has the (sometimes intended) effect of limiting the obtainable results. However, recent work [1] has shown that the number of (significantly different) normal forms may increase non-elementarily in the size of the original proof. An investigation of cut-elimination as non-deterministic computation can be found in [37,38], including a case study of a non-confluent proof in [37].…”

Section: Introductionmentioning

confidence: 99%

On the form of witness terms

2010

Arch. Math. Logic

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International audienceWe investigate the development of terms during cut-elimination in first-order logic and Peano arithmetic for proofs of existential formulas. The form of witness terms in cut-free proofs is characterized in terms of structured combinations of basic substitutions. Based on this result, a regular tree grammar computing witness terms is given and a class of proofs is shown to have only elementary cut-elimination

“…Note that we do not impose any restriction on the strategy that can be applied. Therefore this set of reduction rules is not confluent, see [4] for a strongly non-confluent example. It is also not strongly normalising by allowing the doublecontraction example found e.g.…”

Section: Cut-eliminationmentioning

confidence: 99%

“…[7,23], to be used as a programming language this effect is intended. However, from the foundational point of view that asks for the constructive content of a mathematical proof in classical logic it has the unfortunate consequence of strongly reducing the degree of generality in which the original proof is considered, see [4].…”

Section: Cut-eliminationmentioning

confidence: 99%

A Sequent Calculus with Implicit Term Representation

2010

Computer Science Logic

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We investigate a modification of the sequent calculus which separates a first-order proof into its abstract deductive structure and a unifier which renders this structure a valid proof. We define a cutelimination procedure for this calculus and show that it produces the same cut-free proofs as the standard calculus, but, due to the implicit representation of terms, it provides exponentially shorter normal forms. This modified calculus is applied as a tool for theoretical analyses of the standard calculus and as a mechanism for a more efficient implementation of cut-elimination.