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On the form of witness terms
 ARCH. MATHEMATICAL LOGIC
, 2010
"... We investigate the development of terms during cutelimination in firstorder logic and Peano arithmetic for proofs of existential formulas. The form of witness terms in cutfree proofs is characterized in terms of structured combinations of basic substitutions. Based on this result, a regular tree ..."
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Cited by 4 (3 self)
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We investigate the development of terms during cutelimination in firstorder logic and Peano arithmetic for proofs of existential formulas. The form of witness terms in cutfree 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 cutelimination.
HerbrandConfluence for Cut Elimination in Classical First Order Logic
"... We consider cutelimination in the sequent calculus for classical firstorder logic. It is well known that this system, in its most general form, is neither confluent nor strongly normalizing. In this work we take a coarser (and mathematically more realistic) look at cutfree proofs. We analyze whic ..."
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Cited by 3 (3 self)
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We consider cutelimination in the sequent calculus for classical firstorder logic. It is well known that this system, in its most general form, is neither confluent nor strongly normalizing. In this work we take a coarser (and mathematically more realistic) look at cutfree proofs. We analyze which witnesses they choose for which quantifiers, or in other words: we only consider the Herbranddisjunction of a cutfree proof. Our main theorem is a confluence result for a natural class of proofs: all (possibly infinitely many) normal forms of the nonerasing reduction lead to the same Herbranddisjunction.
Notre Dame Journal of Formal Logic The Computational Content of Arithmetical Proofs
"... Abstract For any extension T of IΣ1 having a cutelimination property extending that of IΣ1, the number of different proofs that can be obtained by cutelimination from a single Tproof cannot be bound by a function which is provably total in T. 1 ..."
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Abstract For any extension T of IΣ1 having a cutelimination property extending that of IΣ1, the number of different proofs that can be obtained by cutelimination from a single Tproof cannot be bound by a function which is provably total in T. 1
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 Sequent Calculus with Implicit Term Representation
, 2010
"... We investigate a modification of the sequent calculus which separates a firstorder 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 cutfree proofs as the s ..."
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We investigate a modification of the sequent calculus which separates a firstorder 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 cutfree 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 cutelimination.
The Isomorphism Between Expansion Proofs and MultiFocused Sequent Proofs
, 2012
"... 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 sequent calculus proofs. 1
HerbrandConfluence
, 2013
"... We consider cutelimination in the sequent calculus for classical firstorder logic. It is well known that this system, in its most general form, is neither confluent nor strongly normalizing. In this work we take a coarser (and mathematically more realistic) look at cutfree proofs. We analyze whic ..."
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We consider cutelimination in the sequent calculus for classical firstorder logic. It is well known that this system, in its most general form, is neither confluent nor strongly normalizing. In this work we take a coarser (and mathematically more realistic) look at cutfree proofs. We analyze which witnesses they choose for which quantifiers, or in other words: we only consider the Herbranddisjunction of a cutfree proof. Our main theorem is a confluence result for a natural class of proofs: all (possibly infinitely many) normal forms of the nonerasing reduction lead to the same Herbranddisjunction. 1
Author manuscript, published in "CSL 2012 (2012)" DOI: 10.4230/LIPIcs.CSL.2012.320 HerbrandConfluence for Cut Elimination in Classical First Order Logic
, 2012
"... We consider cutelimination in the sequent calculus for classical firstorder logic. It is well known that this system, in its most general form, is neither confluent nor strongly normalizing. In this work we take a coarser (and mathematically more realistic) look at cutfree proofs. We analyze whic ..."
Abstract
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We consider cutelimination in the sequent calculus for classical firstorder logic. It is well known that this system, in its most general form, is neither confluent nor strongly normalizing. In this work we take a coarser (and mathematically more realistic) look at cutfree proofs. We analyze which witnesses they choose for which quantifiers, or in other words: we only consider the Herbranddisjunction of a cutfree proof. Our main theorem is a confluence result for a natural class of proofs: all (possibly infinitely many) normal forms of the nonerasing reduction lead to the same Herbranddisjunction.