Proof Nets for Herbrand’s Theorem

McKinley, Richard

doi:10.1145/2422085.2422090

Cited by 15 publications

(16 citation statements)

References 20 publications

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“…An important difference of expansion proofs with respect to other formalisms, such as proof forests [13] and Herbrand nets [19], is that the same @-expansion can occur multiple times. This phenomenon is very natural, as soon as one realizes that the weak regularity condition that we have imposed corresponds to an interpretation of eigenvariables as Skolem functions.…”

Section: (Eigenvariable Condition) For Every @-Expansion`α In P the mentioning

confidence: 99%

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Expansion trees with cut

Aschieri

Hetzl

Weller

2019

Math. Struct. Comp. Sci.

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Herbrand's theorem is one of the most fundamental insights in logic. From the syntactic point of view, it suggests a compact representation of proofs in classical first-and higher-order logic by recording the information of which instances have been chosen for which quantifiers. This compact representation is known in the literature as Miller's expansion tree proof. It is inherently analytic and hence corresponds to a cut-free sequent calculus proof. Recently several extensions of such proof representations to proofs with cuts have been proposed. These extensions are based on graphical formalisms similar to proof nets and are limited to prenex formulas. In this paper we present a new syntactic approach that directly extends Miller's expansion trees by cuts and covers also non-prenex formulas. We describe a cut-elimination procedure for our expansion trees with cut that is based on the natural reduction steps and show that it is weakly normalizing.

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Section: (Eigenvariable Condition) For Every @-Expansion`α In P the mentioning

confidence: 99%

“…Two proof formalisms manipulating only formula instances and incorporating a notion of cut have recently been proposed: proof forests [13] and Herbrand nets [19]. While some definitions in the setting of proof forests are motivated by the game semantics for classical arithmetic [8], Herbrand nets are based on methods for proof nets [12].…”

Section: Introductionmentioning

confidence: 99%

Expansion trees with cut

Aschieri

Hetzl

Weller

2019

Math. Struct. Comp. Sci.

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“…If one refrains from fixing a reduction strategy one may still obtain a strongly normalizing though non-confluent system by using sufficiently strong local reductions [38,39]. Another approach is to carry out cut-elimination in a more abstract formalism, similar to a proof-net, on the level of quantifiers [15,28]. The reduction in such a setting is typically not confluent and strong normalization is open [28] or known not to hold [15].…”

Section: Related Workmentioning

confidence: 99%

“…Another approach is to carry out cut-elimination in a more abstract formalism, similar to a proof-net, on the level of quantifiers [15,28]. The reduction in such a setting is typically not confluent and strong normalization is open [28] or known not to hold [15]. Confluence (up to the equivalence relation of having the same expansion tree) as well as normalization can be recovered for a class of proofs [20] by considering a maximal abstract reduction based on tree grammars [18] which contains all concrete reductions.…”

Section: Related Workmentioning

confidence: 99%

A multi-focused proof system isomorphic to expansion proofs

Chaudhuri

Hetzl

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.

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“…Structures similar to the above Bpπq have been investigated also in [Hei10] and [McK13] where they form the basis of proof net like formalisms using local reductions for quantifiers in classical first-order logic. Our aim in this work is however quite different: we use these structures for a global analysis of the sequent calculus.…”

Section: Proofs and Grammarsmentioning

confidence: 99%

Herbrand-Confluence

Hetzl¹,

Straßburger²

2013

Logical Methods in Computer Science

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Abstract. We consider cut-elimination in the sequent calculus for classical first-order 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 cut-free proofs. We analyze which witnesses they choose for which quantifiers, or in other words: we only consider the Herbrand-disjunction of a cut-free proof. Our main theorem is a confluence result for a natural class of proofs: all (possibly infinitely many) normal forms of the non-erasing reduction lead to the same Herbrand-disjunction.

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Proof Nets for Herbrand’s Theorem

Cited by 15 publications

References 20 publications

Expansion trees with cut

Expansion trees with cut

A multi-focused proof system isomorphic to expansion proofs

Herbrand-Confluence

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