What Is the Problem with Proof Nets for Classical Logic?

Straßburger, Lutz

doi:10.1007/978-3-642-13962-8_45

Cited by 4 publications

(2 citation statements)

References 20 publications

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“…The first approach rejects the involutive negation, which results in negation having a computational content that can be reified in the λµ calculus with a semantics in terms of control categories (see [16] for a survey). The second approach rejects the Cartesian structure for conjunctions, which requires a variant of proof-nets called flow graphs for the proofs and a semantics in terms of enriched Boolean categories [23,37].…”

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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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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“…Proof nets were originally introduced by Jean-Yves Girard with the intent to provide an alternative, more perspicuous syntax for linear logic [7,8]. Later developments have led to numerous variants, and proof-net systems are now available for a larger range of non-classical logics as well as for classical logic [11,14,16,17,22,23,31]. In general, proof nets are distinguished by the fact that they do not require the rigid sequentiality imposed by familiar sequent calculi; one can represent sequent proofs more freely modulo trivial permutations of the rules.…”

Section: Introductionmentioning

confidence: 99%

Complementary Proof Nets for Classical Logic

Pulcini

2023

Preprint

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A complementary system for a given logic is a proof system whose theorems are exactly the formulas that are not valid according to the logic in question. This article is a contribution to the complementary proof theory of classical propositional logic. In particular, we present a complementary proof-net system, CPN, that is sound and complete with respect to the set of all classically invalid (one-side) sequents. We also show that cut elimination in CPN enjoys strong normalization along with strong confluence (and, hence, uniqueness of normal forms).

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Complementary Proof Nets for Classical Logic

Pulcini,

Varzi

2023

Log. Univers.

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A complementary system for a given logic is a proof system whose theorems are exactly the formulas that are not valid according to the logic in question. This article is a contribution to the complementary proof theory of classical propositional logic. In particular, we present a complementary proof-net system, $$\textsf{CPN}$$ CPN , that is sound and complete with respect to the set of all classically invalid (one-side) sequents. We also show that cut elimination in $$\textsf{CPN}$$ CPN enjoys strong normalization along with strong confluence (and, hence, uniqueness of normal forms).

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What Is the Problem with Proof Nets for Classical Logic?

Abstract: Abstract. This paper is an informal (and nonexhaustive) overview over some existing notions of proof nets for classical logic, and gives some hints why they might be considered to be unsatisfactory.

Cited by 4 publications

References 20 publications

A multi-focused proof system isomorphic to expansion proofs

A multi-focused proof system isomorphic to expansion proofs

Complementary Proof Nets for Classical Logic

Complementary Proof Nets for Classical Logic

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