Expansion Nets: Proof-Nets for Propositional Classical Logic

McKinley, Richard

doi:10.1007/978-3-642-16242-8_38

Cited by 7 publications

(13 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We have sequentialization into LK * (Theorem 49), and weakly normalizing cut-elimination directly on expansionnets (Theorem 61). The last two of these results are new to the paper (although the former was sketched in [23]); their proofs rely on the characterization of subnets of expansion nets, including the new notion of contiguous subnet defined in this paper. In addition to these properties, expansion-nets also identify a more natural set of sequent derivations than do the previously existing notions of abstract proof.…”

Section: Resultsmentioning

confidence: 98%

“…In addition, we present a cut-elimination procedure for expansion nets (proof transformations which we prove, in Propositions 56 -59 to preserve correctness) which are weakly normalizing (Lemma 60 and Theorem 61 detail a strategy for reducing any net with cuts to a cut-free net). This result was absent from [23]: with it, we can see that expansion nets have polynomial-time proof checking, sequentialization into a sequent calculus and cut-elimination preserving sequent-calculus correctness -the first notion of abstract proof for propositional classical logic to satisfy all of these properties.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Canonical proof nets for classical logic

McKinley

2013

Annals of Pure and Applied Logic

Self Cite

View full text Add to dashboard Cite

Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proof-net, they are in essence the same proof. Providing a convincing proof-net counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs. By convincing, we mean that (a) there should be a canonical function from sequent proofs to proof nets, (b) it should be possible to check the correctness of a net in polynomial time, (c) every correct net should be obtainable from a sequent calculus proof, and (d) there should be a cut-elimination procedure which preserves correctness.Previous attempts to give proof-net-like objects for propositional classical logic have failed at least one of the above conditions. In [23], the author presented a calculus of proof nets (expansion nets) satisfying (a) and (b); the paper defined a sequent calculus corresponding to expansion nets but gave no explicit demonstration of (c). That sequent calculus, called LK * in this paper, is a novel one-sided sequent calculus with both additively and multiplicatively formulated disjunction rules. In this paper (a self-contained extended version of [23]) , we give a full proof of (c) for expansion nets with respect to LK * , and in addition give a cut-elimination procedure internal to expansion nets -this makes expansion nets the first notion of proof-net for classical logic satisfying all four criteria.

show abstract

Section: Resultsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Canonical proof nets for classical logic

McKinley

2013

Annals of Pure and Applied Logic

Self Cite

View full text Add to dashboard Cite

show abstract

“…Generally speaking, such answers are limited to the propositional fragment, and are primarily concerned with abstracting the propositional structure of sequent proofs [13,35,24,22,27,33]. In the first-order case, it is more common to ignore the propositional structure and instead consider only the first-order content of proofs.…”

Section: Related Workmentioning

confidence: 99%

A multi-focused proof system isomorphic to expansion proofs

Chaudhuri

Hetzl

Miller

2014

J Logic Computation

View full text Add to dashboard Cite

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.

show abstract

“…We look also to extend our system beyond prenex normal form, first to encompass a treatment of the propositional connectives. The paper [17] gives a multiplicative treatment of classical proof nets which improves on [20] by replacing contraction (binary, defined on all formulae) by expansion (n-ary, defined only on positive formulae). Contraction on negative atoms (needed for completeness) is handled by the same basic binding structure used here to model quantification.…”

Section: Conclusion and Further Workmentioning

confidence: 99%

Proof Nets for Herbrand’s Theorem

McKinley

2013

ACM Trans. Comput. Logic

Self Cite

View full text Add to dashboard Cite

This paper explores the connection between two central results in the proof theory of classical logic: Gentzen's cut-elimination for the sequent calculus and Herbrands "fundamental theorem". Starting from Miller's expansion-tree-proofs, a highly structured way presentation of Herbrand's theorem, we define a calculus of weakening-free proof nets for (prenex) first-order classical logic, and give a weakly-normalizing cut-elimination procedure. It is not possible to formulate the usual counterexamples to confluence of cutelimination in this calculus, but it is nonetheless nonconfluent, lending credence to the view that classical logic is inherently nonconfluent.

show abstract

Expansion Nets: Proof-Nets for Propositional Classical Logic

Cited by 7 publications

References 12 publications

Canonical proof nets for classical logic

Canonical proof nets for classical logic

A multi-focused proof system isomorphic to expansion proofs

Proof Nets for Herbrand’s Theorem

Contact Info

Product

Resources

About