Canonical Sequent Proofs via Multi-Focusing

Chaudhuri, Kaustuv; Miller, Dale; Saurin, Alexis

doi:10.1007/978-0-387-09680-3_26

Cited by 41 publications

(52 citation statements)

References 15 publications

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“…While proof-nets and focalized proofs are the results of diverging choices of prooftheoretical design (parallelism versus hypersequentiality), this suggests that they actually may be dierent aspects of the same phenomenon as already advocated in the study of multi-focusing [34].…”

Section: Resultsmentioning

confidence: 95%

On the Dependencies of Logical Rules

Bagnol

Doumane

Saurin

2015

Lecture Notes in Computer Science

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Abstract. Many correctness criteria have been proposed since linear logic was introduced and it is not clear how they relate to each other. In this paper, we study proof-nets and their correctness criteria from the perspective of dependency, as introduced by Mogbil and Jacobé de Naurois. We introduce a new correctness criterion, called DepGraph, and show that together with Danos' contractibility criterion and Mogbil and Naurois criterion, they form the three faces of a notion of dependency which is crucial for correctness of proof-structures. Finally, we study the logical meaning of the dependency relation and show that it allows to recover and characterize some constraints on the ordering of inferences which are implicit in the proof-net.

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Section: Resultsmentioning

confidence: 95%

On the Dependencies of Logical Rules

Bagnol

Doumane

Saurin

2015

Lecture Notes in Computer Science

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“…A recent branch of work on maximal multi-focusing (Chaudhuri, Miller, and Saurin 2008a;Chaudhuri, Hetzl, and Miller 2012) has demonstrated that focused proofs can be further restricted to become even more canonical: in each application to a specific logic, the resulting representations are equivalent to existing representations capturing the identity of proofs -proof nets for linear logic, expansion proofs for classical logic. Scherer and Rémy (2015) applied focusing to the λ-calculus with non-empty sums.…”

Section: Focusingmentioning

confidence: 99%

“…Logicians introduced maximal multi-focusing (Chaudhuri, Miller, and Saurin 2008a) to quotient over those reorderings, and Scherer and Rémy (2015) expressed this in a programming setting as saturation. The idea of maximal multi-focusing is to force each noninvertible phase to happen as early as possible in a term, in parallel, removing the potential for reordering them.…”

Section: Saturated Focused λ-Calculusmentioning

confidence: 99%

Deciding equivalence with sums and the empty type

Scherer

2017

Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages

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The logical technique of focusing can be applied to the λ-calculus; in a simple type system with atomic types and negative type formers (functions, products, the unit type), its normal forms coincide with βη-normal forms. Introducing a saturation phase gives a notion of quasi-normal forms in presence of positive types (sum types and the empty type). This rich structure let us prove the decidability of βη-equivalence in presence of the empty type, the fact that it coincides with contextual equivalence, and a finite model property.

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“…Approaches such as focusing [3], [5] reduce the number of instances of duplicated search, but do not ultimately solve the problem. The authors are not aware of an inductive, bottom-up search algorithm that matches our complexity.…”

Section: Efficient Provability and Proof Searchmentioning

confidence: 99%

Complexity Bounds for Sum-Product Logic via Additive Proof Nets and Petri Nets

Heijltjes

Hughes

2015

2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science

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Abstract-We investigate efficient algorithms for the additive fragment of linear logic. This logic is an internal language for categories with finite sums and products, and describes concurrent two-player games of finite choice. In the context of session types, typing disciplines for communication along channels, the logic describes the communication of finite choice along a single channel.We give a simple linear time correctness criterion for unit-free propositional additive proof nets via a natural construction on Petri nets. This is an essential ingredient to linear time complexity of the second author's combinatorial proofs for classical logic.For full propositional additive linear logic, including the units, we give a proof search algorithm that is linear-time in the product of the source and target formula, and an algorithm for proof net correctness that is of the same time complexity. We prove that proof search in first-order additive linear logic is NP-complete.

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Canonical Sequent Proofs via Multi-Focusing

Cited by 41 publications

References 15 publications

On the Dependencies of Logical Rules

On the Dependencies of Logical Rules

Deciding equivalence with sums and the empty type

Complexity Bounds for Sum-Product Logic via Additive Proof Nets and Petri Nets

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