Focused Proof-search in the Logic of Bunched Implications

V., Gheorghiu, Alexander; Marin, Sonia

doi:10.1007/978-3-030-71995-1_13

Cited by 7 publications

(7 citation statements)

References 34 publications

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“…For the remaining linear logic connectives, polarization and focusing is well understood. Weakening can be absorbed into the rules and it seems possible to restrict contraction to "neutral" structures such as proposed in [14], meaning that it can be applied only before focusing on a formula. The only point of atention would be associativity, and this is under investigation at the moment.…”

Section: Discussionmentioning

confidence: 99%

“…This not only gives the LL based system a more modern presentation (based on nested systems, like e.g. in [10,15]), but it also brings the notation closer to the one adopted by the Lambek community, like in [25]. Finally, it also uniformly extends several LL based systems present in the literature, as Example 8 in the next section shows.…”

Section: Introductionmentioning

confidence: 85%

“…Indeed, since the structural rules are circular, CacLL Σ is not adequate for proof search. But they can be "tamed" by eliminating the application of structural rules over structures, and restricting the application of the structural subexponential rules to neutral formulae, in the same way as done in [14] for contraction.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Non-associative, Non-commutative Multi-modal Linear Logic

Blaisdell

Kanovich

Kuznetsov

et al. 2022

Automated Reasoning

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Adding multi-modalities (called subexponentials) to linear logic enhances its power as a logical framework, which has been extensively used in the specification of e.g. proof systems, programming languages and bigraphs. Initially, subexponentials allowed for classical, linear, affine or relevant behaviors. Recently, this framework was enhanced so to allow for commutativity as well. In this work, we close the cycle by considering associativity. We show that the resulting system ($$\mathsf {acLL}_\varSigma $$ acLL Σ ) admits the (multi)cut rule, and we prove two undecidability results for fragments/variations of $$\mathsf {acLL}_\varSigma $$ acLL Σ .

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 85%

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Non-associative, Non-commutative Multi-modal Linear Logic

Blaisdell

Kanovich

Kuznetsov

et al. 2022

Automated Reasoning

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show abstract

“…Unfortunately, formalizing this would require dealing with variable binders, which we decided to forgo in this paper. It would also be natural to look at extensions such as GBI [19], extensions of BI with various modalities that are used in separation logic [7,16], or the recently proposed polarized sequent calculus for BI [21].…”

Section: Discussionmentioning

confidence: 99%

Semantic cut elimination for the logic of bunched implications, formalized in Coq

Frumin

2022

Proceedings of the 11th ACM SIGPLAN International Conference on Certified Programs and Proofs

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The logic of bunched implications (BI) is a substructural logic that forms the backbone of separation logic, the much studied logic for reasoning about heap-manipulating programs. Although the proof theory and metatheory of BI are mathematically involved, the formalization of important metatheoretical results is still incipient. In this paper we present a self-contained formalized, in the Coq proof assistant, proof of a central metatheoretical property of BI: cut elimination for its sequent calculus.The presented proof is semantic, in the sense that is obtained by interpreting sequents in a particular łuniversalž model. This results in a more modular and elegant proof than a standard Gentzen-style cut elimination argument, which can be subtle and error-prone in manual proofs for BI. In particular, our semantic approach avoids unnecessary inversions on proof derivations, or the uses of cut reductions and the multi-cut rule.Besides modular, our approach is also robust: we demonstrate how our method scales, with minor modifications, to (i) an extension of BI with an arbitrary set of simple structural rules, and (ii) an extension with an S4-like □ modality.

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“…Unfortunately, formalizing this would require dealing with variable binders, which we decided to forgo in this paper. It would also be natural to look at extensions such as GBI [18], extensions of BI with various modalities that are used in separation logic [26,42], or the recently proposed polarized sequent calculus for BI [43].…”

Section: Discussionmentioning

confidence: 99%

Semantic Cut Elimination for the Logic of Bunched Implications and Structural Extensions, Formalized in Coq

Frumin

2022

Preprint

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The logic of bunched implications (BI) is a substructural logic that forms the backbone of separation logic, the much studied logic for reasoning about heap-manipulating programs.Although the proof theory and metatheory of BI are mathematically involved, the formalization of important metatheoretical results is still incipient.In this paper we present a self-contained formalized, in the Coq proof assistant, proof of a central metatheoretical property of BI: cut elimination for its sequent calculus, as well the extension of cut elimination to sequent calculus with arbitrary structural rules. The presented proof is semantic, in the sense that is obtained by interpreting sequents in a particular ``universal'' model.This results in a more modular and elegant proof than a standard Gentzen-style cut elimination argument, which can be subtle and error-prone in manual proofs for BI.In particular, our semantic approach avoids unnecessary inversions on proof derivations, or the uses of cut reductions and the multi-cut rule. Our prof is modular and also robust. We demonstrate how our method scales to (i) all extensions of BI with arbitrary structural rules, and (ii) an extension with an S4-like ▯ modality.

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Focused Proof-search in the Logic of Bunched Implications

Cited by 7 publications

References 34 publications

Non-associative, Non-commutative Multi-modal Linear Logic

Non-associative, Non-commutative Multi-modal Linear Logic

Semantic cut elimination for the logic of bunched implications, formalized in Coq

Semantic Cut Elimination for the Logic of Bunched Implications and Structural Extensions, Formalized in Coq

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