An Algebraic Glimpse at Bunched Implications and Separation Logic

Jipsen, Peter; Litak, Tadeusz

doi:10.1007/978-3-030-76920-8_5

Cited by 3 publications

(2 citation statements)

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“…Similarly, the close connection between inquisitive logic and dependence logic that has been studied e.g., in [10][11] suggests that a similar semantics might be developed for the system of propositional dependence logic [46]. This direction, originally hinted at in [1], also raises the issue of possible connections between the present framework and the setting of residuated lattices which is employed to give a semantics to separation logic and related formalisms [32].…”

Section: The Methods Of Nd-extensionsmentioning

confidence: 72%

An Algebraic Approach to Inquisitive and -Logics

Bezhanishvili

Grilletti

Quadrellaro

2021

The Review of Symbolic Logic

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This article provides an algebraic study of the propositional system $\mathtt {InqB}$ of inquisitive logic. We also investigate the wider class of $\mathtt {DNA}$ -logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, $\mathtt {DNA}$ -varieties. We prove that the lattice of $\mathtt {DNA}$ -logics is dually isomorphic to the lattice of $\mathtt {DNA}$ -varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff’s classic variety theorems. We also introduce locally finite $\mathtt {DNA}$ -varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of $\mathtt {InqB}$ is dually isomorphic to the ordinal $\omega +1$ and give an axiomatisation of these logics via Jankov $\mathtt {DNA}$ -formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].1

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Section: The Methods Of Nd-extensionsmentioning

confidence: 72%

An Algebraic Approach to Inquisitive and -Logics

Bezhanishvili

Grilletti

Quadrellaro

2021

The Review of Symbolic Logic

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“…Unfortunately, the finite model property proof in[18] does not hold. The argument there relies on a version of the Curry's lemma (limiting a number of contractions that can occur in a given sequent in a proof search) which does not hold in BI (see[41]).…”

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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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