Explanatory relations in arbitrary logics based on satisfaction systems, cutting and retraction

Aiguier, Marc; Atif, Jamal; Bloch, Isabelle; Pérez, Ramón Pino

doi:10.1016/j.ijar.2018.07.014

Cited by 11 publications

(13 citation statements)

References 28 publications

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“…By Zorn's lemma, there exists a maximally consistent set of formulas Γ which contains Γ when Γ is consistent. 4 Proposition 11. Let ∆ be a maximally consistent set of formulas.…”

Section: Morpho-logic: Interpretation Of Modalities In Toposmentioning

confidence: 99%

“…Related extensions can be found, on the logical side, in the context of institutions [5] and satisfaction systems [3,4], thus encompassing many different logics in a federative framework. Applications to typical reasoning problems (revision, abduction, spatial reasoning) were instantiated in this framework.…”

Section: Bmentioning

confidence: 99%

“…The motivation of this extension is mainly logical. Indeed, the motivations of the paper is to apply MM to logic, so-called morpho-logic (see Section 5), which has been proved useful to model knowledge, beliefs or preferences, and to model classical reasoning methods such as revision, fusion, abduction or spatial reasoning [3,4,5,13,21,24].…”

Section: Structuring Neighborhoods: Internal Topologymentioning

confidence: 99%

“…Candidate explanations of ϕ with respect to T are many. In a logical framework, following our previous work in [4] where abduction has been studied independently of a given logic, we will also study abduction as a form of inference. Intuitively, finding candidate explanations for ϕ with respect to T consists in cutting/eroding the class of models of T while remaining consistent with ϕ.…”

Section: Abductionmentioning

confidence: 99%

“…Necessity ✷ and possibility ✸ modalities are then interpreted by the two basic MM operators, namely erosion and dilation [19,48,37]. This interpretation allows for easy formulations of non-classical reasoning, including revision, merging, abduction [3,4,20,33], and spatial reasoning [2,5,13].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Morpho-logic from a Topos Perspective: Application to symbolic AI

Aiguier¹,

Bloch²,

Salim³

et al. 2023

Preprint

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Modal logics have proved useful for many reasoning tasks in symbolic artificial intelligence (AI), such as belief revision, spatial reasoning, among others. On the other hand, mathematical morphology (MM) is a theory for non-linear analysis of structures, that was widely developed and applied in image analysis. Its mathematical bases rely on algebra, complete lattices, topology. Strong links have been established between MM and mathematical logics, mostly modal logics. In this paper, we propose to further develop and generalize this link between mathematical morphology and modal logic from a topos perspective, i.e. categorial structures generalizing space, and connecting logics, sets and topology. Furthermore, we rely on the internal language and logic of topos. We define structuring elements, dilations and erosions as morphisms. Then we introduce the notion of structuring neighborhoods, and show that the dilations and erosions based on them lead to a constructive modal logic, for which a sound and complete proof system is proposed. We then show that the modal logic thus defined (called morpho-logic here), is well adapted to define concrete and efficient operators for revision, merging, and abduction of new knowledge, or even spatial reasoning.

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“…By Zorn's lemma, there exists a maximally consistent set of formulas Γ which contains Γ when Γ is consistent. 4 Proposition 11. Let ∆ be a maximally consistent set of formulas.…”

Section: Morpho-logic: Interpretation Of Modalities In Toposmentioning

confidence: 99%