Morpho-logic from a Topos Perspective: Application to symbolic AI
- URL: http://arxiv.org/abs/2303.04895v1
- Date: Wed, 8 Mar 2023 21:24:25 GMT
- Title: Morpho-logic from a Topos Perspective: Application to symbolic AI
- Authors: Marc Aiguier, Isabelle Bloch, Salim Nibouche and Ramon Pino Perez
- Abstract summary: Modal logics have proved useful for many reasoning tasks in symbolic artificial intelligence (AI)
We propose to further develop and generalize this link between mathematical morphology and modal logic from a topos perspective.
We show that the modal logic is well adapted to define concrete and efficient operators for revision, merging, and abduction of new knowledge, or even spatial reasoning.
- Score: 2.781492199939609
- License: http://creativecommons.org/licenses/by-nc-nd/4.0/
- Abstract: 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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