从拓扑观点看形态逻辑：对符号AI的应用
Morpho-logic from a Topos Perspective: Application to symbolic AI
摘要：
模态逻辑已被证明适用于符号人工智能（AI）中的许多推理任务，例如信念修正，空间推理等。另一方面，数学形态学（MM）是一种用于结构非线性分析的理论，在图像分析中得到了广泛的发展和应用。它的数学基础依赖于代数，完备格，拓扑。MM和数理逻辑之间建立了很强的联系，主要是模态逻辑。在本文中，我们建议进一步发展和推广数学形态学和模态逻辑之间的联系，从topos的角度来看，即范畴结构推广空间，并连接逻辑，集合和拓扑。此外，我们依赖于内部语言和逻辑的拓扑。我们定义结构元素，膨胀和腐蚀作为态射。然后，我们引入结构化邻域的概念，并表明基于它们的膨胀和腐蚀导致一个建设性模态逻辑，为它提出了一个健全和完整的证明系统。然后，我们表明，这样定义的模态逻辑（这里称为形态逻辑），是很好地适应定义具体和有效的运营商的修订，合并，和绑架的新知识，甚至空间推理。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
