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.

翻译：模态逻辑在符号人工智能的诸多推理任务中（如信念修正、空间推理等）已被证明具有实用性。另一方面，数学形态学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发展和应用，其数学基础依赖于代数、完备格和拓扑学。数学形态学与数理逻辑（主要是模态逻辑）之间已建立了紧密联系。本文提出从拓扑斯视角出发——即推广空间概念并连接逻辑、集合与拓扑的范畴结构——进一步发展和推广数学形态学与模态逻辑之间的联系。此外，我们利用拓扑斯的内在语言和逻辑，将结构元素、膨胀和腐蚀定义为态射。进而引入结构邻域的概念，并证明基于这些概念的膨胀和腐蚀会导致一种构造性模态逻辑，为此提出了一个可靠且完备的证明系统。最后，我们证明这样定义的模态逻辑（本文称之为形态逻辑）非常适合定义用于知识修正、合并、溯因推理乃至空间推理的具体且高效的算子。