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

arXiv: https://arxiv.org/abs/2303.04895 · HTML: https://mlsanity.com/paper/2303.04895/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Spatial and embodied reasoning, one of 17 directions in LLM reasoning.
- Builds on 2 of its 66 references that are in the map.
- First posted on arXiv 8 Mar 2023.

## Placed in

- LLM reasoning / [Spatial and embodied reasoning](https://mlsanity.com/lr/spatial-and-embodied-reasoning/)

## Builds on (most cited first, 2 of 2)

- [Morphologic for knowledge dynamics: revision, fusion, abduction](https://mlsanity.com/paper/1802.05142/) (2018)
- [Explanatory relations in arbitrary logics based on satisfaction systems, cutting and retraction](https://mlsanity.com/paper/1803.01571/) (2018)

## Abstract (arXiv)

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.