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

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.

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

The aim of this paper is to introduce a new framework for defining abductive reasoning operators based on a notion of retraction in arbitrary logics defined as satisfaction systems. We show how this framework leads to the design of explanatory relations satisfying properties of abductive reasoning, and discuss its application to several logics. This extends previous work on propositional logics where retraction was defined as a morphological erosion. Here weaker properties are required for retraction, leading to a larger set of suitable operators for abduction for different logics.

Belief Revision, Minimal Change and Relaxation: A General Framework based on Satisfaction Systems, and Applications to Description Logics

Belief revision of knowledge bases represented by a set of sentences in a given logic has been extensively studied but for specific logics, mainly propositional, and also recently Horn and description logics. Here, we propose to generalize this operation from a model-theoretic point of view, by defining revision in an abstract model theory known under the name of satisfaction systems. In this framework, we generalize to any satisfaction systems the characterization of the well known AGM postulates given by Katsuno and Mendelzon for propositional logic in terms of minimal change among interpretations. Moreover, we study how to define revision, satisfying the AGM postulates, from relaxation notions that have been first introduced in description logics to define dissimilarity measures between concepts, and the consequence of which is to relax the set of models of the old belief until it becomes consistent with the new pieces of knowledge. We show how the proposed general framework can be instantiated in different logics such as propositional, first-order, description and Horn logics. In particular for description logics, we introduce several concrete relaxation operators tailored for the description logic $\ALC{}$ and its fragments $\EL{}$ and $\ELext{}$, discuss their properties and provide some illustrative examples.

Relaxation-based revision operators in description logics

As ontologies and description logics (DLs) reach out to a broader audience, several reasoning services are developed in this context. Belief revision is one of them, of prime importance when knowledge is prone to change and inconsistency. In this paper we address both the generalization of the well-known AGM postulates, and the definition of concrete and well-founded revision operators in different DL families. We introduce a model-theoretic version of the AGM postulates with a general definition of inconsistency, hence enlarging their scope to a wide family of non-classical logics, in particular negation-free DL families. We propose a general framework for defining revision operators based on the notion of relaxation, introduced recently for defining dissimilarity measures between DL concepts. A revision operator in this framework amounts to relax the set of models of the old belief until it reaches the sets of models of the new piece of knowledge. We demonstrate that such a relaxation-based revision operator defines a faithful assignment and satisfies the generalized AGM postulates. Another important contribution concerns the definition of several concrete relaxation operators suited to the syntax of some DLs (ALC and its fragments EL and ELU).