Dual Logic Concepts based on Mathematical Morphology in Stratified Institutions: Applications to Spatial Reasoning

TL;DR

This paper introduces a novel abstract framework using mathematical morphology within stratified institutions to model dual logical operators, with applications to spatial reasoning.

Contribution

It defines dual logical operators as morphological erosion and dilation in stratified institutions, extending the abstract logical framework for spatial reasoning.

Findings

Operators studied on sets of states and models

Framework accommodates open sentences in stratified institutions

Potential applications to spatial reasoning highlighted

Abstract

Several logical operators are defined as dual pairs, in different types of logics. Such dual pairs of operators also occur in other algebraic theories, such as mathematical morphology. Based on this observation, this paper proposes to define, at the abstract level of institutions, a pair of abstract dual and logical operators as morphological erosion and dilation. Standard quantifiers and modalities are then derived from these two abstract logical operators. These operators are studied both on sets of states and sets of models. To cope with the lack of explicit set of states in institutions, the proposed abstract logical dual operators are defined in an extension of institutions, the stratified institutions, which take into account the notion of open sentences, the satisfaction of which is parametrized by sets of states. A hint on the potential interest of the proposed framework for spatial reasoning is also provided.