Satisfaction, Restriction and Amalgamation of Constraints in the Framework of M-Adhesive Categories

TL;DR

This paper investigates how satisfaction, restriction, and amalgamation of constraints behave within M-adhesive categories, highlighting that initial satisfaction is compatible with these operations while general satisfaction is not.

Contribution

It demonstrates the compatibility of initial satisfaction with restriction and amalgamation in M-adhesive categories, providing theoretical insights into constraint handling.

Findings

Initial satisfaction is compatible with restriction and amalgamation.

General satisfaction is not compatible with restriction and amalgamation.

Compatibility relies on the van Kampen property in M-adhesive categories.

Abstract

Application conditions for rules and constraints for graphs are well-known in the theory of graph transformation and have been extended already to M-adhesive transformation systems. According to the literature we distinguish between two kinds of satisfaction for constraints, called general and initial satisfaction of constraints, where initial satisfaction is defined for constraints over an initial object of the base category. Unfortunately, the standard definition of general satisfaction is not compatible with negation in contrast to initial satisfaction. Based on the well-known restriction of objects along type morphisms, we study in this paper restriction and amalgamation of application conditions and constraints together with their solutions. In our main result, we show compatibility of initial satisfaction for positive constraints with restriction and amalgamation, while general satisfaction fails in general. Our main result is based on the compatibility of composition via pushouts with restriction, which is ensured by the horizontal van Kampen property in addition to the vertical one that is generally satisfied in M-adhesive categories.