Minimality Notions via Factorization Systems

TL;DR

This paper introduces an abstract framework for system minimality using factorization systems in category theory, unifying state reduction and merging in coalgebraic models, with criteria ensuring their properties and interactions.

Contribution

It develops a general categorical notion of minimality and minimization, providing criteria for their existence, uniqueness, and functoriality, applicable to coalgebraic systems.

Findings

Criteria for uniqueness, existence, and functoriality of minimization.

Instantiation of results to reachability and observability in coalgebras.

Analysis of interaction and sequencing of minimization aspects.

Abstract

For the minimization of state-based systems (i.e. the reduction of the number of states while retaining the system's semantics), there are two obvious aspects: removing unnecessary states of the system and merging redundant states in the system. In the present article, we relate the two aspects on coalgebras by defining an abstract notion of minimality. The abstract notion minimality and minimization live in a general category with a factorization system. We will find criteria on the category that ensure uniqueness, existence, and functoriality of the minimization aspects. The proofs of these results instantiate to those for reachability and observability minimization in the standard coalgebra literature. Finally, we will see how the two aspects of minimization interact and under which criteria they can be sequenced in any order, like in automata minimization.