Minimality Notions via Factorization Systems and Examples

TL;DR

This paper introduces an abstract framework for state minimization in coalgebras using factorization systems, unifying removal and merging of states and analyzing their interaction.

Contribution

It defines a general notion of minimality in categories with factorization systems and provides criteria for their uniqueness, existence, and functoriality, extending coalgebra minimization theory.

Findings

Criteria for minimality in categories with factorization systems

Unified approach to reachability and observability minimization

Conditions for sequencing different 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 minimization aspects on coalgebras by defining an abstract notion of minimality. The abstract notions 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.