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Minimality Notions via Factorization Systems and Examples

Thorsten Wißmann

TL;DR

The paper develops an abstract notion of minimality in categories equipped with an (E,M)-factorization system and applies it to coalgebras, showing how reachability (minimal subobjects) and observability (simple quotients) arise as natural instances. It proves lifting results for factorization systems to Coalg(F) when F preserves M and discusses dual liftings to Alg(F) and EM categories, establishing conditions for existence, uniqueness, and functoriality of minimizations. It then analyzes the interaction between the two minimization notions, introducing well-pointed coalgebras and showing that, under suitable assumptions, the two minimization tasks can be performed in either order with equivalent outcomes. The work unifies classical coalgebra minimization results and provides a general framework for designing new minimization techniques and understanding their algorithmic implications.

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.

Minimality Notions via Factorization Systems and Examples

TL;DR

The paper develops an abstract notion of minimality in categories equipped with an (E,M)-factorization system and applies it to coalgebras, showing how reachability (minimal subobjects) and observability (simple quotients) arise as natural instances. It proves lifting results for factorization systems to Coalg(F) when F preserves M and discusses dual liftings to Alg(F) and EM categories, establishing conditions for existence, uniqueness, and functoriality of minimizations. It then analyzes the interaction between the two minimization notions, introducing well-pointed coalgebras and showing that, under suitable assumptions, the two minimization tasks can be performed in either order with equivalent outcomes. The work unifies classical coalgebra minimization results and provides a general framework for designing new minimization techniques and understanding their algorithmic implications.

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.
Paper Structure (9 sections, 16 theorems, 10 equations, 8 figures)

This paper contains 9 sections, 16 theorems, 10 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coalgebra
  4. Factorization Systems
  5. Factorization System for Coalgebras
  6. Minimality in a Category
  7. Interplay of minimality notions
  8. Conclusions
  9. Proofs of standard results

Key Result

Lemma 2.6

The forgetful functor $U\colon \mathsf{Coalg}(F)\to \mathcal{C}$ creates all colimits. That is, the colimit of a diagram $D\colon \mathcal{D}\to \mathsf{Coalg}(F)$ exists, if $U\cdot D\colon \mathcal{D}\to \mathcal{C}$ has a colimit, and moreover, there is a unique coalgebra structure on $\operatorn

Figures (8)

  • Figure 1: Examples of $F$-coalgebras for different $\mathsf{Set}\xspace$-functors $F$
  • Figure 2: Examples of an $F$-coalgebra morphisms for $FX=2\times X^{\{a,b\}}$
  • Figure 5: Factorization $h=m\cdot e$ in $\mathsf{Coalg}(2\times(-)^{\{a,b\}})$ of $h$ from \ref{['figExDfaHom']}
  • Figure 6: Equivalent characterizations of $\mathcal{M}$-minimality and $\mathcal{E}$-minimality in a category $\mathcal{K}$
  • Figure 7: Examples of simple quotients in $F$-coalgebras
  • ...and 3 more figures

Theorems & Definitions (57)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Lemma 2.6
  • Definition 2.7
  • Example 2.8
  • Definition 2.9
  • Definition 2.10: joyofcats
  • ...and 47 more