Further results on Minimal and Minimum Cylindrical Algebraic Decompositions

2026-01-14

AI Summary

This study addresses the problem of redundancy elimination in cylindrical algebraic decomposition (CAD) for families of semi-algebraic sets, aiming to achieve minimal representations. By integrating poset analysis, rewriting systems, and confluence theory, the authors prove that a minimal CAD always exists in one- and two-dimensional cases, whereas it may not exist in three or higher dimensions. They establish necessary and sufficient conditions for the existence of a minimal CAD and provide practical criteria for its verification. Building on these theoretical results, an effective reduction algorithm is proposed to compute minimal CADs, which is validated through multiple examples, clearly revealing the intrinsic minimal CAD structure of the corresponding semi-algebraic sets.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We consider cylindrical algebraic decompositions (CADs) as a tool for representing semi-algebraic subsets of $\mathbb{R}^n$. In this framework, a CAD $\mathscr{C}$ is adapted to a given set $S$ if $S$ is a union of cells of $\mathscr{C}$. Different algorithms computing an adapted CAD may produce different outputs, usually with redundant cell divisions. In this paper we analyse the possibility to remove the superfluous data. We thus consider the set $\text{CAD}^r(\mathcal{F})$ of CADs of class $C^r$ ($r \in \mathbb{N} \cup \{\infty, \omega\}$) that are adapted to a finite family $\mathcal{F}$ of semi-algebraic sets of $\mathbb{R}^n$, endowed with the refinement partial order and we study the existence of minimal and minimum element in $\text{CAD}^r(\mathcal{F})$. We show that for every such $\mathcal{F}$ and every $\mathscr{C} \in \text{CAD}^r(\mathcal{F})$, there is a minimal CAD of class $C^r$ adapted to $\mathcal{F}$ and smaller (i.e. coarser) than or equal to $\mathscr{C}$. In dimension $n=1$ or $n=2$, this result is strengthened by proving the existence of a minimum element in $\text{CAD}^r(\mathcal{F})$. In contrast, for any $n \geq 3$, we provide explicit examples of semi-algebraic sets whose associated poset of adapted CADs does not admit a minimum. We then introduce a reduction relation on $\text{CAD}^r(\mathcal{F})$ in order to define an algorithm for the computation of minimal CADs and we characterise those semi-algebraic sets $\mathcal{F}$ for which $\text{CAD}^r(\mathcal{F})$ has a minimum by means of confluence of the associated reduction system. We finally provide practical criteria for deciding if a semi-algebraic set does admit a minimum CAD and apply them to describe various concrete examples of semi-algebraic sets, along with their minimum CAD of class $C^r$.

Problem

• Cylindrical Algebraic Decomposition
• semi-algebraic sets
• minimal CAD
• minimum CAD
• refinement order

Innovation

• Cylindrical Algebraic Decomposition
• Minimal CAD
• Minimum CAD
• Reduction System
• Confluence