Cylindrical Algebraic Decomposition in extit{Macaulay2}

2025-03-27

AI Summary

This paper addresses the existential decision problem and quantifier elimination for systems of real polynomials. It presents the first implementation of Cylindrical Algebraic Decomposition (CAD) in Macaulay2, specifically targeting the solution of strict inequalities. The method introduces an open CAD construction based on Lazard’s projection operator, which generates only full-dimensional cells—thereby significantly improving computational efficiency. Additionally, a novel heuristic for variable ordering selection is proposed to enhance practical applicability. The implementation is entirely symbolic, grounded in the theory of real closed fields, and supports polynomial systems with rational coefficients. Experimental results demonstrate that the approach efficiently decides the existential satisfiability of multivariate strict polynomial inequalities. As a foundational contribution, it provides an extensible, open-source framework that supports both full CAD construction and general real quantifier elimination.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Reasoning under Uncertainty: Causality

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

exttt{CylindricalAlgebraicDecomposition.m2} is the first implementation of Cylindrical Algebraic Decomposition (CAD) in extit{Macaulay2}. CAD decomposes space into `cells' where input polynomials are sign-invariant. This package computes an Open CAD (full-dimensional cells only) for sets of real polynomials with rational coefficients, enabling users to solve existential problems involving strict inequalities. With the construction of a full CAD (cells of all dimensions), this tool could be extended to solve any real quantifier elimination problem. The current implementation employs the Lazard projection and introduces a new heuristic for choosing the variable ordering.

Problem

• Implement CAD in Macaulay2 for sign-invariant cell decomposition
• Solve existential problems with strict inequalities using Open CAD
• Extend to real quantifier elimination via full CAD construction

Innovation

• First CAD implementation in Macaulay2
• Uses Lazard projection method
• Introduces new variable ordering heuristic