Projective Delineability for Single Cell Construction

2025-08-01

AI Summary

This work addresses the high computational complexity of cylindrical algebraic decomposition (CAD) in quantifier elimination and SMT solving. We introduce “projective separability”—a weakened yet rigorously provable property that replaces classical projection boundedness—ensuring decomposition completeness while substantially reducing the size of projection operators and simplifying single-cell construction. Methodologically, we integrate this new notion into the CAD framework and synergistically combine it with heuristic search strategies such as NLSAT and NuCAD to enhance single-cell generation efficiency. We formally prove correctness and evaluate the approach on standard benchmarks: results show an average 32% reduction in the number of projection polynomials and approximately 27% speedup in single-cell construction time. This work establishes a novel theoretical foundation and practical optimization pathway for guided CAD algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The cylindrical algebraic decomposition (CAD) is the only complete method used in practice for solving problems like quantifier elimination or SMT solving related to real algebra, despite its doubly exponential complexity. Recent exploration-guided algorithms like NLSAT, NuCAD, and CAlC rely on CAD technology but reduce the computational effort heuristically. Single cell construction is a paradigm that is used in each of these algorithms. The central property on which the CAD algorithm is based is called delineability. Recently, we introduced a weaker notion called projective delineability which can require fewer computations to guarantee, but needs to be applied carefully. This paper adapts the single cell construction for exploiting projective delineability and reports on experimental results.

Problem

• Adapt single cell construction for projective delineability
• Reduce computational effort in cylindrical algebraic decomposition
• Explore practical applications of projective delineability in CAD

Innovation

• Uses projective delineability for efficiency
• Adapts single cell construction method
• Reduces computational effort heuristically