Problem formulation for truth-table invariant cylindrical algebraic decomposition by incremental triangular decomposition

TL;DR

This paper introduces heuristics for optimal problem formulation in a new CAD algorithm that uses truth-table invariance and incremental triangular decomposition, enhancing practical efficiency.

Contribution

It develops heuristics for constraint ordering and problem formulation tailored to a novel CAD algorithm based on regular chains technology.

Findings

Heuristics improve the efficiency of the CAD algorithm in practice.

Optimal constraint ordering significantly impacts CAD performance.

Revisiting problem formulation choices enhances the algorithm's applicability.

Abstract

Cylindrical algebraic decompositions (CADs) are a key tool for solving problems in real algebraic geometry and beyond. We recently presented a new CAD algorithm combining two advances: truth-table invariance, making the CAD invariant with respect to the truth of logical formulae rather than the signs of polynomials; and CAD construction by regular chains technology, where first a complex decomposition is constructed by refining a tree incrementally by constraint. We here consider how best to formulate problems for input to this algorithm. We focus on a choice (not relevant for other CAD algorithms) about the order in which constraints are presented. We develop new heuristics to help make this choice and thus allow the best use of the algorithm in practice. We also consider other choices of problem formulation for CAD, as discussed in CICM 2013, revisiting these in the context of the new algorithm.