Towards Incremental Cylindrical Algebraic Decomposition in Maple

TL;DR

This paper presents a proof of concept implementation of an incremental CAD algorithm in Maple, designed to efficiently refine solutions as new polynomial constraints are added, benefiting SMT solving.

Contribution

It introduces the first incremental CAD algorithm integrated into Maple, utilizing Lazard projection, and demonstrates its computational advantages over re-computation methods.

Findings

Incremental CAD reduces computation time compared to re-computation.

Implementation in Maple shows practical feasibility.

Lazard projection scheme enhances the CAD refinement process.

Abstract

Cylindrical Algebraic Decomposition (CAD) is an important tool within computational real algebraic geometry, capable of solving many problems for polynomial systems over the reals. It has long been studied by the Symbolic Computation community and has found recent interest in the Satisfiability Checking community. The present report describes a proof of concept implementation of an Incremental CAD algorithm in Maple, where CADs are built and then refined as additional polynomial constraints are added. The aim is to make CAD suitable for use as a theory solver for SMT tools who search for solutions by continually reformulating logical formula and querying whether a logical solution is admissible. We describe experiments for the proof of concept, which clearly display the computational advantages compared to iterated re-computation. In addition, the project implemented this work under the recently verified Lazard projection scheme (with corresponding Lazard valuation).