# Cylindrical Algebraic Decomposition with Equational Constraints

## TL;DR

This paper reviews how equational constraints can optimize Cylindrical Algebraic Decomposition (CAD), reducing complexity and computational costs, and introduces new theoretical insights and practical methods for leveraging these constraints.

## Contribution

It provides a comprehensive tutorial on CAD with ECs, introduces a new complexity analysis, and demonstrates how ECs can reduce both polynomial count and degree in CAD.

## Key findings

- Double exponential complexity bound reduces with more ECs.
- ECs decrease the number of polynomials and their degrees.
- Leverages McCallum's reduced projection theory for efficiency.

## Abstract

Cylindrical Algebraic Decomposition (CAD) has long been one of the most important algorithms within Symbolic Computation, as a tool to perform quantifier elimination in first order logic over the reals. More recently it is finding prominence in the Satisfiability Checking community as a tool to identify satisfying solutions of problems in nonlinear real arithmetic. The original algorithm produces decompositions according to the signs of polynomials, when what is usually required is a decomposition according to the truth of a formula containing those polynomials. One approach to achieve that coarser (but hopefully cheaper) decomposition is to reduce the polynomials identified in the CAD to reflect a logical structure which reduces the solution space dimension: the presence of Equational Constraints (ECs). This paper may act as a tutorial for the use of CAD with ECs: we describe all necessary background and the current state of the art. In particular, we present recent work on how McCallum's theory of reduced projection may be leveraged to make further savings in the lifting phase: both to the polynomials we lift with and the cells lifted over. We give a new complexity analysis to demonstrate that the double exponent in the worst case complexity bound for CAD reduces in line with the number of ECs. We show that the reduction can apply to both the number of polynomials produced and their degree.