An implementation of CAD in Maple utilising problem formulation, equational constraints and truth-table invariance

Cylindrical algebraic decomposition (CAD) is an important tool for the investigation of semi-algebraic sets, with applications within algebraic geometry and beyond. We recently reported on a new implementation of CAD in Maple which implemented the original algorithm of Collins and the subsequent improvement to projection by McCallum. Our implementation was in contrast to Maple's in-built CAD command, based on a quite separate theory. Although initially developed as an investigative tool to compare the algorithms, we found and reported that our code offered functionality not currently available in any other existing implementations. One particularly important piece of functionality is the ability to produce order-invariant CADs. This has allowed us to extend the implementation to produce CADs invariant with respect to either equational constraints (ECCADs) or the truth-tables of sequences of formulae (TTICADs). This new functionality is contained in the second release of our code, along with commands to consider problem formulation which can be a major factor in the tractability of a CAD. In the report we describe the new functionality and some theoretical discoveries it prompted. We describe how the CADs produced using equational constraints are able to take advantage of not just improved projection but also improvements in the lifting phase. We also present an extension to the original TTICAD algorithm which increases both the applicability of TTICAD and its relative benefit over other algorithms. The code and an introductory Maple worksheet / pdf demonstrating the full functionality of the package are freely available online.Comment: 12 pages; University of Bath, Dept. Computer Science Technical Report Series, 2013-02, 201

Cylindrical Algebraic Sub-Decompositions

Cylindrical algebraic decompositions (CADs) are a key tool in real algebraic geometry, used primarily for eliminating quantifiers over the reals and studying semi-algebraic sets. In this paper we introduce cylindrical algebraic sub-decompositions (sub-CADs), which are subsets of CADs containing all the information needed to specify a solution for a given problem. We define two new types of sub-CAD: variety sub-CADs which are those cells in a CAD lying on a designated variety; and layered sub-CADs which have only those cells of dimension higher than a specified value. We present algorithms to produce these and describe how the two approaches may be combined with each other and the recent theory of truth-table invariant CAD. We give a complexity analysis showing that these techniques can offer substantial theoretical savings, which is supported by experimentation using an implementation in Maple.Comment: 26 page

Validity proof of Lazard's method for CAD construction

In 1994 Lazard proposed an improved method for cylindrical algebraic decomposition (CAD). The method comprised a simplified projection operation together with a generalized cell lifting (that is, stack construction) technique. For the proof of the method's validity Lazard introduced a new notion of valuation of a multivariate polynomial at a point. However a gap in one of the key supporting results for his proof was subsequently noticed. In the present paper we provide a complete validity proof of Lazard's method. Our proof is based on the classical parametrized version of Puiseux's theorem and basic properties of Lazard's valuation. This result is significant because Lazard's method can be applied to any finite family of polynomials, without any assumption on the system of coordinates. It therefore has wider applicability and may be more efficient than other projection and lifting schemes for CAD.Comment: 21 page

An Incremental Algorithm for Computing Cylindrical Algebraic Decompositions

In this paper, we propose an incremental algorithm for computing cylindrical algebraic decompositions. The algorithm consists of two parts: computing a complex cylindrical tree and refining this complex tree into a cylindrical tree in real space. The incrementality comes from the first part of the algorithm, where a complex cylindrical tree is constructed by refining a previous complex cylindrical tree with a polynomial constraint. We have implemented our algorithm in Maple. The experimentation shows that the proposed algorithm outperforms existing ones for many examples taken from the literature

Truth Table Invariant Cylindrical Algebraic Decomposition by Regular Chains

A new algorithm to compute cylindrical algebraic decompositions (CADs) is presented, building on two recent advances. Firstly, the output is truth table invariant (a TTICAD) meaning given formulae have constant truth value on each cell of the decomposition. Secondly, the computation uses regular chains theory to first build a cylindrical decomposition of complex space (CCD) incrementally by polynomial. Significant modification of the regular chains technology was used to achieve the more sophisticated invariance criteria. Experimental results on an implementation in the RegularChains Library for Maple verify that combining these advances gives an algorithm superior to its individual components and competitive with the state of the art

Truth table invariant cylindrical algebraic decomposition

When using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is likely not the signs of those polynomials that are of paramount importance but rather the truth values of certain quantifier free formulae involving them. This observation motivates our article and definition of a Truth Table Invariant CAD (TTICAD). In ISSAC 2013 the current authors presented an algorithm that can efficiently and directly construct a TTICAD for a list of formulae in which each has an equational constraint. This was achieved by generalising McCallum's theory of reduced projection operators. In this paper we present an extended version of our theory which can be applied to an arbitrary list of formulae, achieving savings if at least one has an equational constraint. We also explain how the theory of reduced projection operators can allow for further improvements to the lifting phase of CAD algorithms, even in the context of a single equational constraint. The algorithm is implemented fully in Maple and we present both promising results from experimentation and a complexity analysis showing the benefits of our contributions.Comment: 40 page

Special Algorithm for Stability Analysis of Multistable Biological Regulatory Systems

We consider the problem of counting (stable) equilibriums of an important family of algebraic differential equations modeling multistable biological regulatory systems. The problem can be solved, in principle, using real quantifier elimination algorithms, in particular real root classification algorithms. However, it is well known that they can handle only very small cases due to the enormous computing time requirements. In this paper, we present a special algorithm which is much more efficient than the general methods. Its efficiency comes from the exploitation of certain interesting structures of the family of differential equations.Comment: 24 pages, 5 algorithms, 10 figure

CAD Adjacency Computation Using Validated Numerics

We present an algorithm for computation of cell adjacencies for well-based cylindrical algebraic decomposition. Cell adjacency information can be used to compute topological operations e.g. closure, boundary, connected components, and topological properties e.g. homology groups. Other applications include visualization and path planning. Our algorithm determines cell adjacency information using validated numerical methods similar to those used in CAD construction, thus computing CAD with adjacency information in time comparable to that of computing CAD without adjacency information. We report on implementation of the algorithm and present empirical data.Comment: 20 page

Improving the use of equational constraints in cylindrical algebraic decomposition

When building a cylindrical algebraic decomposition (CAD) savings can be made in the presence of an equational constraint (EC): an equation logically implied by a formula. The present paper is concerned with how to use multiple ECs, propagating those in the input throughout the projection set. We improve on the approach of McCallum in ISSAC 2001 by using the reduced projection theory to make savings in the lifting phase (both to the polynomials we lift with and the cells lifted over). We demonstrate the benefits with worked examples and a complexity analysis