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

Using the Regular Chains Library to build cylindrical algebraic decompositions by projecting and lifting

Cylindrical algebraic decomposition (CAD) is an important tool, both for quantifier elimination over the reals and a range of other applications. Traditionally, a CAD is built through a process of projection and lifting to move the problem within Euclidean spaces of changing dimension. Recently, an alternative approach which first decomposes complex space using triangular decomposition before refining to real space has been introduced and implemented within the RegularChains Library of Maple. We here describe a freely available package ProjectionCAD which utilises the routines within the RegularChains Library to build CADs by projection and lifting. We detail how the projection and lifting algorithms were modified to allow this, discuss the motivation and survey the functionality of the package

Formulating problems for real algebraic geometry

We discuss issues of problem formulation for algorithms in real algebraic geometry, focussing on quantifier elimination by cylindrical algebraic decomposition. We recall how the variable ordering used can have a profound effect on both performance and output and summarise what may be done to assist with this choice. We then survey other questions of problem formulation and algorithm optimisation that have become pertinent following advances in CAD theory, including both work that is already published and work that is currently underway. With implementations now in reach of real world applications and new theory meaning algorithms are far more sensitive to the input, our thesis is that intelligently formulating problems for algorithms, and indeed choosing the correct algorithm variant for a problem, is key to improving the practical use of both quantifier elimination and symbolic real algebraic geometry in general.Comment: To be presented at The "Encuentros de \'Algebra Computacional y Aplicaciones, EACA 2014" (Meetings on Computer Algebra and Applications) in Barcelon

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

Choosing a variable ordering for truth-table invariant cylindrical algebraic decomposition by incremental triangular decomposition

Cylindrical algebraic decomposition (CAD) is a key tool for solving problems in real algebraic geometry and beyond. In recent years a new approach has been developed, where regular chains technology is used to first build a decomposition in complex space. We consider the latest variant of this which builds the complex decomposition incrementally by polynomial and produces CADs on whose cells a sequence of formulae are truth-invariant. Like all CAD algorithms the user must provide a variable ordering which can have a profound impact on the tractability of a problem. We evaluate existing heuristics to help with the choice for this algorithm, suggest improvements and then derive a new heuristic more closely aligned with the mechanics of the new algorithm

Using the distribution of cells by dimension in a cylindrical algebraic decomposition

We investigate the distribution of cells by dimension in cylindrical algebraic decompositions (CADs). We find that they follow a standard distribution which seems largely independent of the underlying problem or CAD algorithm used. Rather, the distribution is inherent to the cylindrical structure and determined mostly by the number of variables. This insight is then combined with an algorithm that produces only full-dimensional cells to give an accurate method of predicting the number of cells in a complete CAD. Since constructing only full-dimensional cells is relatively inexpensive (involving no costly algebraic number calculations) this leads to heuristics for helping with various questions of problem formulation for CAD, such as choosing an optimal variable ordering. Our experiments demonstrate that this approach can be highly effective.Comment: 8 page

An implementation of Sub-CAD in Maple

Cylindrical algebraic decomposition (CAD) is an important tool for the investigation of semi-algebraic sets, with applications in algebraic geometry and beyond. We have previously reported on an implementation of CAD in Maple which offers the original projection and lifting algorithm of Collins along with subsequent improvements. Here we report on new functionality: specifically the ability to build cylindrical algebraic sub-decompositions (sub-CADs) where only certain cells are returned. We have implemented algorithms to return cells of a prescribed dimensions or higher (layered {\scad}s), and an algorithm to return only those cells on which given polynomials are zero (variety {\scad}s). These offer substantial savings in output size and computation time. The code described and an introductory Maple worksheet / pdf demonstrating the full functionality of the package are freely available online at http://opus.bath.ac.uk/43911/.Comment: 9 page

Speeding up Cylindrical Algebraic Decomposition by Gr\"obner Bases

Gr\"obner Bases and Cylindrical Algebraic Decomposition are generally thought of as two, rather different, methods of looking at systems of equations and, in the case of Cylindrical Algebraic Decomposition, inequalities. However, even for a mixed system of equalities and inequalities, it is possible to apply Gr\"obner bases to the (conjoined) equalities before invoking CAD. We see that this is, quite often but not always, a beneficial preconditioning of the CAD problem. It is also possible to precondition the (conjoined) inequalities with respect to the equalities, and this can also be useful in many cases.Comment: To appear in Proc. CICM 2012, LNCS 736

Recent advances in real geometric reasoning

In the 1930s Tarski showed that real quantifier elimination was possible, and in 1975 Collins gave a remotely practicable method, albeit with doubly-exponential complexity, which was later shown to be inherent. We discuss some of the recent major advances in Collins method: such as an alternative approach based on passing via the complexes, and advances which come closer to "solving the question asked" rather than "solving all problems to do with these polynomials"

Applying machine learning to the problem of choosing a heuristic to select the variable ordering for cylindrical algebraic decomposition

Cylindrical algebraic decomposition(CAD) is a key tool in computational algebraic geometry, particularly for quantifier elimination over real-closed fields. When using CAD, there is often a choice for the ordering placed on the variables. This can be important, with some problems infeasible with one variable ordering but easy with another. Machine learning is the process of fitting a computer model to a complex function based on properties learned from measured data. In this paper we use machine learning (specifically a support vector machine) to select between heuristics for choosing a variable ordering, outperforming each of the separate heuristics.Comment: 16 page