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

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

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

Dataset supporting the paper: Truth table invariant cylindrical algebraic decomposition

The files in this data set support the following paper: ########################################################################################## Truth table invariant cylindrical algebraic decomposition. Russel Bradford, James H. Davenport, Matthew England, Scott McCallum and David Wilson. http://opus.bath.ac.uk/38146/ ########################################################################################## Please find included the following: ############################## 1a) A Maple worksheet: Section1to7-Maple.mw 1b) A pdf printout of the worksheet: Section1to7-Maple.pdf 1c) A Maple Library file: ProjectionCAD.mpl These files concern the Maple results for the worked examples throughout Sections 1-7 of the paper. To run the Maple worksheet you will need a copy of the commercial computer algebra software Maple. This is currently available from: http://www.maplesoft.com/products/maple/ The examples were run in Maple 16 (released Spring 2012). It is likely that the same results would be obtained in Maple 17, 18, 2015 and future versions, but this cannot be guaranteed. An additional code package, developed at the University of Bath, is required. To use it we need to read the Maple Library file within Maple as follows: >>> read("ProjectionCAD.mpl"): >>> with(ProjectionCAD): More details on this Maple package are available in the technical report at http://opus.bath.ac.uk/43911/ and in the following publication: M. England, D. Wilson, R. Bradford and J.H. Davenport. Using the Regular Chains Library to build cylindrical algebraic decompositions by projecting and lifting. Proc ICMS 2014 (LNCS 8593). DOI: 10.1007/978-3-662-44199-2_69 If you do not have a copy of Maple you can still read the pdf printout of the worksheet. ############################## 2) A zipped directory WorkedExamples-Qepcad.zip This directory also concerns the worked examples from Sections 1-7 of the paper, this time when studied with Qepcad-B. Qepcad-B is a free piece of software for Linux which can be obtained from: http://www.usna.edu/CS/qepcadweb/B/QEPCAD.html All the files in the zipped directory end in either "-in.txt" or "-out.txt". The former give input for Qepcad and the latter record output. Hence readers without access to Qepcad (e.g. on a Windows system) can still observe the output in the latter files. To verify the output readers should use the following bash command to run a Qepcad input file "Ex-in.txt" and record the output in "Ex-out.txt". >>> qepcad +N500000000 +L200000 Ex-out.txt Windows users without Linux access can still read the existing output files in the folder. ############################## 3a) The text file: Section82-ExampleSet.txt 3b) A Maple worksheet: Section82-ExampleSet.mw 3b) A pdf printout of the worksheet: Section82-ExampleSet.pdf The textfile defines the example set which is the subject of the experiments in Section 8.2, whose results were summarised in Table 2. Within the file the 29 examples are defined in the following syntax: (a) First a line starting with "#" giving the full example name followed in brackets by the shortened name used in Table 2. (b) Then a second line in which the example is defined as a list of two sublists: i) The first sublist defines the polynomials used. They are sorted into further lists, one for each formulae in the example. Each of these has two entries: --- The first is either a polynomial defining an equational constraint (EC); a list of polynomials defining multiple ECs; or an empty list (signalling no ECs). --- The second is a list of any non ECs. ii) The second sublist is the variable ordering from highest (eliminate first in projection) to lowest. Note that Maple algorithms use this order by Qepcad the reverse. This is the syntax used by the TTICAD algorithm that is the subject of the paper. The text file doubles as a Maple function definition. When read into Maple the command GenerateInput is defined which can provide the input in formats suitable for the three Maple algorithms tested. An example is given in the Maple worksheet / pdf. We note that the timings reported in the paper were from running Maple in command line mode. See also the notes for files (1) above. The same example set was tested in Qepcad. Here explicit ECs for a parent formula were entered in dynamically as products of the individual sub-formulae ECs, in cases where an explicit EC exists. See also Qepcad notes for file (2) above. Finally, the example set was also tested in Mathematica. Mathematica's CAD command does not return cell counts - these were obtained upon request to a Mathematica developer. Hence they are not recreatable using the information here (something outside the control of the present authors). ############################## 4a) A Maple worksheet: Section83-Maple.mw 4b) A pdf printout of the worksheet: Section83-Maple.pdf This shows how the numbers in Table 3 from Maple were obtained. See also notes for files (1) above. ############################## 5a) A zipped directory Section83-Qepcad.zipped This shows how the numbers in Table 3 from Qepcad were obtained. See also notes for file (2) above.Cell counts and timings of various CAD algorithms

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

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

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

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

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

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

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