397 research outputs found
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
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
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
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
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
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
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
A "Piano Movers" Problem Reformulated
It has long been known that cylindrical algebraic decompositions (CADs) can
in theory be used for robot motion planning. However, in practice even the
simplest examples can be too complicated to tackle. We consider in detail a
"Piano Mover's Problem" which considers moving an infinitesimally thin piano
(or ladder) through a right-angled corridor.
Producing a CAD for the original formulation of this problem is still
infeasible after 25 years of improvements in both CAD theory and computer
hardware. We review some alternative formulations in the literature which use
differing levels of geometric analysis before input to a CAD algorithm. Simpler
formulations allow CAD to easily address the question of the existence of a
path. We provide a new formulation for which both a CAD can be constructed and
from which an actual path could be determined if one exists, and analyse the
CADs produced using this approach for variations of the problem.
This emphasises the importance of the precise formulation of such problems
for CAD. We analyse the formulations and their CADs considering a variety of
heuristics and general criteria, leading to conclusions about tackling other
problems of this form.Comment: 8 pages. Copyright IEEE 201
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
Adapting Real Quantifier Elimination Methods for Conflict Set Computation
The satisfiability problem in real closed fields is decidable. In the context
of satisfiability modulo theories, the problem restricted to conjunctive sets
of literals, that is, sets of polynomial constraints, is of particular
importance. One of the central problems is the computation of good explanations
of the unsatisfiability of such sets, i.e.\ obtaining a small subset of the
input constraints whose conjunction is already unsatisfiable. We adapt two
commonly used real quantifier elimination methods, cylindrical algebraic
decomposition and virtual substitution, to provide such conflict sets and
demonstrate the performance of our method in practice
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