12 research outputs found
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 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
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
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
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
A Poly-algorithmic Approach to Quantifier Elimination
Cylindrical Algebraic Decomposition (CAD) was the first practical means for
doing real quantifier elimination (QE), and is still a major method, with many
improvements since Collins' original method. Nevertheless, its complexity is
inherently doubly exponential in the number of variables. Where applicable,
virtual term substitution (VTS) is more effective, turning a QE problem in
variables to one in variables in one application, and so on. Hence there
is scope for hybrid methods: doing VTS where possible then using CAD.
This paper describes such a poly-algorithmic implementation, based on the
second author's Ph.D. thesis. The version of CAD used is based on a new
implementation of Lazard's recently-justified method, with some improvements to
handle equational constraints
Lazard-style CAD and Equational Constraints
McCallum-style Cylindrical Algebra Decomposition (CAD) is a major improvement
on the original Collins version, and has had many subsequent advances, notably
for total or partial equational constraints. But it suffers from a problem with
nullification. The recently-justified Lazard-style CAD does not have this
problem. However, transporting the equational constraints work to Lazard-style
does reintroduce nullification issues. This paper explains the problem, and the
solutions to it, based on the second author's Ph.D. thesis and the
Brown--McCallum improvement to Lazard.
With a single equational constraint, we can gain the same improvements in
Lazard-style as in McCallum-style CAD . Moreover, our approach does not fail
where McCallum would due to nullification. Unsurprisingly, it does not achieve
the same level of improvement as it does in the non-nullified cases. We also
consider the case of multiple equational constraints.Comment: 9 page
A “piano movers” problem reformulated
Abstract-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
Layered Cylindrical Algebraic Decomposition
In this report the idea of a Layered CAD is introduced: atruncation of a CAD to cells of dimension higher than a prescribedvalue. Limiting to full-dimensional cells has already beeninvestigated in the literature, but including more levels is shown toalso be beneficial for applications. Alongside a direct algorithm, arecursive algorithm is provided. A related topological property isdefined and related to robot motion planning. The distribution of celldimensions in a CAD is investigated and layered CAD ideas are combinedwith other research. All research is fully implemented within a freelyavailable Maple package, and all results are corroborated withexperimental results