8 research outputs found
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
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
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
Cylindrical algebraic decomposition with equational constraints
Cylindrical Algebraic Decomposition (CAD) has long been one of the most
important algorithms within Symbolic Computation, as a tool to perform
quantifier elimination in first order logic over the reals. More recently it is
finding prominence in the Satisfiability Checking community as a tool to
identify satisfying solutions of problems in nonlinear real arithmetic.
The original algorithm produces decompositions according to the signs of
polynomials, when what is usually required is a decomposition according to the
truth of a formula containing those polynomials. One approach to achieve that
coarser (but hopefully cheaper) decomposition is to reduce the polynomials
identified in the CAD to reflect a logical structure which reduces the solution
space dimension: the presence of Equational Constraints (ECs).
This paper may act as a tutorial for the use of CAD with ECs: we describe all
necessary background and the current state of the art. In particular, we
present recent work on how McCallum's theory of reduced projection may be
leveraged to make further savings in the lifting phase: both to the polynomials
we lift with and the cells lifted over. We give a new complexity analysis to
demonstrate that the double exponent in the worst case complexity bound for CAD
reduces in line with the number of ECs. We show that the reduction can apply to
both the number of polynomials produced and their degree.Comment: Accepted into the Journal of Symbolic Computation. arXiv admin note:
text overlap with arXiv:1501.0446
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