90 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
Combinatorial complexity in o-minimal geometry
In this paper we prove tight bounds on the combinatorial and topological
complexity of sets defined in terms of definable sets belonging to some
fixed definable family of sets in an o-minimal structure. This generalizes the
combinatorial parts of similar bounds known in the case of semi-algebraic and
semi-Pfaffian sets, and as a result vastly increases the applicability of
results on combinatorial and topological complexity of arrangements studied in
discrete and computational geometry. As a sample application, we extend a
Ramsey-type theorem due to Alon et al., originally proved for semi-algebraic
sets of fixed description complexity to this more general setting.Comment: 25 pages. Revised version. To appear in the Proc. London Math. So
A CDCL-style calculus for solving non-linear constraints
In this paper we propose a novel approach for checking satisfiability of
non-linear constraints over the reals, called ksmt. The procedure is based on
conflict resolution in CDCL style calculus, using a composition of symbolical
and numerical methods. To deal with the non-linear components in case of
conflicts we use numerically constructed restricted linearisations. This
approach covers a large number of computable non-linear real functions such as
polynomials, rational or trigonometrical functions and beyond. A prototypical
implementation has been evaluated on several non-linear SMT-LIB examples and
the results have been compared with state-of-the-art SMT solvers.Comment: 17 pages, 3 figures; accepted at FroCoS 2019; software available at
<http://informatik.uni-trier.de/~brausse/ksmt/
Using cylindrical algebraic decomposition and local Fourier analysis to study numerical methods: two examples
Local Fourier analysis is a strong and well-established tool for analyzing
the convergence of numerical methods for partial differential equations. The
key idea of local Fourier analysis is to represent the occurring functions in
terms of a Fourier series and to use this representation to study certain
properties of the particular numerical method, like the convergence rate or an
error estimate.
In the process of applying a local Fourier analysis, it is typically
necessary to determine the supremum of a more or less complicated term with
respect to all frequencies and, potentially, other variables. The problem of
computing such a supremum can be rewritten as a quantifier elimination problem,
which can be solved with cylindrical algebraic decomposition, a well-known tool
from symbolic computation.
The combination of local Fourier analysis and cylindrical algebraic
decomposition is a machinery that can be applied to a wide class of problems.
In the present paper, we will discuss two examples. The first example is to
compute the convergence rate of a multigrid method. As second example we will
see that the machinery can also be used to do something rather different: We
will compare approximation error estimates for different kinds of
discretizations.Comment: The research was funded by the Austrian Science Fund (FWF): J3362-N2
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
Digital Collections of Examples in Mathematical Sciences
Some aspects of Computer Algebra (notably Computation Group Theory and
Computational Number Theory) have some good databases of examples, typically of
the form "all the X up to size n". But most of the others, especially on the
polynomial side, are lacking such, despite the utility they have demonstrated
in the related fields of SAT and SMT solving. We claim that the field would be
enhanced by such community-maintained databases, rather than each author
hand-selecting a few, which are often too large or error-prone to print, and
therefore difficult for subsequent authors to reproduce.Comment: Presented at 8th European Congress of Mathematician
Delta-Decision Procedures for Exists-Forall Problems over the Reals
Solving nonlinear SMT problems over real numbers has wide applications in
robotics and AI. While significant progress is made in solving quantifier-free
SMT formulas in the domain, quantified formulas have been much less
investigated. We propose the first delta-complete algorithm for solving
satisfiability of nonlinear SMT over real numbers with universal quantification
and a wide range of nonlinear functions. Our methods combine ideas from
counterexample-guided synthesis, interval constraint propagation, and local
optimization. In particular, we show how special care is required in handling
the interleaving of numerical and symbolic reasoning to ensure
delta-completeness. In experiments, we show that the proposed algorithms can
handle many new problems beyond the reach of existing SMT solvers
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