507 research outputs found
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
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
Synthesizing Switching Controllers for Hybrid Systems by Continuous Invariant Generation
We extend a template-based approach for synthesizing switching controllers
for semi-algebraic hybrid systems, in which all expressions are polynomials.
This is achieved by combining a QE (quantifier elimination)-based method for
generating continuous invariants with a qualitative approach for predefining
templates. Our synthesis method is relatively complete with regard to a given
family of predefined templates. Using qualitative analysis, we discuss
heuristics to reduce the numbers of parameters appearing in the templates. To
avoid too much human interaction in choosing templates as well as the high
computational complexity caused by QE, we further investigate applications of
the SOS (sum-of-squares) relaxation approach and the template polyhedra
approach in continuous invariant generation, which are both well supported by
efficient numerical solvers
Panorama of p-adic model theory
ABSTRACT. We survey the literature in the model theory of p-adic numbers since\ud
Denefâs work on the rationality of PoincarĂ© series. / RĂSUMĂ. Nous donnons un aperçu des dĂ©veloppements de la thĂ©orie des modĂšles\ud
des nombres p-adiques depuis les travaux de Denef sur la rationalité de séries de Poincaré,\ud
par une revue de la bibliographie
Embedded Finite Models beyond Restricted Quantifier Collapse
We revisit evaluation of logical formulas that allow both uninterpreted
relations, constrained to be finite, as well as interpreted vocabulary over an
infinite domain: denoted in the past as embedded finite model theory. We extend
the analysis of "collapse results": the ability to eliminate first-order
quantifiers over the infinite domain in favor of quantification over the finite
structure. We investigate several weakenings of collapse, one allowing
higher-order quantification over the finite structure, another allowing
expansion of the theory. We also provide results comparing collapse for unary
signatures with general signatures, and new analyses of collapse for natural
decidable theories
Computing Solution Operators of Boundary-value Problems for Some Linear Hyperbolic Systems of PDEs
We discuss possibilities of application of Numerical Analysis methods to
proving computability, in the sense of the TTE approach, of solution operators
of boundary-value problems for systems of PDEs. We prove computability of the
solution operator for a symmetric hyperbolic system with computable real
coefficients and dissipative boundary conditions, and of the Cauchy problem for
the same system (we also prove computable dependence on the coefficients) in a
cube . Such systems describe a wide variety of physical
processes (e.g. elasticity, acoustics, Maxwell equations). Moreover, many
boundary-value problems for the wave equation also can be reduced to this case,
thus we partially answer a question raised in Weihrauch and Zhong (2002).
Compared with most of other existing methods of proving computability for PDEs,
this method does not require existence of explicit solution formulas and is
thus applicable to a broader class of (systems of) equations.Comment: 31 page
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
- âŠ