9,342 research outputs found
Computing with semi-algebraic sets represented by triangular decomposition
This article is a continuation of our earlier work [3], which introduced triangular decompositions of semi-algebraic systems and algorithms for computing them. Our new contributions include theoretical results based on which we obtain practical improvements for these decomposition algorithms. We exhibit new results on the theory of border polynomials of parametric semi-algebraic systems: in particular a geometric characterization of its 'true boundary' (Definition 2). In order to optimize these algorithms, we also propose a technique, that we call relaxation, which can simplify the decomposition process and reduce the number of redundant components in the output. Moreover, we present procedures for basic set-theoretical operations on semi-algebraic sets represented by triangular decomposition. Experimentation confirms the effectiveness of our techniques. ? 2011 ACM.EI
An Incremental Algorithm for Computing Cylindrical Algebraic Decompositions
In this paper, we propose an incremental algorithm for computing cylindrical
algebraic decompositions. The algorithm consists of two parts: computing a
complex cylindrical tree and refining this complex tree into a cylindrical tree
in real space. The incrementality comes from the first part of the algorithm,
where a complex cylindrical tree is constructed by refining a previous complex
cylindrical tree with a polynomial constraint. We have implemented our
algorithm in Maple. The experimentation shows that the proposed algorithm
outperforms existing ones for many examples taken from the literature
Generic Regular Decompositions for Parametric Polynomial Systems
This paper presents a generalization of our earlier work in [19]. In this
paper, the two concepts, generic regular decomposition (GRD) and
regular-decomposition-unstable (RDU) variety introduced in [19] for generic
zero-dimensional systems, are extended to the case where the parametric systems
are not necessarily zero-dimensional. An algorithm is provided to compute GRDs
and the associated RDU varieties of parametric systems simultaneously on the
basis of the algorithm for generic zero-dimensional systems proposed in [19].
Then the solutions of any parametric system can be represented by the solutions
of finitely many regular systems and the decomposition is stable at any
parameter value in the complement of the associated RDU variety of the
parameter space. The related definitions and the results presented in [19] are
also generalized and a further discussion on RDU varieties is given from an
experimental point of view. The new algorithm has been implemented on the basis
of DISCOVERER with Maple 16 and experimented with a number of benchmarks from
the literature.Comment: It is the latest version. arXiv admin note: text overlap with
arXiv:1208.611
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
Polynomial Interrupt Timed Automata
Interrupt Timed Automata (ITA) form a subclass of stopwatch automata where
reachability and some variants of timed model checking are decidable even in
presence of parameters. They are well suited to model and analyze real-time
operating systems. Here we extend ITA with polynomial guards and updates,
leading to the class of polynomial ITA (PolITA). We prove the decidability of
the reachability and model checking of a timed version of CTL by an adaptation
of the cylindrical decomposition method for the first-order theory of reals.
Compared to previous approaches, our procedure handles parameters and clocks in
a unified way. Moreover, we show that PolITA are incomparable with stopwatch
automata. Finally additional features are introduced while preserving
decidability
A Characterization of Reduced Forms of Linear Differential Systems
A differential system , with
is said to be in reduced form if where
is the Lie algebra of the differential Galois group of
. In this article, we give a constructive criterion for a system to be in
reduced form. When is reductive and unimodular, the system is in
reduced form if and only if all of its invariants (rational solutions of
appropriate symmetric powers) have constant coefficients (instead of rational
functions). When is non-reductive, we give a similar characterization via
the semi-invariants of . In the reductive case, we propose a decision
procedure for putting the system into reduced form which, in turn, gives a
constructive proof of the classical Kolchin-Kovacic reduction theorem.Comment: To appear in : Journal of Pure and Applied Algebr
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
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