14 research outputs found
Validity proof of Lazard's method for CAD construction
In 1994 Lazard proposed an improved method for cylindrical algebraic
decomposition (CAD). The method comprised a simplified projection operation
together with a generalized cell lifting (that is, stack construction)
technique. For the proof of the method's validity Lazard introduced a new
notion of valuation of a multivariate polynomial at a point. However a gap in
one of the key supporting results for his proof was subsequently noticed. In
the present paper we provide a complete validity proof of Lazard's method. Our
proof is based on the classical parametrized version of Puiseux's theorem and
basic properties of Lazard's valuation. This result is significant because
Lazard's method can be applied to any finite family of polynomials, without any
assumption on the system of coordinates. It therefore has wider applicability
and may be more efficient than other projection and lifting schemes for CAD.Comment: 21 page
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
Cylindrical Algebraic Sub-Decompositions
Cylindrical algebraic decompositions (CADs) are a key tool in real algebraic
geometry, used primarily for eliminating quantifiers over the reals and
studying semi-algebraic sets. In this paper we introduce cylindrical algebraic
sub-decompositions (sub-CADs), which are subsets of CADs containing all the
information needed to specify a solution for a given problem.
We define two new types of sub-CAD: variety sub-CADs which are those cells in
a CAD lying on a designated variety; and layered sub-CADs which have only those
cells of dimension higher than a specified value. We present algorithms to
produce these and describe how the two approaches may be combined with each
other and the recent theory of truth-table invariant CAD.
We give a complexity analysis showing that these techniques can offer
substantial theoretical savings, which is supported by experimentation using an
implementation in Maple.Comment: 26 page
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
An Implementation of CAD in Maple Utilising McCallum Projection
Cylindrical algebraic decomposition (CAD) is an important tool for the
investigation of semi-algebraic sets. Originally introduced by Collins in the
1970s for use in quantifier elimination it has since found numerous
applications within algebraic geometry and beyond. Following from his original
work in 1988, McCallum presented an improved algorithm, CADW, which offered a
huge increase in the practical utility of CAD. In 2009 a team based at the
University of Western Ontario presented a new and quite separate algorithm for
CAD, which was implemented and included in the computer algebra system Maple.
As part of a wider project at Bath investigating CAD and its applications,
Collins and McCallum's CAD algorithms have been implemented in Maple. This
report details these implementations and compares them to Qepcad and the
Ontario algorithm.
The implementations were originally undertaken to facilitate research into
the connections between the algorithms. However, the ability of the code to
guarantee order-invariant output has led to its use in new research on CADs
which are minimal for certain problems. In addition, the implementation
described here is of interest as the only full implementation of CADW, (since
Qepcad does not currently make use of McCallum's delineating polynomials), and
hence can solve problems not admissible to other CAD implementations.Comment: 9 pages; University of Bath, Dept. Computer Science Technical Report
Series, 2013-02, 201
Deciding the consistency of non-linear real arithmetic constraints with a conflict driven search using cylindrical algebraic coverings
We present a new algorithm for determining the satisfiability of conjunctions
of non-linear polynomial constraints over the reals, which can be used as a
theory solver for satisfiability modulo theory (SMT) solving for non-linear
real arithmetic. The algorithm is a variant of Cylindrical Algebraic
Decomposition (CAD) adapted for satisfiability, where solution candidates
(sample points) are constructed incrementally, either until a satisfying sample
is found or sufficient samples have been sampled to conclude unsatisfiability.
The choice of samples is guided by the input constraints and previous
conflicts.
The key idea behind our new approach is to start with a partial sample;
demonstrate that it cannot be extended to a full sample; and from the reasons
for that rule out a larger space around the partial sample, which build up
incrementally into a cylindrical algebraic covering of the space. There are
similarities with the incremental variant of CAD, the NLSAT method of Jovanovic
and de Moura, and the NuCAD algorithm of Brown; but we present worked examples
and experimental results on a preliminary implementation to demonstrate the
differences to these, and the benefits of the new approach