14 research outputs found
Cylindrical Algebraic Decomposition Using Local Projections
We present an algorithm which computes a cylindrical algebraic decomposition
of a semialgebraic set using projection sets computed for each cell separately.
Such local projection sets can be significantly smaller than the global
projection set used by the Cylindrical Algebraic Decomposition (CAD) algorithm.
This leads to reduction in the number of cells the algorithm needs to
construct. We give an empirical comparison of our algorithm and the classical
CAD algorithm
TheoryGuru: A Mathematica Package to Apply Quantifier Elimination Technology to Economics
We consider the use of Quantifier Elimination (QE) technology for automated
reasoning in economics. There is a great body of work considering QE
applications in science and engineering but we demonstrate here that it also
has use in the social sciences. We explain how many suggested theorems in
economics could either be proven, or even have their hypotheses shown to be
inconsistent, automatically via QE.
However, economists who this technology could benefit are usually unfamiliar
with QE, and the use of mathematical software generally. This motivated the
development of a Mathematica Package TheoryGuru, whose purpose is to lower the
costs of applying QE to economics. We describe the package's functionality and
give examples of its use.Comment: To appear in Proc ICMS 201
Need Polynomial Systems Be Doubly-Exponential?
Polynomial Systems, or at least their algorithms, have the reputation of
being doubly-exponential in the number of variables [Mayr and Mayer, 1982],
[Davenport and Heintz, 1988]. Nevertheless, the Bezout bound tells us that that
number of zeros of a zero-dimensional system is singly-exponential in the
number of variables. How should this contradiction be reconciled?
We first note that [Mayr and Ritscher, 2013] shows that the doubly
exponential nature of Gr\"{o}bner bases is with respect to the dimension of the
ideal, not the number of variables. This inspires us to consider what can be
done for Cylindrical Algebraic Decomposition which produces a
doubly-exponential number of polynomials of doubly-exponential degree.
We review work from ISSAC 2015 which showed the number of polynomials could
be restricted to doubly-exponential in the (complex) dimension using McCallum's
theory of reduced projection in the presence of equational constraints. We then
discuss preliminary results showing the same for the degree of those
polynomials. The results are under primitivity assumptions whose importance we
illustrate.Comment: Extended Abstract for ICMS 2016 Presentation. arXiv admin note: text
overlap with arXiv:1605.0249
CAD Adjacency Computation Using Validated Numerics
We present an algorithm for computation of cell adjacencies for well-based
cylindrical algebraic decomposition. Cell adjacency information can be used to
compute topological operations e.g. closure, boundary, connected components,
and topological properties e.g. homology groups. Other applications include
visualization and path planning. Our algorithm determines cell adjacency
information using validated numerical methods similar to those used in CAD
construction, thus computing CAD with adjacency information in time comparable
to that of computing CAD without adjacency information. We report on
implementation of the algorithm and present empirical data.Comment: 20 page
SMT-Solving Induction Proofs of Inequalities
This paper accompanies a new dataset of non-linear real arithmetic problems
for the SMT-LIB benchmark collection. The problems come from an automated proof
procedure of Gerhold--Kauers, which is well suited for solution by SMT. The
problems of this type have not been tackled by SMT-solvers before. We describe
the proof technique and give one new such proof to illustrate it. We then
describe the dataset and the results of benchmarking. The benchmarks on the new
dataset are quite different to the existing ones. The benchmarking also brings
forward some interesting debate on the use/inclusion of rational functions and
algebraic numbers in the SMT-LIB.Comment: Presented at the 2022 SC-Square Worksho
Comparing machine learning models to choose the variable ordering for cylindrical algebraic decomposition
There has been recent interest in the use of machine learning (ML) approaches
within mathematical software to make choices that impact on the computing
performance without affecting the mathematical correctness of the result. We
address the problem of selecting the variable ordering for cylindrical
algebraic decomposition (CAD), an important algorithm in Symbolic Computation.
Prior work to apply ML on this problem implemented a Support Vector Machine
(SVM) to select between three existing human-made heuristics, which did better
than anyone heuristic alone. The present work extends to have ML select the
variable ordering directly, and to try a wider variety of ML techniques.
We experimented with the NLSAT dataset and the Regular Chains Library CAD
function for Maple 2018. For each problem, the variable ordering leading to the
shortest computing time was selected as the target class for ML. Features were
generated from the polynomial input and used to train the following ML models:
k-nearest neighbours (KNN) classifier, multi-layer perceptron (MLP), decision
tree (DT) and SVM, as implemented in the Python scikit-learn package. We also
compared these with the two leading human constructed heuristics for the
problem: Brown's heuristic and sotd. On this dataset all of the ML approaches
outperformed the human made heuristics, some by a large margin.Comment: Accepted into CICM 201
The Complexity of Cylindrical Algebraic Decomposition with Respect to Polynomial Degree
Cylindrical algebraic decomposition (CAD) is an important tool for working
with polynomial systems, particularly quantifier elimination. However, it has
complexity doubly exponential in the number of variables. The base algorithm
can be improved by adapting to take advantage of any equational constraints
(ECs): equations logically implied by the input. Intuitively, we expect the
double exponent in the complexity to decrease by one for each EC. In ISSAC 2015
the present authors proved this for the factor in the complexity bound
dependent on the number of polynomials in the input. However, the other term,
that dependent on the degree of the input polynomials, remained unchanged.
In the present paper the authors investigate how CAD in the presence of ECs
could be further refined using the technology of Groebner Bases to move towards
the intuitive bound for polynomial degree