19 research outputs found
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
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
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
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
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
Problem formulation for truth-table invariant cylindrical algebraic decomposition by incremental triangular decomposition
Cylindrical algebraic decompositions (CADs) are a key tool for solving problems in real algebraic geometry and beyond. We recently presented a new CAD algorithm combining two advances: truth-table invariance, making the CAD invariant with respect to the truth of logical formulae rather than the signs of polynomials; and CAD construction by regular chains technology, where first a complex decomposition is constructed by refining a tree incrementally by constraint. We here consider how best to formulate problems for input to this algorithm. We focus on a choice (not relevant for other CAD algorithms) about the order in which constraints are presented. We develop new heuristics to help make this choice and thus allow the best use of the algorithm in practice. We also consider other choices of problem formulation for CAD, as discussed in CICM 2013, revisiting these in the context of the new algorithm
Recent advances in real geometric reasoning
In the 1930s Tarski showed that real quantifier elimination was possible, and
in 1975 Collins gave a remotely practicable method, albeit with
doubly-exponential complexity, which was later shown to be inherent. We discuss
some of the recent major advances in Collins method: such as an alternative
approach based on passing via the complexes, and advances which come closer to
"solving the question asked" rather than "solving all problems to do with these
polynomials"
Computation in Real Closed Infinitesimal and Transcendental Extensions of the Rationals.
Abstract. Recent applications of decision procedures for nonlinear real arithmetic (the theory of real closed fields, or RCF) have presented a need for reasoning not only with polynomials but also with transcendental constants and infinitesimals. In full generality, the algebraic setting for this reasoning consists of real closed transcendental and infinitesimal extensions of the rational numbers. We present a library for computing over these extensions. This library contains many contributions, including a novel combination of Thom’s Lemma and interval arithmetic for representing roots, and provides all core machinery required for building RCF decision procedures. We describe the abstract algebraic setting for computing with such field extensions, present our concrete algorithms and optimizations, and illustrate the library on a collection of examples. 1 Overview and Related Work Decision methods for nonlinear real arithmetic are essential to the formal verification of cyber-physical systems and formalized mathematics. Classically, thes
The growth of regular functions on algebraic sets
We are concerned with the set of all growth exponents of regular functions on an algebraic subset V of . We show that its elements form an increasing sequence of rational numbers and we study the dependence of its structure on the geometric properties of V