Ryoya Fukasaku

Real Quantifier Elimination by Computation of Comprehensive Gröbner Systems

A real quantifier elimination method based on the theory of real root counting and the computation of comprehensive Gröbner systems introduced by V. Weispfenning is studied in more detail. We introduce a simpler and more intuitive algorithm which is shown to be an improvement of the original algorithm. Our algorithm is implemented on the computer algebra system Maple using a recent algorithm to compute comprehensive Gröbner systems together with several simplification techniques. According to our computation experiments, our program is superior to other existing implementations for many examples which contain many equalities.

On the Implementation of CGS Real QE

A CGS real QE method is a real quantifier elimination (QE) method which is composed of the computation of comprehensive Grobner systems (CGSs) based on the theory of real root counting. Its fundamental algorithm was first introduced by Weispfenning in 1998. We further improved the algorithm in 2015 so that we can make a satisfactorily practical implementation. For its efficient implementation, there are several key issues we have to take into account. In this extended abstract we introduce them together with some important techniques for making an efficient CGS real QE implementation.

Improving a CGS-QE Algorithm

A real quantifier elimination algorithm based on computation of comprehensive Grobner systems introduced by Weispfenning and recently improved by us has a weak point that it cannot handle a formula with many inequalities. In this paper, we further improve the algorithm so that we can handle more inequalities.

Race Against the Teens --- Benchmarking Mechanized Math on Pre-university Problems

This paper introduces a benchmark problem library for mechanized math technologies including computer algebra and automated theorem proving. The library consists of pre-university math problems taken from exercise problem books, university entrance exams, and the International Mathematical Olympiads. It thus includes problems in various areas of pre-university math and with a variety of difficulty. Unlike other existing benchmark libraries, this one contains problems that are formalized so that they are obtainable as the result of mechanical translation of the original problems expressed in natural language. In other words, the library is designed to support the integration of the technologies of mechanized math and natural language processing towards the goal of end-to-end automatic math problem solving. The paper also presents preliminary experimental results of our prototype reasoning component of an end-to-end system on the library. The library is publicly available through the Internet.

CGSQE/SyNRAC: a real quantifier elimination package based on the computation of comprehensive Gröbner systems

CGSQE is a Maple package for real quantifier elimination (QE) we are developing. It works cooperating with SyNRAC which is also a Maple package for real QE one of the authors is developing. For a given first order formula, CGSQE eliminates all possible quantifiers using the underlying equational constraints by the computation of comprehensive Gröbner systems (CGSs). In case all quantifiers are not removable, it transforms the given formula into a formula which contains only strict inequalities of quantified variables, then uses a cylindrical algebraic decomposition based real QE program of SyNRAC to remove the remaining quantifiers. The core algorithm of CGSQE is a CGS real QE algorithm which was first introduced by Weispfenning in 1998 and further improved by us in 2015 so that we can make a satisfactorily practical implementation. CGSQE is superior to other real QE implementations for many examples which contain many equational constraints. In the software presentation, we would like to show high-performance computation of CGSQE.

On Real Roots Counting for Non-radical Parametric Ideals

An algorithm we have introduced has a great effect on quantifier elimination of a first order formula containing many equalities. When the parametric ideal generated by the underlying equalities is not radical, however, our algorithm tends to produce an unnecessarily complicated formula. In this short paper, we show a result concerning Hermitian quadratic forms. It enables us to improve our algorithm so that we can get a simple formula without any radical computation.

On QE Algorithms over an Algebraically Closed Field Based on Comprehensive Gröbner Systems

We introduce new algorithms for quantifier eliminations (QE) in the domain of an algebraically closed field. Our algorithms are based on the computation of comprehensive Gröbner systems (CGS). We study Suzuki–Sato’s CGS computation algorithm and its successors in more detail and modify them into an optimal form for applying to QE. Based on this modified algorithm, we introduce two QE algorithms. One is pursuing the simplest output quantifier free formula which employs only the computations of CGS. The other consists of parallel computations of CGS and GCD of parametric unary polynomials. It achieves faster computation time with reasonably simple output quantifier free formulas. Our implementation shows that in many examples our algorithms are superior to other existing algorithms such as a QE algorithm adopted in the Mathematica package Reduce and Resolve or a QE algorithm adopted in the Maple package Projection which are the most efficient existing implementations as far as we know.

QE Software Based on Comprehensive Gröbner Systems

We introduce two quantifier elimination softwares, one is in the domain of an algebraically closed field and another is of a real closed field. Both softwares are based on the computations of comprehensive Grobner systems.