Hidenao Iwane

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.

SyNRAC: A Toolbox for Solving Real Algebraic Constraints

We introduce various aspects of the design and the implementation of a symbolic/symbolic-numeric computation toolbox, called SyNRAC. SyNRAC is a package of commands written in the Maple language and the C language. This package indeed provides an environment for dealing with first-order formulas over the reals.

Construction of explicit optimal value functions by a symbolic-numeric cylindrical algebraic decomposition

Recently parametric treatment of constraint solving and optimization problems has received considerable attention in science and engineering. In this paper we show an efficient and systematic algorithm for parametric programming, i.e. computing exact optimal value functions, based on a specialized symbolic-numeric cylindrical algebraic decomposition. We also present some practical application examples from system and control theory.

Semantic Parsing of Pre-university Math Problems

We have been developing an end-to-end math problem solving system that accepts natural language input. The current paper focuses on how we analyze the problem sentences to produce logical forms. We chose a hybrid approach combining a shallow syntactic analyzer and a manually-developed lexicalized grammar. A feature of the grammar is that it is extensively typed on the basis of a formal ontology for pre-university math. These types are helpful in semantic disambiguation inside and across sentences. Experimental results show that the hybrid system produces a well-formed logical form with 88% precision and 56% recall.

Efficient Subformula Orders for Real Quantifier Elimination of Non-prenex Formulas

In this paper we study speeding up real quantifier elimination QE methods for non-prenex formulas. Our basic strategy is to solve non-prenex first-order formulas by performing QE for subformulas constituting the input non-prenex formula. We propose two types of methods heuristic methods/machine learning based methods to determine an appropriate ordering of QE computation for the subformulas. Then we empirically examine their effectiveness through experimental results over more than 2,000 non-trivial example problems. Our experiment results suggest machine learning can save much effort spent to design effective heuristics by trials and errors without losing efficiency of QE computation.

An effective implementation of a symbolic-numeric cylindrical algebraic decomposition for optimization problems

With many applications in engineering and in scientific fields, quantifier elimination (QE) has been attracting more attention these days. Cylindrical algebraic decomposition (CAD) is used as a basis for a general QE algorithm. We propose an effective symbolic-numeric cylindrical algebraic decomposition (SNCAD) algorithm for solving polynomial optimization problems. The main ideas are a bounded CAD construction approach and utilization of sign information. The bounded CAD constructs CAD only in restricted admissible regions to remove redundant projection factors and avoid lifting cells where truth values are constant over the region. By utilization of sign information we can avoid symbolic computation in the lifting phase. Techniques for implementation are also presented. These techniques help reduce the computing time. We have examined our implementation by solving many example problems. Experimental results show that our implementation significantly improves efficiency compared to our previous work.

A Symbolic-Numeric Approach to Multi-Objective Optimization in Manufacturing Design

Some important classes of optimization problems originating from the optimal design of semiconductor memories such as SRAM, aiming at boosting the yield rate, are studied. New optimization methods for the classes based on a symbolic algorithm called quantifier elimination, combined with numerical computation, are proposed. The total efficiency of the design process is improved by reducing the number of numerical yield-rate evaluations. In addition, useful information such as the explicit relations among design variables, objective functions, and the yield rate, is provided.

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.

Policy gradient reinforcement learning method for discrete-time linear quadratic regulation problem using estimated state value function

In this paper, we propose a policy gradient reinforcement learning method which directly estimates the gradient of the state value function (V-function) with respect to a feedback coefficient matrix using measurable data and uses it for policy improvement. The proposed method can be applicable to the case where the state-action value function (Q-function) is difficult to estimate, and can update the policy in an effective direction for reducing the accumulated cost.