Quantifier Elimination Problem Research Articles

BEAMS ON TENSIONLESS ELASTIC FOUNDATION: APPROXIMATE QUANTIFIER ELIMINATION WITH CHEBYSHEV SERIES

The well-known Sturm's theorem (based on Sturm's sequences) for the determination of the number of distinct real zeros of polynomials in a finite or infinite real interval has been already used in elementary quantifier elimination problems including applied mechanics and elasticity problems. Here it is further suggested that this theorem can also be used for quantifier elimination, but in more complicated problems where the functions involved are not simply polynomials, but they may contain arbitrary transcendental functions. In this case, it is suggested that the related transcendental equations/inequalities can be numerically approximated by polynomial equations/inequalities with the help of Chebyshev series expansions in numerical analysis. The classical problem of a straight isotropic elastic beam on a tensionless elastic foundation, where the deflection function (incorporating both the exponential function and trigonometric functions) should be continuously positive (this giving rise to a quantifier elimination problem along the length of the beam) is used as an appropriate vehicle for the illustration of the present mixed (symbolic–numerical) approach. Two such elementary beam problems are considered in some detail (with the help of the Maple V computer algebra system) and the related simple quantifier-free formulae are established and seen to coincide with those already available in the literature for the same beam problems. More complicated problems, probably necessitating the use of more advanced computer algebra techniques (together with Sturm's theorem), such as the Collins well-known and powerful cylindrical algebraic decomposition method for quantifier elimination, can also easily be employed in the present approximate (because of the use of Chebyshev series expansions) symbolic–numerical computational environment.

Computing over the Reals with Addition and Order: Higher Complexity Classes

This paper deals with issues of structural complexity in a linear version of the Blum-Shub-Smale model of computation over the real numbers. Real versions of PSPACE and of the polynomial time hierarchy are defined, and their properties are investigated. Mainly two types of results are presented: •Equivalence between quantification over the real numbers and over {0, 1};•Characterizations of recognizable subsets of {0, 1}* in terms of familiar discrete complexity classes. The complexity of the decision and quantifier elimination problems in the theory of the reals with addition and order is also studied.

Applying Quantifier Elimination to Stability Analysis of Difference Schemes

Stability analysis is an important tool for constructing time-stepping finite difference schemes for partial differential equations. This paper describes how von Neumann stability analysis can be reduced to a quantifier elimination problem over the reals. We report our experience in analysing some difference schemes by using a quantifier package based on the partial cylindrical algebraic decomposition algorithm.

Herbrand style proof procedures for modal logic

ABSTRACT In this paper we state and prove Herbrand's properties for two modal systems, namely T and S4, thus adapting a previous result obtained for the system D [CIA 86a] to such theories. These properties allow the first order extension—along the lines of [CIA 91]—of the resolution method defined in [ENJ 86] for the corresponding propositional modal systems. In fact, the Herbrand-style procedures proposed here treat quantifiers in a uniform way, that suggests the definition of a restricted notion of unification for quantifier free modal formulae obtained by means of a Skolem-like procedure. The differences among the three systems considered, D, T and S4, are taken into account by preliminary propositional transformations of the formulae, in such a way that the problem of quantifier elimination can be treated within the same pattern in the different systems. The methods employed to prove the fundamental theorems are purely proof-theoretical and make use of the formalization of the considered modal theori...

Angus Macintyre, Kenneth McKenna, and Lou van den Dries. Elimination of quantifiers in algebraic structures. Advances in mathematics, vol. 47 (1983), pp. 74–87. - L. P. D. van den Dries. A linearly ordered ring whose theory admits elimination of quantifiers is a real closed field. Proceedings of the American Mathematical Society, vol. 79 (1980), pp. 97–100. - Bruce I. Rose.

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Solving Diophantine Problems Over All Residue Class Fields of a Number Field and All Finite Fields

0. Introduction 203 A. History and explanation of the problem 203 B. Background from logic and elimination theory 208 1. Generalizing the quantifier elimination problem . 210 A. Notations and terminology 210 B. The Frobenius symbol 212 C. Galois stratification and generalization of the diophantine problem .213 2. The intersection-union process 215 A. The intersection-union process over a finite field 215 B. The intersection-union process over a perfect field 217 3. A generalization of the theorems of Bertini and Noether . 219 4. Diophantine problems over all residue class fields of a number field . . .225 5. Diophantine problems over finite fields . 230 A. Over all extensions of a fixed finite field 230 B. Over all finite fields 231