Fast Theorem-Proving and Wu's Method

In this paper, the possibility of fast algorithm is discussed for me-chanical theorem proving, where the degeneracy condition are considered in designingof these algorithms. It is found that all of the methods depend seriously on some prin-ciples appearing in Wu's Method. In other words, some principles in Wu's Methodare the instinctive properties in these new fast algorithms of theorem proving.

A Study on Computer Consciousness on Intuitive Geometry Based on Mathematics Experiments and Statistical Analysis

In this paper, we present our research on building computing machines consciousness about intuitive geometry based on mathematics experiments and statistical inference. The investigation consists of the following five steps. At first, we select a set of geometric configurations and for each configuration we construct a large amount of geometric data as observation data using dynamic geometry programs together with the pseudo-random number generator. Secondly, we refer to the geometric predicates in the algebraic method of machine proof of geometric theorems to construct statistics suitable for measuring the approximate geometric relationships in the observation data. In the third step, we propose a geometric relationship detection method based on the similarity of data distribution, where the search space has been reduced into small batches of data by pre-searching for efficiency, and the hypothetical test of the possible geometric relationships in the search results has be performed. In the fourth step, we explore the integer relation of the line segment lengths in the geometric configuration in addition. At the final step, we do numerical experiments for the pre-selected geometric configurations to verify the effectiveness of our method. The results show that computer equipped with the above procedures can find out the hidden geometric relations from the randomly generated data of related geometric configurations, and in this sense, computing machines can actually attain certain consciousness of intuitive geometry as early civilized humans in ancient Mesopotamia.

ON THE FOUNDATION OF ALGEBRAIC DIFFERENTIAL GEOMETRY

An algebraic differential variety is defined as the zero-set of a differentialpolynomial set,and algebraic differential geometry is devoted to the study of such varieties.We give various decomposition formulas for the structures of such zero-sets which imply inparticular,the unique decomposition of an algebraic differential variety into its irreduciblecomponents.These formulas will find applications in various directions including mechanicaltheorem-proving of differential geometries.

Simplification and normalization of indexed differentials involving coordinate transformation

In nD differential geometry, basic geometric structures and properties are described locally by differentiable functions and equations with indices that obey Einstein summation convention. Although symbolic manipulation of such indexed functions is one of the oldest research topics in computer algebra, so far there exists no normal form reduction algorithm to judge whether two indexed polynomials involving indices of different coordinate systems are equal or not. It is a challenging task in computer algebra. In this paper, for a typical framework—the partial derivatives in coordinate transformation matrix involved are of order no more than two (such as local computations of ordinary curvatures and tor-sion), we put forward two algorithms, one on elimination of all redundant dummy indices of indexed polynomials, the other on normalization of such indexed polynomials, by which we can judge whether two indexed polynomials are equal or not. We implement the algorithms with Maple V.10 and use them to solve tensor verification problems in differential geometry, and to derive automatically the transformation rules of locally defined indexed functions under the change of local coordinates.

On the Satisfiability of Linear Systems of Equations for Quantifiers

In this paper, the satisfiability problem of a linear system of equations for quantifiers is discussed. This problem arises from the mechanical theorem-proving on modules. A discriminating condition for this problem is obtained which places no restriction on quantifiers. Some applications to Abelian groups, elementary number theory, vector spaces, ect. are discussed.