Efficient Algorithms for Guessing, Inequalities, Summation

Computers can not just calculate with numbers but also with complicated mathematical expressions that may contain for instance also variables and functions. For doing so, there are specialized computer programs. In order to develop or to enhance such computer programs, one first needs a theoretical understanding of systematic procedures with which mathematical expressions can be processed. The funded project deals with such systematic procedures for three central problem areas. The first is known as guessing. It concerns automatic procedures that a computer can use in order to make meaningful predictions how a given sequence of numbers should continue, similar as in certain intelligence tests. Such procedures play an important role in experimental mathematics, where they are used for detecting conjectured relations. The second problem area is about inequalities. While automated methods for the treatment of equalities are already quite well understood, there still remains a considerable need for research and development as far as inequalities are concerned. In the third problem area, we deal with the simplification of expressions involving sums and integrals. For mathematical expressions of this kind, automated procedures are of particular interest because traditional methods (paper and pencil) are limited to rather simple problems. In all three areas the goal of the project consists of analyzing and improving known procedures, inventing and developing new procedures, turning the theoretical procedures into actual computer programs, investigating the efficiency of these programs, and finally applying all this to the solution of open problems in other parts of mathematics. This project is an Austrian-French cooperation that on the Austrian side involves scientists from JKU Linz and the Radon Institute for Computational and Applied Mathematics, and on the french side scientists from INRIA and Sorbonne University.