Recently quantifier elimination (QE) has been of great interest in many fields of science and engineering. In this paper an effective symbolic-numeric cylindrical algebraic decomposition (SNCAD) algorithm and its variant specially designed for QE are proposed based on the authors' previous work and our implementation of those is reported. Based on analysing experimental performances, we are improving our design/synthesis of the SNCAD for its practical realization with existing efficient computational techniques and several newly introduced ones. The practicality of the SNCAD is now examined by a number of experimental results including practical engineering problems, which also reveals the quality of the implementation.

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.

When David Hilbert started so called "Hilbert's program" (formalization of mathematics) in the early 20th century to give a solid foundation to mathematics, he unintentionally introduced the possibility of automatization of mathematics. Theoretically, the possibility was denied by Gödel's incompleteness theorem. However, an interesting issue remains: is "mundane mathematics" automatizable? We are developing a system that solves a wide range of math problems written in natural language, as a part of the Todai Robot Project, an AI challenge to pass the university entrance examination. We give an overview and report on the progress of our project, and the theoretical and methodological difficulties to be overcome.