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Abstract. We address satisfiability checking for the first-order theory of the real-closed field (RCF) using satisfiability-modulo-theories (SMT) solving. SMT solvers combine a SAT solver to resolve the Boolean structure of a given formula with theory solvers to verify the consistency of sets of theory constraints. In this paper, we report on an integration of Gröbner bases as a theory solver so that it conforms with the requirements for efficient SMT solving: (1) it allows the incremental adding and removing of polynomials from the input set and (2) it can compute an inconsistent subset of the input constraints if the Gröbner basis contains 1. We modify Buchberger's algorithm by implementing a new update operator to optimize the Gröbner basis and provide two methods to handle inequalities. Our implementation uses special data structures tuned to be efficient for huge sets of sparse polynomials. Besides solving, the resulting module can be used to simplify constraints before being passed to other RCF theory solvers based on, e.g., the cylindrical algebraic decomposition.

Abstract. We present the growing C++ library GiNaCRA, which provides efficient and easy-to-integrate data structures and methods for real algebra. It is based on the C++ library GiNaC, supporting the symbolic representation and manipulation of polynomials. In contrast to other similar tools, our open source library aids exact, real algebraic computations based on an appropriate data type representing real zeros of polynomials. The only non-standard library GiNaCRA depends on is GiNaC, which makes the installation and usage of our library simple. Our longterm goal is to integrate decision procedures for real algebra within the Satisfiability-Modulo-Theories (SMT) context and thereby provide tool support for many applied formal methods.

Though there is an increasing need for computer scientists in our society, gifted young people with strong mathematical background, who would be well-suited for a study of computer science, often do not consider this choice because they have a wrong picture of computer science in mind. In this paper we present a new concept of collaborative learning to introduce high-school students to the field of theoretical computer science. In particular, we choose the demanding field of real-time systems in order to illustrate the applicability of theoretical methods on real-world systems. Statistical results show that with our concept we are able to point out the manifoldness, beauty and challenge of this scientific area and can convince high-school students to consider computer science as a choice of study.