An Overview of A Formal Framework For Managing Mathematics

Mathematics is a process of creating, exploring, and connecting mathematical models. This paper presents an overview of a formal framework for managing the mathematics process as well as the mathematical knowledge produced by the process. The central idea of the framework is the notion of a biform theory which is simultaneously an axiomatic theory and an algorithmic theory. Representing a collection of mathematical models, a biform theory provides a formal context for both deduction and computation. The framework includes facilities for deriving theorems via a mixture of deduction and computation, constructing sound deduction and computation rules, and developing networks of biform theories linked by interpretations. The framework is not tied to a specific underlying logic; indeed, it is intended to be used with several background logics simultaneously. Many of the ideas and mechanisms used in the framework are inspired by the imps Interactive Mathematical Proof System and the Axiom computer algebra system

Transformers for Symbolic Computation and Formal Deduction

. A transformer is a function that maps expressions to expressions. Many transformational operators|such as expression evaluators and simpliers, rewrite rules, rules of inference, and decision procedures|can be represented by transformers. Computations and deductions can be formed by applying sound transformers in sequence. This paper introduces machinery for dening sound transformers in the context of an axiomatic theory in a formal logic. The paper is intended to be a rst step in a development of an integrated framework for symbolic computation and formal deduction. 1 Introduction Mechanized mathematics is the study of how the computer can be used to support, improve, and automate the mathematical reasoning process. The eld is divided into two quite separated camps: computer algebra and theorem proving. Computer algebra focuses on nonbranching symbolic computations over concrete structures implemented by fast, but not necessarily, sound algorithms. Theorem proving focus..

Integration of the Signum, Piecewise and Related Functions

When a computer algebra system has an assumption facility, it is possible to distinguish between integration problems with respect to a real variable, and those with respect to a complex variable. Here, a class of integration problems is defined in which the integrand consists of compositions of continuous functions and signum functions, and integration is with respect to a real variable. Algorithms are given for evaluating such integrals. 1 Introduction In recent years, `assume' or `declare' facilities have been implemented in most of the available computer algebra systems (CAS). As well, such facilities have been gaining wider acceptance within the user community. The presence of these facilities has altered the way CAS behave, and many established areas of symbolic computation need to be reconsidered. The topic of this paper is an example of the impact on one traditional field of computer algebra, namely, symbolic integration. Because the early versions of many present-day CAS cou..