Algebraic Processors

Abstract

<p> Algebraic simplification is the task of reducing an algebraic expression to a simpler form without changing the meaning of the expression. Simplification is generally a difficult task and may have different meanings according to what the subject considers as "simple" . This thesis starts off by reverse-engineering the concept of algebraic processors in the IMPS interactive mathematical proof system - which is responsible for handling all the algebraic simplification tasks - and discusses its algorithm and usage in detail. Then it explores the idea of algebraic processors as generic programs that can be configured for any type of algebraic structure to simplify expressions of that type by first formalizing the theory of algebraic processors of IMPS and then extending it to provide solutions for related topics. Algebraic processors can be defined for any user-defined algebra, as long as it conforms to the structure defined in this paper. The processors are defined as external units that can communicate with other mechanized mathematics systems in a trustable fashion and provide a program and a proof of correctness for any requests of simplification. Finally, some related processors such as one for simplification in partial orders and equivalence classes are outlined with some discussion of possible future expansions.</p>