Abstract
Equations and formulas occur frequently in mathematics, logic, and computer science. This chapter reviews the main results concerning equations and the methods available for reasoning about them and computing with them. Reasoning about equations may involve deciding if an equation follows or is a consequence of a given set of equations or axioms, or if an equation is true in a given theory. Equations may also be used as definitions. The use of equations as definitions is well known in computer science because the programs written in applicative languages, abstract interpreter definitions, and algebraic data type definitions are clearly of this nature. Computing with equations and reasoning about equations are closely related. One method for reasoning about equations consists in compiling them into rewrite rules and using them to reduce expressions to canonical form. It is useful to start with a clear understanding of the various semantics that one can associate with equations and with rewrite rules. The chapter presents various semantic definitions and then discusses their associated proof theories.
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