Equations and rewrite rules: a survey

[摘要] Equations occur frequently in mathematics, logic and computer science. In this paper, we survey the main results concerning equations, and the methods available for reasoning about them and computing with them. The survey is self-contained and unified, using traditional abstract algebra. Reasoning about equations may involve deciding if an equation follows from a given set of equations (axioms), or if an equation is true in a given theory. When used in this manner, equations state properties that hold between objects. Equations may also be used as definitions; this use is well known in computer science: programs written in applicative languages, abstract interpreter definitions, and algebraic data type definitions are clearly of this nature. When these equations are regarded as oriented "rewrite rules," we may actually use them to compute. In addition to covering these topics, we discuss the problem of "solving" equations (the "unification" problem), the problem of proving termination of sets of rewrite rules, and the decidability and complexity of word problems and of combinations of equational theories. We restrict ourselves to first-order equations, and do not treat equations which define non-terminating computations or recent work on rewrite rules applied to equational congruence classes.

[学科分类] 计算机科学（综合）