Interation Theories

Written both for graduate students and research scientists in theoretical computer science and mathematics, this book provides a detailed investigation of the properties of the fixed point or iteration operation. Iteration plays a fundamental role in the theory of computation: for example, in the theory of automata, in formal language theory, in the study of formal power series, in the semantics of flowchart algorithms and programming languages, and in circular data type definitions. It is shown that in all structures that have been used as semantic models, the equational properties of the fixed point operation are captured by the axioms describing iteration theories. These structures include ordered algebras, partial functions, relations, finitary and infinitary regular languages, trees, synchronization trees, 2-categories, and others. The book begins with a gentle introduction to the study of universal algebra in the framework of algebraic theories. A remarkably useful calculus is developed for manipulating algebraic theory terms. The reader is then guided through a vast terrain of theorems and applications by means of detailed proofs, examples, and exercises, with the emphasis on equational proofs. The last chapter shows that the familiar topic of correctness logic is a special case of the equational logic of iteration theories. Several significant open problems are scattered throughout the text.