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The subject of this book is string-rewriting systems. It is generally accepted that string-rewriting was first introduced by Axel Thue in the early part of this century. In the 1960´s and early 1970´s, it received renewed attention due to interest in formal language theory. In the 1980´s and 1990´s, it has received more interest since it can be viewed as a special case of term rewriting, a subject that has become important in the study of automated deduction. Today, string-rewriting is studied by researchers in theoretical computer science and also by researchers interested in the foundations of artificial intelligence. A sketch of the way that the subject has developed is contained in Chapter 0, and the reader is advised to begin with that chapter. Both authors have been active in the field and have lectured on the subject in several universities. Lecture notes have been produced and dis tributed. This monograph is a result of revising and rewriting those notes. It represents an attempt by the authors to present the concepts that the authors consider to be most fundamental and to gather together the most useful results in such a way that they can be understood and used in studies relating to more general rewriting, to automated deduction, and to algo rithmic problems of algebraic structures. This monograph is written for independent study by researchers in the oretical computer science or in the foundations of artificial intelligence.

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0 Introduction.- 0.1 Historical Development.- 0.2 An Outline of Recent Developments.- 0.3 Contents of the Monograph.- 1 Preliminaries.- 1.1 Abstract Reduction Systems.- 1.2 Reduction Modulo an Equivalence Relation.- 1.3 Strings, Languages and Automata.- 1.4 Some Turing Machine Constructions.- 1.5 Bibliographic Remarks.- 2 String-Rewriting Systems.- 2.1 Rewriting Systems for Strings.- 2.2 Computing Normal Forms.- 2.3 Testing for Local Confluence.- 2.4 The Knuth-Bendix Completion Procedure.- 2.5 Some Undecidable Properties.- 2.6 Bibliographic Remarks.- 3 Length as the Basis for Reduction.- 3.1 Basic Properties.- 3.2 Testing for Confluence.- 3.3 Confluence on a Single Class.- 3.4 Equivalent Systems.- 3.5 Church-Rosser Congruences.- 3.6 Other Systems Based on Length.- 3.7 Bibliographic Remarks.- 4 Monadic String-Rewriting Systems.- 4.1 Basic Properties.- 4.2 Specification of Formal Languages.- 4.3 A Decision Procedure.- 4.4 Applications of the Decision Procedure.- 4.5 Limitations of the Decision Procedure.- 4.6 Bibliographic Remarks.- 5 Length-Reducing Non-Monadic String-Rewriting Systems.- 5.1 Presenting Recursively Enumerable Languages.- 5.2 Some Undecidability Results.- 5.3 Some Questions on Congruential Languages.- 5.4 Bibliographic Remarks.- 6 Algebraic Protocols.- 6.1 Basic Properties.- 6.2 Security and Cascade Protocols.- 6.3 Security and Name-Stamp Protocols.- 6.4 Bibliographic Remarks.- 7 Algebraic Properties.- 7.1 Finite Monoid-Presentations.- 7.2 Tietze Transformations.- 7.3 Some Undecidability Results.- 7.4 The Free Monoid Problem.- 7.5 The Group Problem.- 7.6 Bibliographic Remarks.- References.

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