Replacement systems, such as term rewriting systems, tree manipulat ing systems, and graph grammars, have been used in Computer Science in the context of theorem proving, program optimization, abstract data types, algebraic simplification, and symbolic comput ation. Replacement systems for strings arose about seventy years earlier in the area of combinatory logic and group theory. The most natural and appropriate formalism for dealing with string rewriting is the notion of a semi-Thue system and this monograph treats its central aspects. The reduction relation is here defined firstly by the direction of the rules and secondly by some metric that yields efficient algorithms. These systems are general enough to discuss the basic notions of arbitrary replacement systems, such as termination, confluence, and the Church-Rosser property in its original meaning. Confluent semi-Thue systems in which each and every derivation consists of finitely many steps only are called complete; they guarantee the existence of unique normal forms as canonical representatives of the Thue congruence classes. Each such system can be considered a nondeterministic algorithm for the word problem which works correctly without backtracking. This is often conceptually simpler and more elegant than an ad hoc construction. In many cases a replace ment system can be altered to a complete system by the Knuth-Bendix completion method.

Klappentext

This monograph treats comprehensively central aspects of string rewriting systems in the form of semi-Thue systems. These are so general as to enable the discussion of all the basic notions and questions that arise in arbitrary replacement systems as used in various areas of computer science. The Church-Rosser property is used in its original meaning and the existence of complete monoid and group presentations is the central point of discussion. Decidability problems with their complexity are surveyed and congruential languages including the deterministic context-free NTS languages are discussed. The book contains a number of generalizations of results published elsewhere, e.g., the uniqueness of complete string rewriting systems with respect to the underlying order. Completely new and unpublished results which serve as an exposition of techniques and new methods are discussed in detail. With the help of semi-Thue systems it is shown in which situations the famous Knuth-Bendix completion method does not terminate and why, and that in general complete replacement systems cannot always be used as algorithms to solve the word problem. It is suggested how these situations can be stated by using a certain control under which the rewriting is to be performed. This monograph is a reference for graduate students and active researchers in theoretical computer science. The reader is led to the forefront of current research in the area of string rewriting and monoid presentations.

Inhalt

1. Basic Definitions.- 1.1 General Reduction Systems.- 1.2 String Rewriting.- 2. Decision Problems.- 2.1 Undecidability Results.- 2.2 Decidability Results.- 3. Congruential Languages Specified by Semi-Thue Systems.- 4. Complete STSs, Groups, and Monoids.- 4.1 Introduction and General Results.- 4.2 The STS S1 = {(abba, ?)}.- 5. The Special One-Relator STSs Snfor n > 1 and the Groups Gn.- 5.1 Definitions, Notation, and Characterization of Gnas a Semidirect Product.- 5.2 Matrix Representation of Gn in GL2(?).- 5.3 Normal Forms, Conjugates, and Homomorphisms.- 5.4 Commutativity and Unique Divisibility in Gn.- 5.5 Complete Presentation for Gn.- 5.6 Sn Has No Preperfect Equivalent.- 5.7 Irreducible Strings in STSs Equivalent to Sn.- References and Further Reading.