A Course in Formal Languages, Automata and Groups

This book is based on notes for a master's course given at Queen Mary, University of London, in the 1998/9 session. Such courses in London are quite short, and the course consisted essentially of the material in the ?rst three chapters, together with a two-hour lecture on connections with group theory. Chapter 5 is a considerably expanded version of this. For the course, the main sources were the books by Hopcroft and Ullman ([20]), by Cohen ([4]), and by Epstein et al. ([7]). Some use was also made of a later book by Hopcroft and Ullman ([21]). The ulterior motive in the ?rst three chapters is to give a rigorous proof that various notions of recursively enumerable language are equivalent. Three such notions are considered. These are: generated by a type 0 grammar, recognised by a Turing machine (deterministic or not) and de?ned by means of a Godel ¨ numbering, having de?ned recursively enumerable for sets of natural numbers. It is hoped that this has been achieved without too many ar- ments using complicated notation. This is a problem with the entire subject, and it is important to understand the idea of the proof, which is often quite simple. Two particular places that are heavy going are the proof at the end of Chapter 1 that a language recognised by a Turing machine is type 0, and the proof in Chapter 2 that a Turing machine computable function is partial recursive.

Most books on formal languages and automata are written for undergraduates in computer science; in contrast, this book provides a rigorous text aimed at the postgraduate-level mathematician with precise definitions and clear and succinct proofs. The book examines the interplay between group theory and formal languages, and is the first to include an account of the significant Muller-Schupp theorem. Includes a clear account of deterministic context-free languages and their connection with LR(k) grammars. A complete solutions manual is available to lecturers via the web. Includes supplementary material: sn.pub/extras Request lecturer material: sn.pub/lecturer-material

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Based on the author's lecture notes for an MSc course, this text combines formal language and automata theory and group theory, a thriving research area that has developed extensively over the last twenty-five years.

The aim of the first three chapters is to give a rigorous proof that various notions of recursively enumerable language are equivalent. Chapter One begins with languages defined by Chomsky grammars and the idea of machine recognition, contains a discussion of Turing Machines, and includes work on finite state automata and the languages they recognise. The following chapters then focus on topics such as recursive functions and predicates; recursively enumerable sets of natural numbers; and the group-theoretic connections of language theory, including a brief introduction to automatic groups. Highlights include:

• A comprehensive study of context-free languages and pushdown automata in Chapter Four, in particular a clear and complete account of the connection between LR(k) languages and deterministic context-free languages.

• A self-contained discussion of the significant Muller-Schupp result on context-free groups.

Enriched with precise definitions, clear and succinct proofs and worked examples, the book is aimed primarily at postgraduate students in mathematics but will also be of great interest to researchers in mathematics and computer science who want to learn more about the interplay between group theory and formal languages.

A solutions manual is available to instructors via www.springer.com.

Contenu
Preface.- Contents.- 1. Grammars and Machine Recognition.- 2. Recursive Functions.- 3. Recursively Enumerable Sets and Languages.- 4. Context-free language.- 5. Connections with Group Theory.- A. Results and Proofs Omitted in the Text.- B. The Halting Problem and Universal Turing Machines.- C. Cantor's Diagonal Argument.- D. Solutions to Selected Exercises.- References.- Index.