## A Course in Formal Languages, Automata and GroupsThis 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. |

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### Contents

1 | |

Recursive Functions | 21 |

Recursively Enumerable Sets and Languages | 48 |

Contextfree Languages | 59 |

Connections with Group Theory | 93 |

Results and Proofs Omitted in the Text | 131 |

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abacus computable assume Cayley graph Chapter Chomsky normal form class of languages computable functions configuration context-free grammar context-free languages context-sensitive Ð Ð Ð defined Definition deterministic PDA edges element of G exercise final finite index finite set free group free product function f G-derivation Godel numbering grammar G group G halts hence HNN-extension holds in G homomorphism induction infinite input isomorphic label language with alphabet leftmost derivation Lemma length Let G letter LR(k mapping natural numbers Normal Form numerical TM obtained partial function partial recursive functions predicate prefix-free primitive recursive functions primitively recursively closed Proof Pumping Lemma reading reduced word register program regular grammar regular languages replace right-hand side rightmost derivation sequence stack starting string subset Suppose symbol tape description terminal Theorem TM’s transition diagram triangle Turing machine variables vertex vertices word problem