By Ian Chiswell

ISBN-10: 1848009402

ISBN-13: 9781848009400

In keeping with the author’s lecture notes for an MSc direction, this article combines formal language and automata conception and crew conception, a thriving learn sector that has constructed widely during the last twenty-five years.

The objective of the 1st 3 chapters is to offer a rigorous facts that a variety of notions of recursively enumerable language are similar. bankruptcy One starts off with languages outlined through Chomsky grammars and the belief of desktop acceptance, incorporates a dialogue of Turing Machines, and contains paintings on finite kingdom automata and the languages they know. the next chapters then specialize in issues akin to recursive capabilities and predicates; recursively enumerable units of normal numbers; and the group-theoretic connections of language idea, together with a short creation to automated teams.

Highlights include:
* A entire research of context-free languages and pushdown automata in bankruptcy 4, particularly a transparent and entire account of the relationship among LR(k) languages and deterministic context-free languages.
* A self-contained dialogue of the numerous Muller-Schupp outcome on context-free groups.

Enriched with exact definitions, transparent and succinct proofs and labored examples, the e-book is aimed basically at postgraduate scholars in arithmetic yet may also be of significant curiosity to researchers in arithmetic and computing device technology who are looking to research extra concerning the interaction among staff concept and formal languages.

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Extra resources for A Course in Formal Languages, Automata and Groups (Universitext)

Example text

Proof. 10, there is an abacus machine M such that f (x) is defined if and only if (x1 , . . , xn , 0, 0, . ) ϕM is, for x = (x1 , . . , xn ) ∈ Nn , in which case (x1 , . . , xn , 0, 0, . ) ϕM = ( f (x), 0, 0, . ). 21, there is a TM T which simulates M, and T is the required TM. Thus abacus computable implies TM computable, and the corollary follows by Cor. 19. 20. 23 (Kleene Normal Form Theorem). There exist primitive recursive functions ϕ : N → N and ψ : N3 → N such that, if f : N → N is partial recursive, there exists g ∈ N such that f (x) = ϕ (μ t(ψ (g, x,t) = 0)).

Now let T be a numerical TM, and modify it as above to obtain T , so the transitions are words of length 5 in N∗ , and N is linearly ordered. We can therefore order the transitions by the lexicographic ordering, then number them to respect this ordering, say qi ai qi ai Di (1 ≤ i ≤ k) (so this is the ith transition in the lexicographic ordering, and k is the number of transitions). Define k i i i i p5i+2 p5i+3 pD gn(T ) = 2k 3q0 ∏ pq5ii pa5i+1 5i+4 q a i=1 (gn stands for “G¨odel numbering”, an idea discussed in the next chapter).

2 Recursive Functions 47 10. Construct a numerical TM T2 which, started on the tape description u01a 01b 01c , halts with the tape description u01a+1 01b−1 01c+1 , provided b > 0, for any u ∈ {0, 1}∗ and natural numbers a, b, c. 11. Construct a numerical TM T3 which, started on the tape description u01a 01b 0, halts with the tape description u01a+b 01b 0 , for any u ∈ {0, 1}∗ and natural numbers a, b. 12. Construct a numerical TM T4 which, started on the tape description u01a 01b 0 , halts with the tape description u01a+b 0 , for any u ∈ {0, 1}∗ and natural numbers a, b.

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A Course in Formal Languages, Automata and Groups (Universitext) by Ian Chiswell


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