Group Theory

Download e-book for iPad: A Course in Formal Languages, Automata and Groups by Ian Chiswell

By Ian Chiswell

ISBN-10: 1848009402

ISBN-13: 9781848009400

In response to the author’s lecture notes for an MSc direction, this article combines formal language and automata idea and workforce conception, a thriving study sector that has constructed largely over the past twenty-five years.

The target of the 1st 3 chapters is to provide a rigorous evidence that a number of notions of recursively enumerable language are an identical. bankruptcy One starts off with languages outlined by way of Chomsky grammars and the belief of desktop reputation, features a dialogue of Turing Machines, and comprises paintings on finite kingdom automata and the languages they realize. the next chapters then specialize in subject matters equivalent to recursive features and predicates; recursively enumerable units of usual numbers; and the group-theoretic connections of language conception, together with a quick creation to automated teams.

Highlights include:
* A entire examine of context-free languages and pushdown automata in bankruptcy 4, specifically a transparent and whole 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 distinctive definitions, transparent and succinct proofs and labored examples, the e-book is aimed basically at postgraduate scholars in arithmetic yet can also be of serious curiosity to researchers in arithmetic and machine technology who are looking to study extra in regards to the interaction among staff concept and formal languages.

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

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2) The abacus machines of depth n are the words M1 . . Mr , where each Mi is a simple abacus machine of depth at most n, and some Mi has depth exactly n. (3) The simple abacus machines of depth n + 1 are the words (M)k , where M is an abacus machine of depth n and k ≥ 1. An abacus machine is a set of instructions for operating on the registers, as follows. 2 Recursive Functions 33 ak : add 1 to contents of register k; sk : subtract 1 from register k unless it contains 0. M1 . . Mr : execute M1 , .

For exp(x, 0) = 1 exp(x, y + 1) = m(x, exp(x, y)). (4) (factorial) Fac(x) = x! is primitive recursive since Fac(0) = 1, Fac(x + 1) = m(x + 1, Fac(x)). (5) Any constant function Nn → N is primitive recursive. For n = 1, the constant function 0 is z, the constant function 1 is σ ◦ z, the constant function 2 is σ ◦ (σ ◦ z), etc. For general n, the constant function c is c ◦ π1n , where c : N → N is the constant function with value c. (6) (predecessor) We define Pred(x) to be x − 1 if x > 0 and Pred(0) to be 0.

Examples. (1) Cleark = (sk )k (clears the contents of register k). (2) Descopy p,q = Clearq (s p aq ) p (copies contents of register p to register q and clears register p. This is short for “destructive copy” since the contents of register p are destroyed). (3) Copy p,q,r = Clearq (s p aq ar ) p (sr a p )r (if register r is clear, copies register p to register q, leaving registers other than q unchanged). Next we prove some results on the structure of an abacus machine. 8. (1) An abacus machine has the same number of left and right parentheses (all the letters )k , k ≥ 1 are regarded as right parentheses here).

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

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