New PDF release: Algebraic Theory of Automata and Languag

By Masami Ito

ISBN-10: 9810247273

ISBN-13: 9789810247270

Even though there are a few books facing algebraic concept of automata, their contents consist regularly of Krohn–Rhodes concept and similar issues. the themes within the current ebook are relatively assorted. for instance, automorphism teams of automata and the partly ordered units of automata are systematically mentioned. in addition, a few operations on languages and certain periods of normal languages linked to deterministic and nondeterministic directable automata are handled. The publication is self-contained and accordingly doesn't require any wisdom of automata and formal languages.

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M. ,~nrn). Now, we compute the value ~ ( i - l ) ~ First, + ~ . notice that h&, s = 1 , 2 , . . , n, t = 1 , 2 , . . , rn is the [(s- 1)m t]-th entry of p ( f ) . Put here (Q 8 11)(a)= ( t a p ) , a , p = 1 , 2 , . . , nm. since s,t is an and matrix, holds. Consequently, we have fore, we have Thereand thuss CHAPTER 1. e. G = H x K . Moreover, we assume that A = X , SQ) and B = ( K , X , S n ) are a n ( n , H ) - , and a n (m,K)-automaton, respectively. Then A x B is a strongly connected automaton with G ( A 2 B) M G i f and only i f {Q @ I I ( a ) I a E X } is a regular system in Grim.

CHARACTERISTIC MONOIDS A N D INPUT SETS 33 Put T = (12 3 . . n ) be the element of the symmetric group S ( n ) on { 1,2,3,. . , n}. Moreover, we assign Q ( z ) = e F ( q l ) E where e is the identity of G. Thus we can define an (n,G)-automaton A= X,Sq). ( (c, Now, we prove that A is regular. Proof of the strong connectedness of A First, we prove that, for any i , j = 1 , 2 , ,. ,n and all h E H , there exists an element z E X* such that the (i,j)-entry of Q(z) is equal to h. By the assignment of Q ( Y ) ,for any h E H there exist some integers s = 1 , 2 , .

Hence C ( A )= C ( B ) . (G,X , 6,p) be a n (n,G)-automaton. T h e n the characteristic monoid of A is isomorphic to Q f ( X * ) . 2 Let A = Proof Obvious from the fact that, for any x,y E only if @(z)= @(y). X*,x - y if and From the above proposition, to study the structure of the characteristic monoid of a strongly connected automaton, it is enough to study that of the corresponding regular group-matrix type automaton. Thus the following result in [57] can be proved by our method. 5. 3 Let A = ( S , X , 6 ) be a strongly connected automaton.

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Algebraic Theory of Automata and Languag by Masami Ito


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