Get Algebraic Theory of Automata and Languag PDF

By Masami Ito

ISBN-10: 9810247273

ISBN-13: 9789810247270

Even if there are a few books facing algebraic concept of automata, their contents consist normally of Krohn–Rhodes idea and similar subject matters. the themes within the current e-book are particularly varied. for instance, automorphism teams of automata and the partly ordered units of automata are systematically mentioned. furthermore, a few operations on languages and distinct periods of normal languages linked to deterministic and nondeterministic directable automata are handled. The ebook is self-contained and accordingly doesn't require any wisdom of automata and formal languages.

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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. Then C ( A )forms a group if and only if A is a permutation automaton.

9. REGULARITIES 45 Here, assume that Thus we have Consequently, we have: 0 0 ... p‘- t h 0 gyzp’ 0 * * *** * * * *** * * * ... * *** * *** * On the other hand, we have 6q/(p(G),a ) = p(G)Q’(a) = p(G)*(a) i-th 0 ... * *** * * * *** * p‘- t h 0 ... * *** * * *** * * * * *** * *** * By (a), p(b*(g^,u))= &(p(G),u) holds. Thus we have A M A’. Consequently, A’ is a strongly connected (1,H)-automaton. Therefore, G(A’) x H holds. On the other hand, since G(A) M G(A’), we have IG(A)I = / H I . By (2) and by the strong connectedness of A , we can obtain easily IH1 = nIGI.

Moreover, assume that A = X, 6 q ) is an (n,G)-automaton. Then there X, S q l ) such that A M A'. If, exists an (n,GI)-automaton A' = in addition, A is regular, then A' is also regular. (g, Proof (G, Let @ be an isomorphism of G onto GI. We define @'(a) = (6(gPq(a))) for any a E X where 6 is the extension of @ on Go with @(O) = 0 and @ ( a )= ($Jpq(a)). With the above Q', we define A' = X, Sq,). First, we prove that A M A'. To this end, we put p(g) = ( 6 ( g p ) ) for any 6 = ( g p ) E G,. Then p is a bijective mapping of G, onto By the definition of @' and the fact that @ is an isomorphism of G onto GI, it is obvious that p(&p(g,a)) = G ~ / ( p ( g ) , a )This .

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Algebraic Theory of Automata and Languag by Masami Ito

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