Download e-book for kindle: Algebraic Theory of Automata and Languag by Masami Ito

By Masami Ito

ISBN-10: 9810247273

ISBN-13: 9789810247270

Even supposing there are a few books facing algebraic idea of automata, their contents consist frequently of Krohn–Rhodes idea and comparable issues. the subjects within the current publication are particularly diversified. for instance, automorphism teams of automata and the partly ordered units of automata are systematically mentioned. in addition, a few operations on languages and specified sessions of normal languages linked to deterministic and nondeterministic directable automata are handled. The publication is self-contained and for that reason doesn't require any wisdom of automata and formal languages.

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Example text

The following theorem plays a fundamental role in this section. 7. 1 Let A = X , 6,) be a regular (n,G)-automaton and E be a homomorphism of G onto some group. e. Ker(t) = J - l ( ( ( e ) )(e is the identity of G). 1 In the above, we identify G ( A )with G. 1). This assumption will be taken throughout this section. 1 X , G). Then for f^ the following: (z, - Put H = K e r ( J ) & G and A / H = = (fp) E G,, k = ( k p ) E G^, we have h - - f^ = holds if and only if there exists an element E h E H such that f = h($) = p h ( i ) = ( h k p ) .

We will obtain a representation of A by a group-matrix type automaton of order ISl/lG(A)I on G(A). e. P(P> = 4 , p ( q ) = P, p ( t ) = r and p(r) = t. ISI/IG(A)( = 2, x = { a , b , c ) , s = { P , q , r , t ) , s1 = b , q ) , s2 = { t , r ) , T = { p , t ) , J(p,a) = q = p ( p ) , J ( t , a ) = t = e ( t ) , +(a) = t = e(t),b(t,b) = p = e(p), Q(b) = 6(t,c)= r = p ( t ) , ~ ( c=) Then A' = ( G @ ) 2 , X , 6 q ) M A. 22 CHAPTER 1. 4 Equivalence of regular systems In this section, we will deal with the problem as noted in the previous page.

2 Let G be a finite group and let n be a positive integer. T h e n [ I ( G ) / n ] J ( n , G ) 5 [ I ( G ) / n l + p ( n ) where p(1) = 0 and p ( n ) = 1 f o r n 2 2. < Proof Obviously, the theorem holds true for the case n = 1. Therefore, we consider the case n >_ 2. 1. Hence we have to prove the inequality J ( n ,G ) 5 [ I ( G ) / n l P(4. e. [HI = G such that H = {hi I hi E G , i = 1 , 2 , . . , I ( G ) } . Now, put X = Y U { z } where Y = {yi I i = 1 , 2 , .. Moreover, for any i = 1 , 2 , .

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