Products of Automata by Ferenc Gécseg

By Ferenc Gécseg

Both theoretical and sensible concerns encourage the repre­ sentation of items as sure compositions of easier ones. within the idea of automata this commentary has resulted in the recommendations of professional­ ducts and entire platforms of automata. within the common type of the goods of automata the entire part automata are fed again to each other. With this very vast inspiration of goods, the conclusion of automata with huge numbers of states through compositions of uncomplicated elements is a hugely concerned strategy; this raises the potential of mistakes. in an effort to lessen the complexity of feedbacks, a hierarchy of goods referred to as lXi-pro­ ducts was once brought a few 10 years in the past, the place i runs over the set of all non-negative integers. In an IXcproduct the index set of the part automata is linearly ordered. The enter of every automaton within the product might rely on the states of all automata previous it, i. e. , all part automata steer all these automata which persist with them within the product. additionally, at so much the following i-I automata (including itself) could be fed again to the enter of a given part automaton. hence for iXcproducts the lengths of feedbacks are at such a lot i. the purpose of this monograph is to offer a scientific account of iXi-Products. It includes 5 chapters, a reference part, and an index. the 1st bankruptcy comprises the required ideas and effects from common algebra, automata, and sequential machines.

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Under any input signal x, (l = 1, 2), the first component will be set to i + 1 (mod k), and the contents of the 2nd , ... ,(l + mk)th components are shifted one place to the left, and the value of the 2nd component underflows. Moreover, the (l + mk)th component will be filled up with vmk+I' where Vmk+1 is determined according to the three cases below: Case 1. i ¥- k. Then, vmk+1 = Ui+1 or Vmk +1 = u:+1' depending on whether (V mk - i+l''''' vmk) = (ul''''' u) or (V mk - i+l' ... , vmk) = (u~, ...

We. show that Wn E HSPat (%). It is obvious that for every n E N, HSPJP I a1 (%) c:; HSPat (%) contains all counters with n states. 5 Comparison of the Homomorphic Representation Powers 51 //--~ '3 , \ : x,y I I n/ 2 x,y x,y y UOx ()xo Fig. 14. Automaton Ill" Fig. IS. Automaton \B n too. By our assumptions, the following properties: (i) A. = {a p ... , b) with I an' bl' ... , bm } (n, m ~ I). (ii) ~(ai' x) = ai+l(m~dnr _ (iii) b(al' y) = b l and,Jor every i E {2, ... , n}, b(ai' y) = a i + 1 (m~dnr (iv) For each i E [m], b(b p x), b(b i , y) E {b p ...

Finally, we give the. formal definition of the homomorphism 1/1 0[(£ onto m. ,. , Vmk) E C (Xl' ... , Xmk E X) be arbitrary. Then there are uniquely determined integers r E [k] and t E [m] fulfilling i == r (mod k) and tk + i - r ~ mk, such that exactly one of the conditions (1) and (2) holds, depending on whether i ~ k or i < k. Put I/I(b) = l5(a(,k+i-r)/k' Xmk - i+2 ... x mk). Corresponding to the above three cases, one can easily verify that 1/1 is a homomorphism. On the other hand, 1/1 is obviously onto.

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