By Ferenc Gécseg

Both theoretical and useful concerns inspire the repre sentation of gadgets as yes compositions of less complicated ones. within the thought of automata this remark has ended in the techniques of professional ducts and entire structures of automata. within the basic type of the goods of automata all of the part automata are fed again to each other. With this very large suggestion of goods, the conclusion of automata with huge numbers of states via compositions of uncomplicated elements is a hugely concerned approach; this raises the opportunity of blunders. so that it will reduce 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 may perhaps rely on the states of all automata previous it, i. e. , all part automata steer all these automata which keep on with them within the product. moreover, at such a lot the subsequent i-I automata (including itself) should be fed again to the enter of a given part automaton. hence for iXcproducts the lengths of feedbacks are at so much i. the purpose of this monograph is to provide a scientific account of iXi-Products. It involves 5 chapters, a reference part, and an index. the 1st bankruptcy includes the required techniques and effects from common algebra, automata, and sequential machines.

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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 ... , bm }. Denote by Ill' = ({x, y}, A', b') the subautomaton of the direct power 'lin generated by the element (al' ... )) = { 0 if {e p ... , eJ = {a p n I otherwise cl''''' C ... , aJ, (Cel' ... , en) E A') is obviously a homomorphism of Ill' onto Ill i . Next, let (£ = ({x}, [n], b(f) be a counter with n E N.

Be a homomorphism of W onto ~. Take a cycle b p ... , bm in ~ together with the input signals xl' ... ,xm E X, for which bf(b i, x) = bi+l(m~dm) (i E [m]). Set p = XI'" x m ' and let a E A be an arbitrary counter image of b l under I{!. Obviously, there are integers k and I with ~ k < I satisfying b(a, pk) = b(a, i). Consequently, we have a cycle a p ... (a) = bi(m~dmr Thus, min. ff', it is enough to show that if a sequence ao' ... , an (a o = an) is a cycle in an automaton ° 9i = (X, ,4, b) = n m Wi[X, i= 1 cp] (Wi = (Xi' Ai, b;) E %, i = 1, ...

In)) = j if and only if i1 = ... = in = j = 0 or j ~ 1 and there exists atE [n] such that i1 = ... = it = 1, it+1 = ... = in = 0 and i - t == j (mod n), where (i, ip ... , ill) E B' andj E [n]. It is obvious that IjJ is a homomorphism of~' onto Ill.. Thus, we have that Ill n E HSP~oHSP~oP 1~1 (%) ~ HSP ~1 (%). Again, let n EN be arbitrary, and define the automaton ~n = ({xo' ... , XII}' {O, ... lln(i, x) = i + 1 (mod n) (1 ~ i ~ 11, 0 ~ j ~ 11). Fig. 15 shows ~ .. We shall prove that ~II E HSP~l (%).