Minimization of microprograms and algorithm schemes
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It is wel l known (see [1]) that the opera t ion of any m i e r o p r o g r a m can be reduced to the scheme of the i n t e r ac t i on of two au tomata , A and B. The f i r s t , ca l led an opera t iona l automaton, i s a f in i te or inf ini te Moore automaton; the second, a control automaton, is a f in i te Mealy automaton. The output s igna ls (xl ,x 2 . . . . . Xm) of A are used as input s igna ls for B, and the output s igna l s (mic roopera t ions Yl,Y2,. ,Yn) of B a re input s igna ls for A. We a s s u m e that the in i t ia l s ta te b 0 of B is f ixed and that the in i t ia l s ta te of A can be v a r i e d wi thin ce r t a i n l im i t s . The in t e rac t ion in opera t ion of the two au tomata m e a n s that even if it is poss ib le to se lec t any state of A as the in i t ia l s ta te it is not , g ene ra l l y speaking, pos s ib le for all a p r i o r i conceivable sequences of input s igna ls to a r r i v e at the input of B. This c l ea r ly c r ea t e s f u r t h e r pos s ib i l i t i e s for m i n i m i z i n g the control automaton B and also for the m i c r o p r o g r a m r e p r e s e n t e d by it. Let us examine one of the m o r e na tu ra l ways of doing th is . We f i r s t identify the output s ignals x i of the Moore automaton A with the se ts of s ta tes m a r k e d by these s igna ls . F o r any s ignal x i and any m i c r o o p e r a t ion yj we denote by xiy j the union of all se t s x k which conta in s ta tes of the f o r m ayj , where a E x i. F o r any set of output s igna l sM = {x,,, xi . . . . . . xik} of A we denote by Myj the union of al l se ts x~,, Yl . . . . . x,~g~. We have thus defined a f ini te au tomaton C without output s igna ls with the same input alphabet as A, and whose s ta tes a re a r b i t r a r y sets of output s ignals of A. We shal l cal l the automaton C so cons t ruc ted the reduc t ion of A. It is uniquely defined by A and when A is f ini te i t can be cons t ruc ted f rom it a lgor i thmica l ly . Having f o r m e d C, we cons t ruc t au tomata D and F without output s igna ls with the same se t s of s ta tes as C. The input s igna ls of D are all poss ib le pa i r s of output and input s igna l s of A (xi,Yj) and the input s ignals of F a re any sets of such pa i r s . F o r any s ta te d of D the product d(xi, yj) is put equal to xiYj if x i E d and to 0 o therwise . The product dN, where N is any se t of p a i r s (the t r a n s i t i o n funct ion in F ) , is defined as the union of the products dq for all p a i r s q in N. Let us now tu rn to a cons ide ra t ion of the control automaton B. F o r any pa i r of s ta tes (bs ,b r) of this automaton we denote by Bs the set of al l p a i r s (xi,Yj) such that the effect of the input s ignal x i i s to make B pass f r o m s ta te b s to b r and at the same t ime give the output s ignal yj. Suppose that each state k (k = 1 , 2 , . . . ,p) of B is m a r k e d by any set M k of output s igna ls x i of A. We define the new m a r k e r se ts M} by the fo rmu la s
[1] V. M. Glushkov,et al. Automata theory and formal microprogram transformations , 1965 .