Synchronization of Finite Automata

Methods allowing the design of a synchronized sequential machine, working in the fundamental mode, and realizing the behaviour of any given finite automaton are studied from a theoretical point of view; the transformations of the initial flow table lead in a natural way to the definition and to the use of master-slave flip-flops. 1. Introduetion The aim of this paper is to give a theoretical background to the following problem: given a finite automaton .1, how is it possible to simulate its state behaviour by a practical device using e.g. commercial flip-flops. Such a practical device is called hereafter a sequential machine. The present study is restricted to synchronized machines: i.e. machines that must be able to distinguish two consecutive occurrences of the same input letter (1. We do not pay here any attention to the class of asynchronous machines, the input dictionary of which is restricted to words without repetition of letter. The difference between (1 and (1(1 is done by introducing one or several auxiliary input letters which correspond to the typographical blank space; these new input letters are called: synchronizing signals. The problem is studied on the semi-automaton A associated with .1 since no difficulty arises in defining, a posteriori, the output function of the sequential machine. The main purpose of this work is to introduce a transformation of an arbitrary flow table A to the normal flow table of a synchronized machine A *. One of the most interesting features of that transformation is that any state assignment of A leads to a valid STT assignment of A *. A general model of flip-flop is proposed, and it is seen how to use flip-flops in view to design synchronized sequential machines. Preference will be given to methods leading to sequential machines working in the fundamental mode rather than to sequential machines working in the pulse mode. Some familiarity with classical switching and automata theory is assumed. The notations are those of Ginzburg 1). For the sake of simplicity, it will be assumed that zî is a complete automaton; in fact, all the results can be extended to the case of incomplete automata. SYNCHRONIZATION OF FINITE AUTOMATA 127 2. Transformation of the semi-automaton A (1) The semi-automaton A = (S, .E, M,,) is first transformed into the semiautomaton A' = (S, .E', M'",) defined as follows: , .E' = .E U {El' E2' ... , Ek}' 1 ~ k <» = # E; M'E} = f::..., \;/ j = 1 to k; M'" = M", \;/ a E.E; in other words, the flow table defining A' can be deduced from the flow table defining A by adding to it k columns with headings El' E2' ... , Ek, in which all entries correspond to stable states. The synchronizing signals are the members of {El' E2' ... , Ed; the information signals are the letters of .E. (2) The semi-automaton A' is next transformed to yield the semi-automaton A* = (SxS, .E', M*",) defined as follows: (SI' S2) M*" = (SI' SI M',,) = (SI' SI M,,), \;/ a E.E £; E'; (S1>S2) M*E} = (S2' S2), \;/j = 1 to k. The behaviour of A* can be déscribed by the following formal relations: SI+ = sea') S2U s(a') SI; S2+ = s(a') S2U s(a') SI M'",; (I) with (Sl+' S2+) = (SI' S2) M*",; sea') = 1 iff a' E {El' E2' ... ,Ed; sea') = 0 iff a' E.E. Example. Figures 2 and 3 describe the semi-automata A' and A* deduced from the semi-automaton A of fig. 1; k has arbitrarily been chosen equal to two. Proposition 2.1. The flow table defining A * is normal. Proof This is a straightforward consequence of the definition: (SI' S2) M*E} = (S2' S2) and (S2' S2) M*E} = (S2' S2), . \;/j = 1 to k; . (SI' S2) M*" = (SI' SI M,,) and (Sb SI M,,) M*" = (SI' SI M,,), \;/a E.E. Proposition 2.2. The total state {(SI' S2), a'} of A* is stable iff S2 = SI M'",. Proof (1) Assume that {(SI' 82), a'} is stable; then (SI' S2) M*", = (SI' S2); (a) s(a') = 1=> (SI' S2) M*", = (S2,S2) = (Sl,S2) =>S2= 81= SIM'",; (b) sea') = 0 => (SI' s2)M*", = (SI' SI M'",) = (Sl,S2)~S2 =SlM'",. (2) Conversely, assume' that S2= SI M'",; a similar proof yields (SI' S2) M*", = (SI' S2), i.e. {(SI' S2), a'} is stable.