UNIVERSAL SINGLE TRANSITION TIME ASYNCHRONOUS STATE ASSIGNMENTS

In this paper we consider the problem of deriving upper bounds on the number of state variables required for an n-state universal asynchronous state assignment (i.e., a state assignment which is valid for any n-state asynchronous sequential function). We will consider a special class of state assignments called SST assignments which were first derived by Liu [1] and later extended by Tracey [2]. In these assignments all variables which must change in a given transition are allowed to change simultaneously without critical races. The best universal bound known so far has been developed by Liu and requires 2 so -1 state variables, where S0= [log2n], n being the number of states, and [x] being the least integer >x. We shall show how this bound can be substantially improved. We further show that, by generalizing the state assignment to allow multiple codings for states, the bounds can be still further improved. A mathematical statement of the problem is as follows. Define an (i, j)-separating system (SS) for a finite set S to be a family of subsets S1, S2, … St of S such that for any subsets P, QCS with| P| =i, i,| Q| =j and PnQ=ϕ, there is a set Sk of the family for which either PCSk, QnSk=ϕor QCSk, PnSk= Let m=m(i,j,n)de-note the minimum value that t can assume for an (i, j)-SS when| S| =n. Copyright © 1969 by The Institute of Electrical and Electronics Engineers, Inc.