On the linearity of sequential machines

This paper presents a method for determining, from ments in which the correspondence between states and n the flow table of a sequential machine, whether the machine is tuples of secondary variables is one-to-one. linearly realizable using the minimum number of unit delay elements. With regard to the input, two cases are considered: In The method is an extension of a previously presented method for autonomous machines. A linearity algorithm is presented for each the first case, the input r tuple for each input is given. In of two cases: 1) where the output is given and is to be linear, and the second case, the inputs are not specified, except that 2) where the output is either not given or may be nonlinear. The p different inputs are required; hence, an assignment of method is simple, straightforward, and, in general, provides a ready r = [log2 p] input variables will have to be made. In the solution to the linearity problem. following development, no distinction between the two INTRODUCTION cases is necessary; however, both are illustrated for the INEAR SEQUENTIAL machines [1], [2] are a sake of added clarity in the examples. L_ INer mh ines ] [2].ar a The problem of linearity [3], [9] resolves itself to one proper [12] subset of the set of all sequential maof fidn nasgmn o h nenlsae n,i chns an poss seea.sflpoete,sc of finding an assignment for the internal states and, in chines, d possess several useful r p r i s, such as'canonicalforms 3 r4 and transfe r func s 3 some cases, the input states, such that the next state as canonical forms L3 4] transfer functions l3. 4]which arenotenjoye] [ on [ .'.s. IT and output functions are linear. Linear sequential ma4] wh no enoe by nliermcns. Itchines are adequately presented in the references [i]has been suggested [5], [6] that by studying linear [4], and it is assumed that the reader is familiar with machines, we may be able to apply some of their more their basic properties. useful properties to sequential machines in general. A P P major stumbling block exists, unfortunately, which impedes any significant progress in this direction: At the present time, no general method exists, by which it can be determined whether a sequential machine given by its 1, 2 *. p flow table is linear or not. Several approaches [6][8], [13] to thisproblem have been made, but the results produced [6], [7] are applicable only to a limited class SPse1/Z11 SSt/Zat e.. of sequential machines and, furthermore, do not appear S2 S21/Z21 S22/Z22 .. 2p/Z2p to be extendable to the general case. This report extends one previously presented method [8] to include the nonautonomous (or input) case. Alsq Sql/zql Sq2/Zq2 '' Sqp/Zqp though the class of machines considered is still restricted, it is more general than most classes previously Next State/Output considered [6]-[8] and has no apparent restrictions for Fig. 1. Flow table of a sequential machine. its extension to the general case. We consider fixed, finite, deterministic, and synchronous sequential machines [10] whose flow tables are reduced and in the form of Fig. 1. We restrict ourselves to finding minimal Because we are restricted to binary realizations, it is binary realizations for completely specified sequential natural that we consider only those binary partitions machines, where a minimal realization of a machine that can be used in a valid minimal assignment. having q states is a realization using n= [log2 q] unit Definition 1: A qualified binary partition (qbp) over a delays and [N] represents the smallest integer>N. set S, containing q elements, is a partition having two Furthermore, we concern ourselves with those assignblocks, such that neither block contains more than 2n-i elements. Manuscript received November 24, 1964; revised May 11, 1965, We will use the following notation for operations on August 30, 1965, and October 26, 1965. The work presented in this sets: Given two sets A and B, their union is denoted by paper was supported in part by a scholarship from the Defence Research B3oard and the National Research Council of Canada under (AVB), their intersection by (AB), their sum (modulo Grant A-1617. The paper was presented at the Fifth Annulal Sym2) by (A +B), and the complement of the set A with posium on Switching Circuit Theory and Logical Design, Princeton, repc oteuieslse,b .Teepysti N. J., 1964. ' 'rsett h nvra e,b .Teepysti W. A. Davis is with the Defence Research Telecommunications denoted by 4.> Establishment, Shirley Bay, Ottawa, Ontario, Canada, and the UniDeiiin2 b'P,F,..,P hr versity of Ottawa, Ontario, Canada.Deiiin2 m qb' ,P,**, mwhr J. A. Brzozowski is with the Department of Electrical Engineerp =fb.;}3 over the set S containing q states, are said ing, University of California, Berkeley, Calif., on leave from the $ t University of Ottawa. 'to be mutually consistent if, and only if,