410 IRE TRANSACTIONS ON ELECTRONIC COMPUTERS June

orrespondence

Comment on "A Magnetostrictive Nonlinear Feedback 4

Delay-Line Shift Register"* Shift Registers*The above mentioned article' is an in- In 1959 Elspas' considered the theory of

teresting tutorial on a common use of linear feedback shift registers (FSR's) in themagnetostrictive delay lines. However, the general case of multivalued p-nary logic (p /implication that the described technique is prime); in particular he justified the tech-both novel and more practical than previous nique of finding the prime factors of themethods certainly warrants clarification. characteristic polynomial of a transition t / t4

Anderson,' of the Marine Physical Lab- matrix T as a method of discovering what ,3oratory, University of California, performed cycle lengths a particular configurationorigiinal research in this field of delayed wotild produce. Elspas' transition matrix istime compressioni (DELTIC3). Also, since of order nXn where n is the numiiber of cQo) Col)1954, Computer Control Conmpany, Inc., stages in the shift register.had been producinig both commnercial and In this coniinuliication binary logic only t /military equipmeints for this applicationi at is considered, and it is shown that by study-considerably higher pulse rates. In fact, we inig a transitioni matrix of order NXNhave recenitly designed and developed a (N-2"l) it is possible to consider both the t, t2DELTIC usinig all ultrasoinic delay line at linear anid mionlinear cases. The factors ofa clock rate of 16 Mc. This particular delay- the characteristic polynomial of the matrixline shift register ulses only 17 tranisistors as are shown to be intimnately coiinnected with \opposed to the 47 used in Hargrave's equip- the cycles which arise. Coo)meint. Attentioni is also called to the article A shift register' of n stages has N-2 twritten by Rosenibloom.4 states which may be written as n vectors.

WILLIAM P. HORTON These N-states may also be represented byComIIputer Control Co., Inc. N-vectors as showni in (1) below t,

Framliolghaml, Mass.State Representation Fig. 1-The digraph G4.

(n-vector) (N-vector)Author's Reply' (o0 0)' [t0 0]' = aei

I am gratefully to Mr. Horton for calliiig (0 0 * 1)' [O 1 0] 8my attention to the early work of Andersonin the field of delayed time compression, and 1). '

LO 1 aN

to the advances made thereon by the Com- aputer Control Company. Mr. Horton's Note: a prime (') denotes transposition.observations are certainly germane and well As is well known,' 34 a further repre-taken. Since I was unaware of this research sentation of these states, and of the possibleiin the field, the similarity in approach and transitions between them, is by a state di-result between the two efforts is indeed agram in which the N states are represented t7 t8uncanny. by N nodes, and a possible transition from t

I must, however, take exception to Mr. one state to another is represented by a di- C o (01Horton's comparison between the two ap- rected branch between the corresponding (IlO) o IOproaches on the basis of transistor count. If two nodes. Such a state diagram is in fact athe DELTIC employing seventeen tran- digraph, and is denoted here by GN; GNsistors is similar to that described in Fig. 4 contains N nodes and 2N branches. G4 andof Rosenbloom's article4, then it should be Gs are shown in Figs. 1 and 2. De Bruijn' (lot)pointed out that this count includes no has shown how to generate G2N from GN byprovision for the binary countdown and as- a simple process which he calls "doubling" t,6 t3 N/t4sociated circuitry to perform the weight the graph. The digraphs GN have also been 6integral. If these functions were also to be considered by Gill.'omitted from my approach, the resulting A particular FSR of n stages and Co lo)circuitry would contain but thirteen transis- N(=-2n) states will correspond to some sub- tstors. LEE HARGRAVE, graph of GN containing all N nodes, but t3

Sanders Associates, Inc. / \Nashua, N. H. *Received October 17, 1961; revised manuscript (100) (001)

received, February 5, 1962. 2' B. Elspas, "The theory of autonomous linear

sequential netwrorks," IRE TRANS. ON CIRCUIT t \g* Received February 8, 1962. THEORY, vol. CT-6, pp. 45-60; March, 1959.I\ 21 L. E. Hargrave, Jr., TRANS. ON ELECTRONIC 2 We find it convenient to adopt the convention\ /

COMPUTERS, vol. EC-10, pp. 702-708; December, that the register shifts from left to right; this is the \ (ooo)^i96i. opposite of that used by Elspas. __° O

' V. C. Anderson, "The DELTIC Correlator," 3'N. G. De Bruijn, "A combinatorial problem,"Acoustic Res. Lab., Harvard University, Cambridge, Proc. Koninki. Ned. Akad. Wetenschap., vol. 49, pp.Mass., Tech. Memo. No. 37; i956. 758-764; September, 1946. (Also, Indagationes Mathe-

2 This technique was first described in the Marine maticae, vol. 5, p. 461; Auguit, 1946.)Physical Laboratory quarterly report for January 1 4 F. E. Hobo, S. Seshu, and D. D. Aufenkamp,to March 31, 1954. "The theory of nets," IRE TRANS. ON ELECTRONIC

4 M. Rosenbloom, "Using time-compression tech- COMPUTERS, vol. EC-6, pp. 154-161; September, 1957. {nlques and digital correlation," Electronics, vol. 34, ' A. Gill, "Cbaracterizing experiments for finite tpp. 191-193; March 10, 1961. memory automata," IRE TRANS. ON ELECTRONIC

5Received March 5, 1962. COMPUTTERS, vol. EC-9, pp. 469-471; December, 1960. Fig. 2-The digraphl G,.

1962 Correspondence 411

only N of the 2N branches: one and only exactly one "one" in every column as well Thus a nonsinigular machine is completelyone branch will leave each and every node. as every row. Clearly it is singular when specified by N/2 binary digits and so there

Let us denote the NXN connection there is a column of zeros. exist 2NI/2 nonsingular machines. If we num-matrix4'5 of GN by CN; then CN is of the form Theorem 3: When T is nonsingular the ber all machines in terms of their binary

N N corresponding digraph is a set of non- representation X treated as a binary number,1 2 - -+1 .. N touching cycles. say S, the first nonsingular machine occuLrs

2 2 Proof: Since there is one "one" per row when T=(01 01 01, * * 01), i.e., when

-1 1 - 1in T it must be possible to leave every state 2N - 1by one path only. And since there is one S = --- = Si say. (7)

1 1 "2one" per column it must be possible to 3

1 1 3 (2) enter every state from only one other state.Nonsingular machines thenoccur forCN= 1 1 4 (2) Hence in the corresponding digraph each

node has precisely one branch entering and S2 = S1 + 1one leaving and so the digraph consists of a

j N-1 set of nontouching cycles. S3 = Si + 4

_ 1 1 1N Theorem 4: When T is nonsingular and S4 = Si + 4 + 1the corresponding cycles are of length S5 = Si + 16 . (8)

where Cii=1 for i=2p and 2p-1 Pl, P2, * ' *, etc. ( 1pi =N) then S6 = Si + 16 + 1

N det (T - XI) = (Xpl - 1) (XP2 - 1) . . . (4)= p and -+ p

2 Proof: Since the cycles are nontouching, S2N/- Si + Si = 2Si/ N \ the nodes of the digraph may be renum-

p = 1, 2, ,- bered so that the corresponding matrix T is It is of interest to notice that a machine2 in block diagonal form. The ith block Ti specified by a particular r = (tN, tN-1, * * ,

Cii = 0 otherwise. corresponding to the cycle of length pi, is of ti) has the same digraph structure as thatorder pi Xpi and of the form, specified by r'-(t7t1* . tN), i.e., by a re-

The nonzero elements Cii indicate the ' numbering of the nodes the two graphs canexistence of a directed branch from node i . 1] be made identical. This is obvious when theto node j in GN. The product CN'ai [ai 1 two matrices are written out.defined by (1)] gives rise to an N vector 1 A complete solution to the synthesiswhich is the transpose of the ith row of CN; 1 problem can be obtained by using the char-the two "ones" in it correspond to the two T = 1 acteristic polynomial of TGen. The coeffi-nodes which can be reached from node i in cients of this polynomial will be sums ofGN. A particular FSR will be represented by . products of the tI and ii, the coefficient ofa subgraph of GN whose connection matrix XN-r being the sum of all rth order principalT is obtained from CN by replacing exactly _ j minors of TGen. Unfortunately we do notone "one" in each row of CN by zero. Thus know the general form for these coefficients;T will contain exactly one "one" in each The characteristic polyniomial of T1 is the technique of investigating all principalrow, but may have zero columns. It then (-1)Pi(XPi- 1) and so that of T is I1i(XPi- 1). minors of TGen is unsatisfactory except forfollows that T'ai, being the transpose of the Theorem 5: When T is singular the cor- small N. Theorem 7 below gives a method ofith row of T, will contain precisely one responiding digraph is a set of self-starting ohtaining the coefficients by inspection of"one," and will represent the unique node cycles, the digraph, but again this is unsuitablereached from node i in that particular Proof: By Theorem 2, T contains at except for small N. The coefficients forFSR. The matrix T' is thus a transition least one zero column and so the node cor- n=3 (N=8) were obtained by this secondmatrix for the FSR. Notice that the general responding to that column cannot be con- method, and they are tabulated in Table I.form for these transition matrices T is tained in a cycle. Hence from Theorem 1given by the following (see Figs. 1 and 2): the digraph must be as described. TABLE I

Theorem 6: If T is sinigular and describes COEFFICIENTS OF XV IN THE EXPANSION OF1 1 cycles of length pi, P2, ( i <N) DET (TGen -XI) FOR N =8

t2 t2 thenl X8

t3 t3 det (T - xI) = X(N-P1i)(P- 1)(XP2 - 1) . X7 -(11+78)TGn = t4 14 (3) XI6 t/8 -136

'Oen (3) Proof: By a suitable renumbering of the X5 117216+73t678-7476t7-72t315. . n~~~~~~~~iodes of the digraph, the Mpi columns Of XJ_tltt3tt6tt8+2t]74tt66t77+tzlt2tt3Zt5+746t778+72t3t5t8

tN-1 tN_1 T corresponding to nodes contained in -11121315 -12141517 -14161718to~~~~~~~~~~~X -151416718 -t112t3578 ±1l72t41517 +tl141617718L tN tN I cycles may be put into block triangular form. X +7tt2t3t578 +t2t41st7t8 -7t2t451t7 -72t4757718

The blocks on the diagonal denoted by T1 X2f -t172t475t778+tl72147577t8+71t2t475t778+7273t4T5t6t7where 1i =I -ti and t =1 if and only if the correspond to the cycles as in Theorem 4 +12t3ht46617 -/511415/7t8 -72t314151617 -It24ltI576t7

ft272137475t67 +-1st273t4ts7ts7+7212775t,4718-state following state i has a zero at the most above, but some of the columns may contain I+2t71t4456t778-t1t273t47567 -t721 3t415t6t718significant (left-hand) end. GN and T have "ones" below these blocks. The remaining XI -t1t2t3t4t5t6t7 -t2t314t516t7t8 +T1t273t415t6t7

+121314151677t8 +7s1t2t74t516t7 +121314t5161718the following properties: columns of T may be put into a form in -7t2t1273Ws6t7t -712t3747s16t7 -72t374751617t8Theorem 1: A physically realizable which there are no "ones" on or above the -1273h57s77t6.

FSR must have a digraph which contains at main diagonal. The form of the characteris- XO (t112 -tst) (t314 -134) (Is6 s516) (t718 -718)least one cycle. tic polynomial now follows.

Proof: There are N nodes and N branches From this table we may design an FSRand therefore at least one loop exists; any A particular FSR is completely specified of 3 stages which contains for example a self-loops must be directed cycles because there by the binary N-tuple T= (IN, iN- ,is exactly one branch leaving every noDde. 1), for thiS specifies T from (3). Thus there starting cycle of length five as follows:Furthermore every node lies either on a exist 2N physically realizable machines. The The characteristic polynomial must becycle or on a path directed into one. This determinant of T is seen from(:3) to be (X5-1 )X3. Because the coefficients of X7, X6,seonsiutin whe not al ndsaeo deT=(tt-1/i(I/-//)5, X4 are all zero the following equations are

secod s1uatln, wenot al noes ae o detT = tlt2t2t)(t34-t43) ** 'obtained:cycles, may be graphically described as a (Nt NNi.()~/= 8set of self-starting cycles. (ylytt-) 5 1=Ot

Theorem 2: T is singular if and only if Hence for T to be nonsingular we must have X8 /3/6 = 0there is a column of zeros in T.

Proof: When there are no zero columns T /2* =X /25 = .. 6 t4t6t7 = 0 /2t3t5 = 0roust be nonsingular because then there is 2/2_=1,* ,>y 6 X' 1-51212/ = 0 13/473/7 = 0 /4/6/1/5 = 0.

412 IRE TRANSACTIONS ON ELECTRONIC COMPUTERS June

The fact that there are a number of ways of TABLE II the output, direction, and next state forsatisfying these equations only indicates NUMBER OF CYCLES OF LENGTH k IN GN each state-symbol pair. The Turing ma-that there are several ways of making such

kchines presented here move in only one di-

a counter. The final equations obtained bky 1 2 3 4 5 6 7 8 rection, reading the input bits in sequence.making the coefficient of X3 equal minus one n(N) If in addition, the present output is de-are 2(4) 2 1 2 1 pendent on the present state only, then this

3(8) 2 1 2 3 2 3 4 2 Turing machine is a finite automaton. Wet1t2t4t5t7 or t2t4t5t7t8= 1. 4(16) 2 1 2 3 6 7 8 12

5(32) 2 1 2 3 6 9 12 20 will construct an a-state Turing machineOne way of satisfying this last equation is - -- which reads the bits of N and writes thet1=0, t2=1, t4=1, t5=0, ts= 1. These values bits of aN. For a even, this Turing machineof the ti satisfy the equations for X4, X.5 The has shown that when k =2n= N, the term is is a finite automaton. For a odd, the presentadditional values t3= 1, t6=0 t8= 1 satisfy 2N I2/N (i.e., the number of Hamiltonian output is dependent on both the presentthe remaining equations. cycles in GN is 2N/2/N). From this result it is state and the present input symbol. For

easy to see by a consideration of the digraphs this case, Lee3 has shown how to construct.-. T =(11 0 0 1 1 1 0). that when k = N -1, the term is 2N I2+1/N. a finite automaton with 2a +1 states, which

The logical equation for the input is obtained Elspas' has proved the following theorem: is equivalent to our a-state Turing machine.by considering #, where r=(0o0 1 1 0 0 0 1). Theorem 8: For k <n, the number of Thus a-state Turing machine can be inter-Lettering the binary variables in the shift cycles of length k in GN is less than or equal preted in terms of a single-input, single-register A, B, C, from left to right, the log- to /A and, for k = n, equals Ak, given by output sequential machine.ical equation for the input is Z kllk = 2n Multiplying N by a is the same as adding

ABC + ABC + ABC kin N to itself a times. Consider this addition(summed over all k which divide n) where as is the ith digit of N and si is the sum

using the standard symbolism of Boolean of the ith digit column plus the carry fromAlgebra. This is readily simplified to We have h=e2, /2e=1,93=2,/s 4 =3, etc. the previous column. The sum si is used toAB-+BC. Finally, Theorem 9 is also relevant. obtain the output bi and the new carry for

The following theorem is of interest as it Theorem 9: The number of cycles of the next digit column.shows a direct connection between the co- length k in GN is less than or equal to the The states of the Turing machine areefficients and the cycles of the digraph. number of cycles of length k in G2N. denoted by qj where j equals the carry to

Theorem 7: A product of r of the terms Proof: Consider one cycle of length k in the present digit column and ranges from 0ti, 7i will be contained in the coefficient of the digraph GN. Apply the doubling tech- to a-1. For the state qj and input as, denoteXN-r in the characteristic polynomial of nique used by de Bruijn in generating the output as bj a. and the new carry, indi-TGen if and only if the corresponding branches G2N from GN to this cycle alone. Clearly a cating the next state, as Cj,a%. The state-of GN form a set of nontouching cycles. distinct cycle of length k will occur in G2N symbol table will be completely described

Proof: Necessity-The coefficient of for every cycle of that length in GN. when Cia. and bi,ai is specified with ai =0(-)Nf-r is the sum of all rth order principal Theorems 8 and 9 together show that and with a5 =1 for all states qj; 0<j <a -1.minors, and so any product of r of the t,, t, each column of Table II (extended) will be The form of the state-symbol table whichoccurring in this coefficient must arise from a nondecreasing column as n increases, characterizes this Turing machine is shownone such minor and each element of the reaching and holding the value ik for n>k- in Fig. 1. Thus the states of the Turing ma-product must lie on a different row and a In fact it may be shown that the value kk iS chine contain the memory of the system anddifferent column of this minor. Hence by an reachedwhenn=k-1. together with the input determine theargument similar to that used in the proof P. R. BRYANT operation of the machine.of Theorem 3 the product must define a set R. D. KILLICKSymbolof nontouching cycles in the subgraph of r The General Electric Co. Ltd. Symbolnodes defined by this rth order principal Telecommun. Research Labs. State 0submatrix, and so also in GAT. Hirst Research Centre _ _

Sufficiency-A set of r branches of GN Wembley, England qo bo,o Co,o bo,i c0,1forming a set of nontouching cycles will be q bi, o ci,o b,1 c1incident on a set of r nodes. The correspond- q2 b2,0 C2,0 b2,1 c

ing rth order principal minor of TGen will 2contain the product of the corresponding ts,ti and so this product will appear in the co- qj bj,o Cj, bj,i Cj,efficient of XN-r

It is of interest to know the number of An Alpha-State Finite Automatondifferent cycles of a given length k in GN for Multiplication by Alpha- ba0 a b C

since this is the same as the number of ways Fig. I-State-symbol table.of counting up to k with an n-stage shift The problem considered here is to con-register. Table II was obtained by direct con- struct a finite automaton which, when pre- The necessary relations are stated assideration of the digraphs. We know of no sented with an arbitrary number N, will follows:general formula which gives the term in the produce as the output aN where a is a given 1) si = aai Aj(n, k)th position; however Radchenko5 fixed integer. The input number N is as-claims to have proved that for k < 2n no term sumed to be represented in binary form 2 c Si wis zero; in a recent publication7 an independ- with the least significant digit ao presented L2 went proof of this has been given, which in- first. The output will also be a binary num- [x] = greatest integer part of xcludes a design procedure. It has been pointed ber which replaces the input number.

if= is even

out to the authors by a referee that a re- N = aaa2 a aN = bbb2 ... b. 3) bj,i 0 if s is evenport by Golomb, Welch, and Goldstein.8 1 if si is odd.which they have not seen, also contains a Recall that a Turing machine' 2 is de- T c tconstructive proof. Furthermore, de Bruijn3 scrihed by a state-symbol table indicating Ther re aythe iitil fstaeolf thlemnTurn

6 A. N. Radcbenko and v. I. Filippov, "Sbift machine is qs. For this state and input a5 =0,registers witb logical feedback and tbeir use as count- * Received May 3, 1961; revised manuscript re-log and coding devices," Automation and Remote Con- ceived April 4, 1962. Tbe result reported in tbis corre- the output bo, iS 0 and the new carry c5,5,tlrol, vol. 20, pp. 1467-1473; November, 1959. spondence was obtained wbile tbe autbor was at Co- indicating the next state, is 0. Now to form

7P. R. Bryant, F. G. Heatb, and R. D. Killick, lumbia University, New York, N. Y., and M.I.T.'"Counting witb feedback sbift registers by means of a Lincoln Lab., Lexington, Mass. M.I.T. Lincoln Lab, the rest of the state-symbol table, considerjump tecbnique," IRE TRANS. ON ELECTRONIC COM- is operated witb support from tbe U. S. Army, Navy successive states qj and qj+s. The output andPUTERS, vol. 11, pp. 285-286 (Correspondence); and Air Force.April, 1962. 1 T. Bartee, I. Lebow, and I. Reed, "Tbeory and new carry are related as shown in Table I.

8 5, W. Golomb, L. R. Welcb, and R. M. Gold- Design of Digital Systems," McGraw-Hill Book Co.,stein, "Cycles from Non-linear Sbift Registers," Jet, Inc., New York, N. V., cb. 11; 1962. 3 C. Y. Lee, "Automata and finite automata,"Propolsion Lab., California Inst. Tecb., Pasadena 2S. Kleene, "Introduction to Metomatbematics," Bell. .Sys. Tech. J., vol. 39, pp.l1267-1295;,September,Ping. Rept. No. 20-389; August 31, 1959. D. Van Nostrand Co., Inc., New York, N. V.; 1952. 1960.