IEEE TRANSACTIONS ON COMPUTERS, FEBRUARY 1970

Define M'l(a)I(X, Y, S, { M'(yI x) }) by letting

m«i(yI X) ai(yI x)mi(yI x) + (1 - ai(yI x))ni(yI x)

where

ai(yI x) [0, 1], i= 1,2,. * c (2)

for each pair (x, y)EH(X, Y). Post-multiplying (2) by Hand using (1), we get

mi'(y x)H = oi(y | x)g;(y x) + (1 - ai(y x))9i(y I x)= gi(y x) i = 1, 2, . . .*, c

where

gi(y I x) _ ith row of the matrix G(y | x).

Consequently, M'(y x)H= G(y| x) for every pair (x, y)EE(X, Y), and thus M'(a)EEG,(MO) for all ai(ylx)E[o, 1], i=l, 2, * , c, and the assertion is thusproved. Q.E.D.

REFERENCES[1] J. W. Carlyle, "Reduced forms for stochastic sequential ma-

chines," J. Math. Anal. Appi., vol. 7, pp. 167-175, 1963.[21 G. C. Bacon, "Minimal-state stochastic finite state systems,"

IEEE Trans. Circuit Theory (Correspondence), vol. CT-11, pp.307-308, June 1964.

[31 S. Even, "Comments on the minimization of stochastic ma-chines," IEEE Trans. Electronic Computers, vol. EC-14, pp. 634-637, August 1965.

[41 J. W. Carlyle, "State-calculable stochastic sequential machines,equivalences, and events," 1965 IEEE Conf. Rec. on SwitchingTheory and Logical Design, pp. 258-263.

Solution to Harrison's ProblemK. K. NAMBIAR

Abstract-The complete solution to a number theoretic problemwhich has applications in relay contact networks is given. As a corol-lary, a conjecture by Harrison is shown to be true.

Index Terms-Number theoretic problem, relay contact net-works, uniform loading.

While commenting on a letter by Fielder [1] onShannon's almost uniform distribution, Harrison [2] hasshown that the only solution to the congruence

2= 3 mod (n - 1)

is n = 2. After solving this congruence, Harrison statesthat "a problem of more difficulty is the following: anecessary and sufficient condition for uniform loadingin relay contact networks of n variables is that n be asolution of the following congruence:

2n 2 mod (n -1)." 1

Harrison gives some partial results and a conjecture. Inthis note we show that his conjecture is true and givethe complete solution to (1). Since (1) is trivially truefor n=2, we will exclude this solution from furtherconsideration.

Manuscript received April 29, 1969.The author is with the Department of Electrical Engineering,

Drexel Institute of Technology, Philadelphia, Pa. 19104.

Case A, when n is even: We have to solve the congru-ence

2n" 2 mod (n - 1)

or equivalently

2n-=1 mod (n - 1) (2)

since n -1 is odd. Let p5alp2a2 . . .pPra be the prime powerdecomposition of n -1. Note that no pi can be equal to2 since n -1 is odd. As a consequence of Euler's theorem[3 ], we can state that the solution to (2) is given by allthose n for which the ratio

n-i

4(n - 1)is an integer. Here 4(k) represents the well-known Eulerfunction defined as the number of numbers of thesequence

which are relatively prime to k. Since [3],

n-I PIP2.. *Pr

o(n -1) (pl -1)(p2 -1) ... (pr -1)(3)

we have to find those sets of pi which make the right-hand side of (3) an integer. But clearly, it is impossibleto find such a set, since the numerator is odd and thedenominator is even. Hence, we conclude that an evenn cannot be a solution of (1).

Case B, when n is odd: Putting n=2m+1, we haveto solve the congruence

22m+l- 2 mod 2m

or equivalently22m = 1 mod m. (4)

It is easy to see that m cannot be even, since the oddnumber (22m - 1) cannot be divided without a remainderby an even m. Let qlblq22 . . . q,be be the prime powerdecomposition of m. Note that no qi can be equal to 2,since we have already established that m cannot beeven. The solution to (4) is given by all those m forwhich the ratio

2m 2qlq2 q5

¢0(m) (q, -1)(q2 -1) . . .(q - 1)(5)

is an integer. The only way to make the right-hand sideof (5) an integer is to have s = 0 or s = 1 and qi = 3. Hencethe general solution of (4) is

m = 3k, k = 0,1, 2

Theorem: General solution to the congruence

2" _ 2 mod (n - 1)

is

n=1 + 2.3, k = 0,1, 2,**.

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SHORT NOTES

Harrison has conjectured [2 ]that (1) implies

n 1 mod 6.

It is clear from the general solution that this conjectureis true provided we exclude the solution n =3 and thetrivial solution n = 2.

REFERENCES[1] D. C. Fielder, "On Shannon's almost uniform distribution,"

IEEE Trans. Electronic Computers (Correspondence), vol.EC-13, pp. 53-54, February 1964.

[2] M. A. Harrison, "A remark on uniform distribution," IEEETrans. Electronic Computers (Correspondence), vol. EC-13, pp.630-631, October 1964.

[3] J. Hunter, Number Theory. London: Oliver and Boyd, 1964.

General Repetitive Events and MachinesB. G. REYNOLDS

Abstract-A finite automaton is a general repetitive machine ifthere is a path in the state graph from every final state to the startstate. In this paper general repetitive events are defined in terms ofcertain factorization properties of the associated regular expressions,and a one-to-one onto correspondence is shown between generalrepetitive machines and general repetitive events.

Index Terms-Automaton, finite automaton, general repetitiveevent, general repetitive machine, repetitive event, repetitive ma-chine.

INTRODUCTION

Although there are techniques for going from regularexpressions to state graphs and vice versa [1 ], [2 ], thereis no general theory relating forms of regular expressionsto state graph structures. Insight into such relationshipshas been developed through investigations of particularclasses of events and automata [3 ]- [5 ]. In keeping withthis approach we define in this paper a class of autom-ata and a class of events and show the correspondencebetween them. The class of automata was defined byReynolds and Cutlip [6]. The reader is referred toSection 2 of that paper for the notation and terminologyused here.

GENERAL REPETITIvE EVENTS AND MACHINES

The following definition is taken from [6].Definition 1: An automaton A = (S, M, so, F) is a

general repetitive machine (GRM) iff for each state si Fthere is a tape xE* such that M(si, xi) =so.

If for some tape x, M(s, x) so for all s(F, A is arepetitive machine (RM1).A general repetitive machine is thus an automaton

having a path from every final state to the start state.The class of repetitive machines forms a proper subclassof the class of general repetitive machines; in an RM asingle tape x will take A from any final state to s0, but

Manuscript received February 27, 1969; revised August 21, 1969.Material for this paper was taken from the author's Ph.D. disserta-tion [7]. This work was supported by the U. S. Air Force Office ofScientific Research under Contract AFOSR-67-1023.

The author is with Texas Instruments Incorporated, Dallas, Tex.75222.

some GRMs require different tapes for different finalstates.

It was also shown in [6] that every GRM is eitherstrongly connected or consists of a single nonfinal deadstate SD and a strongly connected subset S-{ SD }. Notethat the results of this study apply to strongly con-nected machines, since these are a proper subclass ofGRMs.We now define repetitive and general repetitive

events. To avoid cumbersome notation, we make nonotational distinction between an event and a regularexpression denoting that event.

Definition 2: A repetitive event (RE) is a regular eventP such that, for some tape xG i*,

1) P can be written P=P(xP)*, and2) for any events R and Q, if RCP and RxQCP

then QCP.

The tape x is called a return tape. The class of REs isdenoted by R.

Definition 3: A general repetitive event (GRE) is aregular event R with the properties

1) R = UJ 1 Pi, where each Pi is an RE with returntape xi,

2) for 1 < i, j < m, Pi can be written Pi = (Pjxj) *Pi,and

3) for 1<i, j<m and for any events Ri and Q, ifRiCPi and RixiQCP1 then Qcpj.

The class of GREs is denoted by G.Observe that an RE is a GRE, and hence R is a proper

subclass of G. Also, no finite event belongs to eitherclass.A lemma is needed for later use.Lemma 1: If P is an event and x is a tape, the follow-

ing are equivalent:P = P(xP)*P = (Px)*PPxPC P.

(1)(2)

(3)Proof: The equivalence of (1) and (2) is seen as

follows:

P(xP)* = P[A + xP + xPxP + * * ]

= P + PxP + PxPxP+

= [A + Px+ PxPx+ * *P

= (Px)*P.

That (1) implies (3) is trivial. Conversely, if PxPCP,then (PxP)xPCPxPCP, or P(xP)2CP, and in likefashion we observe that P(xP)nCP for n=0, 1, 2, * *.*where (xP)O =A. Hence P(xP)*CP. Obviously PCP(xP) * so P=P(xP)*. Q.E.D.

The following theorem relates GREs to GRMs. Allautomata are assumed to be reduced.

Theorem 1: Let A (R) = (S, M, so, F) be a finite autom-aton. Then A (R) is a GRM if and only if R is a GRE.

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