102 IEEE TRANSACTIONS ON ELECTRONIC COMPUTERS

Short Notes

Critical Pairs and Set Systems no partition W7r' different from 7ri for which 7rW < ri'.<. We shall nowshow that K[]=L [4)], so it is, at least conceptually, easy to form

ALBERT FRIEDES L [4)] from 4).Karp [1 ] shows that for transition incomplete machines, the set of Lemma 1: For any partition r1i <, where 4 is a set system, there

all critical pairs completely characterizes the set of all partition pairs, exists an element 7r2EK[0] such that w2>1r.just as the Mm pairs did for the transition complete machines. He Proof: Let 7r1 be a partition such that 7r1<. 4. Let the blocks of 7rnfurther shows, in Theorem 1 [1], that the partition pair [p, m(p) ] is be denoted by B1, * * *, B,,. Since irni<., for every block Bj of 7r, wea critical pair if, and only if, p is maximal with respect to some 1i par- can find (at least) one block Aj of 4) which contains all the elements oftition. Thus, as Karp points out, the principal difficulty encountered Bj. (If there is more than one block of 4) which contains all the ele-in computing the critical pairs is that of determining partitions maxi- ments of Bj, choose one such block and denote it by Aj.) Since 7rmal with respect to , partitions. In this note another method is de- contains all states, so will the set of blocks A1, * * *, A,,. If the Aj'sveloped for computing the partitions maximal with respect to the A are not all distinct (the elements of more than one of the Bj's beingpartitions, and the connection is shown between this part of Karp's contained in the same Aj), we can omit copies of the same Aj andpaper and the more recent work of Hartmanis and Stearns [2]. The relabel the remaining set A1', A2%, * * *, Am'. We can then relabel thereader is assumed to be familiar with both [1] and [2]. By's so that Ai' contains all the elements in

A set system 4 on a set S= { si } is a collection of subsets of S ni(called the blocks of 4)) such that the following three conditions are U Bei.met: j=i

1) No block of 4 is properly contained in any other block of 4. Now, by omitting from each Ai' all states not contained in2) Every element of S is in at least one block of 4. If two elements n

si and s, of S are in the same block of 4), we will say si and sj are U Baj,compatible under 4) and denote this by si-sj((4). If si and s, are j=1not in the same block of 4), we will denote this by s~ri-1-sj(). we obtain a partition 7r3>7r.. By the method of construction of 7r3,

3) If r elements s1, S2, , Sr of S are not in the same block of 0, r3EK'[4)] and, therefore, there is an element 7r2CK[q)] such thatthen there are at least two elements si and sj, 1 <i<j<r, such W2.w3 For the case where Ajs are distinct the relabeling step maythat sj"-'s,j(4)). be omitted and the proof proceeds as above.

This definition of a set system is slightly more restrictive than Theorem 1: L[4)]=K[4)].that given by Hartmanis and Stearns [2 ], but in no way affects their Proof: a) Let the partition p be a member of L [4]. Then byresults. Part 3) of the above definition says that every block of a set Lemma 1, there exists a partition irCK[4] such that ir>p. But irsystem 4 must correspond to a maximal compatible [3] under cwhere.coptblt .nei sdfndaoe cannot be >p since this would imply that p was not a member ofwherexompality under(2,124)isas

defined above. -statemachineLL[] contrary to our assumption. Hence 7r=p, and every member of

For example, ¢1t=(Xm, b,eC, 4) defined on a four-state machine L[4) is a member of K[4)].would not be a set system, because blocks 12, , and are not maxi- b) Let the partition p be a member of K [cjj. Then, by Lemma 1mal compatibles under 41. 4)2=(12, 23, 4) defined on a four-state and the definition of K[4], there can be no partition 7r such thatmachine is a set system. p<w7r<. Since, in addition, every member of K[4)] is <4), pCL[4)].

The subdivision of S into blocks of a set system may be thought F a cof as being induced on S by a relation which is reflexive and sym- partins mavximal hith r et to tollows:metric but not, in general, transitive. A partition is then a special case 1) Compute all ,u-partitions as in [i].of a set system. Thus, the symmetric reflexive relation 0 among the 2) Forueah / -partition asc p [II aelements~ ~of,dfndo ae58o ar 1Icnb huh fa

2) For each ,i-partition ,u;, compute M(lAi) as a set system. To doelements of S defined on page 508 of Karp [i] can be thought of as this, let the inpuits of an arbitrary machine 9 be denoted by Ik, anddefining a set system O ons . let the next state entry of state si of YZ under input Ik be denoted by

Now consider a.set system 4). Denote by K'[4)] the set of all par- N(si, Ik). Then note that if 4) is any set system defined on the statestitions which can be formed from 4 by deleting all occurrences except of nZ,one of each state. Then, if we exclude all partitions ir of K'[I] forwhich there exists a partition 7r' in K'[4)] such that ir'>ir, we get ss - s,(M(4)) if and only if N(ss, Ik) - N(s1, Ik)(4)another set of partitions which we will denote by K[4]. for all inputs Ik for which N(si, Ik) and N(sj, Ik) are both specified.

Example 1: Hence for any ,3-partition Aij we can find all pairs of states which are

compatible under M(lAj). These compatibles may then be combined= (1, 3; 3, 4, 5; 2) to form maximal compatibles under M(lAi) and so yield the blocks

____ of M(lAi).K'[4] = {(1,3; 4, 5; 2), (1, 3,4,5; 2)} = K[4]. 3) Since K [M(Ai)]= L [M(/i3)] we can obtain the set L[M(ui)]

Example 2: by the method previously described for computing K[M(^3i)]. Thefollowing theorem now shows that a partition p is maximal with

4)= (1, 3; 2, 4; 2, 3) respect to a ,u-partition .,u if, and only if, pEL [M(/3s) ] and, therefore,K'[)] { 1,32,4, (,3;2;4),(1;2,4 3) (1 ; }. completes our procedure for computing partitions maximal withK'[f = (1 3;, 4, (, 3,2; ), 1; 2 4;3),(1; , 3 4)} . respect to /3-partitions.

K[4)] = {1(1, 3; 2, 4), (1; 2, 3; 4) } # K'[4)]. Theorem 2: A partition p is maximal with respect to some ,u-partitionDefine the set of partitions L [4] as the set of all partitions less pi if, and only if, pEL [M&gs)].

than or equal to the set system 4) such that for all iniGL [4)] there is Proof: a) Let pEL [M(,1) ]. Then, since p .M(/3i), (p, si is a par-tition pair. Furthermore, since M(/3s) is the largest set system suchthat (M(/3j), ,ui) is a system pair, and there is no partition 4) different

Manuscript received May 10, 1965; revised September 23, 1965. The research from p such that p <4.M(,u3), p is maximal with respect to ju.i~reported here was supported by National Science Foundation Grant NSF GP2789. b) Let p be maximal with respect to ,u- Then (p ,u i) is a partitionThe author is with the Department of Electrical Engineering, Columbia Uni- par an fro th deiiino (3) .(3) o suigta

versity, New York, N. Y. pl,adfo h eillno (u) (u) o suigta

I1 I2 13 I4

1 3 2 4 3

2 1 - 2 3

3 4 1 2 5 Fig. 1. Flow table of sequential machinies.

4 -5-2

5 1 3 3 4

Associated Maximal,u-Partition M(Ai) Partitions p (L[M(,4i)]) m (p)

0 0 0 0*

I I I |1*

(1,3;2,4; 5) (1,2;3;4;5) (1,2;3;4;5) (1,3;2,4;5)*

_ _4

_ (1;2,4;3;5) (1;2,3;4;5)*

(1;2,3,5;4) (1,4;2,4;3;5) (i2;3;5) (1;2,3,5-;4)*

_______ (1,5;2,4;3) (1,2,3,4;5)*

(1, 2, 5;3;4) (1, 2,3, 4;5)*

(1,4;2;3,5) (1;2,3;4;5) (1;2,3;4;5) (1,4;2;3,5)*

(1;2,3;4;5) (;2,4;3;5) (1;2,4;3;5) (1;2,3;4;5)

(1;2,3,4;5) 1(;2,4;452,5;23) ,3;;5)(1;2,5;3;4) (1;2,3,4;5)*

(1,2,5;3;4) (1;2;3,4;5) (1;2;3,4;5) (1,2,-5;3;4-)*

(1;2,4;3,5) (1;2;3;4,5) (1;2;3;4,5) (1;2,4;3,5)*

(1,3,5;2,4) (1,2;3;4,5) (1,2;3;4,5) (1,3,5;2,4)*

__ __ _ (1;2,4;3;5) (1;2,3;4;5)(1,4;2,3,5) (1,4;2,3;2 4;5) (1,4;23;) J1,4,3,5)*

(1,4;2,3;5) (1,4;2,3,5)*

_ _ _ ~~~~~~~~~~(1,;2;3;5;) (i;2,-3,54,)*

(1;2,3,4,5) (1,4;2,4,4;53)4;2,3;4;5)(1,4;2,45;) (1;2,53;4,5)

(1,2,43,5;) (1,;2,34;,4;5 (1;2,43;45) (1;2,34;35)*

Fig. 2. Computation of critical pairs.

104 IEEE TRANSACTIONS ON ELECTRONIC COMPUTERS FEBRUARY

p is not a member of L [M(pi) ] implies that there exists a partition 0 x

different from p such that p<4<M(pi), and since 4<M(Mi), (0, pi)is a partition pair. Hence it is shown that p is not maximal with _,_ 4, 0

2 3,0 2, 0respect to /ui contrary to our original assumption. Therefore, if p is 3 1,0 3,1maximal with respect to 1-si, p&L[M(Oi)]. 4 1, 1 2,1

Hence, the set of partitions maximal with respect to a ,u-partition Fig. 1. Machine M. Entries refer to next state and output.,ui is just the set of the largest partitions <M( .(i).Example 3: As an example of the aforementioned procedure we com- Initial State Intermediate States Final Statepute the critical pairs for the flow table of Fig. 1. The starred entries 1 4 1 4 1

in column 4 of Fig. 2 indicate that the corresponding entries in 2 3 3 3 1columns 3 and 4 constitute a critical pair. I 4 2 3 3 1

It should be noted that as Hartmanis and Stearns [2 ] have indi- Fig. 2. Possible paths of machine M with inpuit sequence 1010.

cated the Mm system pairs could be used to obtain state assignmentswith reduced dependence for incompletely or completely specified Initial State Final Statemachines just as Mm partition pairs were used for completely speci- 1 - -

fied flow tables. 2 3 13 1-4 1

ACKNOWLEDGMENTFig. 3. Possible paths of machine M with inptut sequence 00.

The author wishes to acknowledge his appreciation for the usefulcomments and careful reading of this work by Prof. S. H. Unger.

xREFERENCES 0 1

[1] R. M. Karp, "Some techniques of state assignment for synchronous sequientialmachines," IEEE Trans. on Electronic Computers, vol. EC-13, pp. 507-518, 1234 ___ 2, 3, 4October 1964.

[21 J. Hartmanis and R. E. Stearns, 'Pair algebra and its application to automata 13 3, 4theory," Information and Control, vol. 7, pp. 485-507, December 1964.

[3] M. C. Paull and S. H. Unger, 'Minimizing the nuimber of states in incompletely 234 1, 3 2, 3specified sequential switching functions," IRE Trans. on Electronic Computers,vol. EC-8, pp. 356-367, September 1959. 23 1,3 2, 3

Fig. 4. Test table for observable synchronizing sequences.

A0

1234 __ _(fii)

Synchronizing Sequences for Incompletely 234 1,3 2, 3Specified Flow Tables 13

23 13 2.3DANIEL J. ROSENKRANTZ, STUDENT MEMBER, IEEE 23 -(D-;-_

A synchronizing sequence for a completely specified flow table isan input sequence which will take the flow table to a fixed state Fig. 5. Test table for controllable synchronizing sequence.regardless of the initial state [1]- [3]. There are two possible gen-eralizations for incompletely specified flow tables; these can be calledcontrollable and observable synchronizing input sequences. x

Unspecified next-state entries in the flow table are considered tocorrespond to prohibited inputs. A controllable synchronizing se- 1 --- 3, 0quence is an input sequence such that for any initial state the final 2 4, 0 2, 0

3 5,0 3,1state is the same, and a machine corresponding to the flow table 4 1, 1 2,1would not pass through an unspecified transition. For instance, 100 5 6, 0 3,1

6 2,1 1,0

is not a controllable synchronizing sequence for the flow table ofFig. 1, since if the initial state were 1 or 3, an unspecified transition Fig. 6. Flow table with observable synchronizing sequence whichwould occur and the final state would not be determined. However, cannot be made controllable by filling in "don't cares."1010 is a controllable synchronizing sequence giving a final state of 1.The possible paths of the machine when subject to the input sequence1010 are shown in Fig. 2.An10admissibl in sequ e w ever, he himself cannot apply 00 in the expectation that this will syn-An admissible input sequence with respect to an initial state s iS chonzth. acieone such that the machine beginning in state s will not have to make chronize themcisnea transition through a "don't care" entry when the input sequence is

applied. An observable synchronIzi sr wIc al minimal length ones can be constructed using the column sets of the

iniia stte wit repc to whc iti .disbeaetesm flow table [i ] [4]. An example of a test for an observable synchroniz-final stae Exmpe for mahn M of Fig 1 are0 and 010 Th ng sequence for machine Mi1S shown in Fig. 4. The stubs (row labels)

posil pah whc h ahn a olwaesonii. 3 If of the test table are sets of states, beginning with the set of all states.the achne soeraing iththeunseciied ntres orrspodin The entry of input i, row r, is the set of next states for the elements of

toprhbie or restricte inu seune an a.0iengigit row r with input i. "Don't cares" are ignored. If an observable syn-the~~~~~~~~~~~~~~mahieanobevrwudkowtaThTia saei.Hw chronizing sequence exists, an entry consisting of a single state will

appear in the test table. A sequence leading to the single row entry iscircled in the table.

Manuscript received June 21, 1965; revrised October 27, 1965. A test table for a controllable synchronizing sequence for machineThe author is with the Department of Electrical Engineering, Columbia Uni- M is soniFg. .5 ahi nee hnvroeoh ttso

versity, New York, N. Y.'M ssonn g.5Adahseneewhnvrneftesaesf