Minimization of Incompletely Specified Sequential Machines

IEEE TRANSACTIONS ON COMPUTERS, VOL. c-24, NO. 11, NOVEMBER 1975

Department, IBM, East Fishkill, N. Y. In 1966, he joined the switching theory, while his main research interest is in fault-tolerantfaculty of the Department of Electrical and Computer Engineering, computing.University of Wisconsin-Madison, where he is currently an Associate Dr. Kime is a member of the Association for Computing Ma-Professor. He spent the 1973-1974 academic year as a Visiting chinery, the American Society of Engineering Education, TauAssociate Professor with the Department of Electrical Engineering Beta Pi, Eta Kappa Nu, and Sigma Xi. He is also presently Sec-and Computer Sciences, University of California, Berkeley. His retary of the IEEE Computer Society Technical Committee onteaching interests are in computer organization, logic design and Fault-Tolerant Computing.

Minimization of Incompletely SpecifiedSequential Machines

C. V. S. RAO AND NRIPENDRA N. BISWAS, SENIOR MEMBER, IEEE

Abstract-A simple yet efficient method for the minimization ofincompletely specified sequential machines (ISSM's) is proposed.Precise theorems are developed, as a consequence of which severalcompatibles can be deleted from consideration at the very first stagein the search for a minimal closed cover. Thus, the computationalwork is significantly reduced. Initial cardinality of the minimalclosed cover is further reduced by a consideration of the maximalcompatibles (MC's) only; as a result the method converges to thesolution faster than the existing procedures. "Rank" of a compatibleis defined. It is shown that ordering the compatibles, in accordancewith their rank, reduces the number of comparisons to be made inthe search for exclusion of compatibles. The new method is simple,systematic, and programmable. It does not involve any heuristics orintuitive procedures. For small- and medium-sized machines, it canle used for hand computation as well. For one of the illustrativeexamples used in this paper, 30 out of 40 compatibles can be ignoredin accordance with the proposed rules and the remaining 10 compati-bles only need be considered for obtaining a minimal solution.

Index Terms-Basic compatible, deletion of compatibles, ex-clusion of compatibles, incompletely specified sequential machine,minimal closed cover, min-max cover, primary compatible, primeclosed sets of compatibles, rank of a compatible, symbolic com-patible.

I. INTRODUCTION

QEQUENTIAL machines are encountered in many areasof human endeavor like computers, digital com-

munication systems, digital control systems, and so on. Inpractical situations, the state tables are hardly evercompletely specified due to a variety of reasons which arewell known. Minimization of the number of states is

Manuscript received February 4, 1975; revised May 12, 1975.C. V. S. Rao is with the School of Automation, Indian Institute of

Science, Bangalore, India, on deputation from the Department ofElectronics and Communication Engineering, Osmania UniversityEngineering College, Hyderabad, India.

N. N. Biswas is with the Depaxtment of Electrical Communica-tion Engineering, Indian Institute of Science, Bangalore, India.

important for several reasons. The complexity of thecircuit reduces, reliability increases, and the lengths oftest sequences become smaller. Furthermore, it plays animportant role in fault-detectable and fault-tolerantdesign of sequential machines.

Several authors [1]-[14] have studied the problem ofminimizing incompletely specified sequential machines(ISSM's). Although the techniques for the generationof maximal compatibles (MC's) are fairly well established,there is, as yet, no known simple, efficient procedure forsolving the problem in its most general form.

Paull and Unger [1] and Unger [2] have developed ageneral theory for incomplete machines. However, theirmethods for obtaining a minimal closed cover involve alarge amount of enumeration and inspection. Further,their implication graph shows only the implications be-tween compatible pairs but obscures vital information inrespect to larger implied compatibles containing morethan two states.McCluskey [31, Pager [4], and Ehrich [5] have de-

veloped ingenious methods for minimizing a restrictedclass of incomplete machines. Their techniques are ap-plicable only to a very special class of machines.

Grasselli and Luccio [6] have presented an interestingapproach by casting the problem in the form of a linearinteger program. However, their minimization processusing prime compatibles and closure-covering table isquite complex and is not attractive for hand computation,but it appears to be suitable for use with a computer.Luccio [7] improved the method by extending the defini-tion of prime compatibility classes and developing certainrules to eliminate some of the prime compatibility classes.His method, however, is quite lengthy and involves tediousprocedures.

Meisel [8] constructed directed trees with prime com-

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IEEE TRANSACTIONS ON COMPUTERS, NOVEMBER 1975

patibles as nodes, from which the shortest paths yieldminimal solutions. In this method, however, there existtoo many paths to be examined and it becomes unwieldyeven for relatively small machines.

DeSarkar, Basu, and Choudhury [9], [10] have simpli-fied the procedure of [6] by tackling the problem of closurefirst and then coverage. They generate prime closed setsfor all prime compatibles and then select a union of primeclosed sets representing a minimal cover. Their method,however, is lengthy as it considers the entire set of primecompatibles.

Bennetts, Washington, and Lewin [11] have developeda program to extract a nearly minimal closed cover. Theirprocedure is analogous to that described in [9]. Thealgorithm, however, uses intuitive criteria for the selectionof MC's or prime compatibles for which the closure func-tion conjuncts are generated. The final disjunctive formobtained from the closure function conjuncts yields theprime closed sets. The method is generally applicable ifone is satisfied with a nearly minimal solution and is notconcerned with the number of optional entries in theactual realization of excitation functions. Further, incases where the cardinal numbers of the minimal closedcover and the nearly minimal closed cover fall on eitherside of an integral power of 2, the method may not bevery useful, especially so if the designer is eventuallyaiming at a fault-tolerant encoding of the machine.

Kella [12] developed a new method which does notrequire the generation of MC's but involves lengthyalgorithms.Yang [13] proposed yet another method in which all

"superseded" compatibles are eliminated, thus reducingsubstantially the number of compatibles to be consideredin the search for a minimal closed cover. However, this ispurely an intuitive basis and there is no formal proof toshow that at least one minimal closed cover containingonly unsuperseded compatibles exists. Further, his pro-

cedure to arrive at the unsuperseded compatibles is quitelaborious.

Biswas [14] proposed a method using "implicationtrees." This method, however, becomes unwieldy if theMC's consist of a large number of states. Furthermore,the "bunching" of compatibles by inspection is tedious,particularly when it involves examination of a largenumber of "secondary trees."

In this paper, a simple method, not involving any

heuristics, is presented to pinpoint the compatibles whichcannot be members of any minimal solution and hencecan be deleted. In addition, the method permits deletionof several other compatibles without in any way affectingour chances of finding a minimal closed cover. The new

method is decidedly more efficient in the sense that some

compatibles which cannot be eliminated in other methodscan be deleted, thereby yielding a smaller set of compati-bles with which to deal.We assume that the MC's are available for our use. They

may-be obtained by any one of several methods, such as

those contained in [l], [15]. The terms "incompletely

specified sequential machine" (ISSM) or simply "in-complete machine," "compatibility class" or simply"compatible," "maximal compatible" (MC), "implica-tion chart," "covering machine," "closed cover," and"4minimal closed cover," are freely used with their usualmeanings [16], [17]. In order to develop a consistentnotation, we need the following definitions.

Definition 1: The closure class set Es of a compatibleCi is a set of all compatibles implied by Ci obtained byrepeated use of transitivity of implication, such that thecompatibles which are subsets of either Ci or any othermember of Ei are removed from the set.

Definition 2: A set Ei dominates a set Ej denoted byEi 2 Ej if and only if every member of Ej is a subsetof at least one member of Es. The equality relation holdswhen Ei and Ej are identical sets, in which case theydominate each other.

Definition 3: A set of compatibles is said to be closedif and only if, for every compatible contained in the set,each implied compatible is also contained in at least onecompatible of the set. In other words, a closed set ofcompatibles is a set of compatibles which is closed withrespect to the binary relation "'implication."

Definition 4: A prime closed set of compatibles is aclosed set of compatibles which cannot be partitionedinto two subsets such that both the subsets are closedwith respect to the binary relation "implication." (Oneof the subsets may be closed.)

Definition 5: For a compatible Ci, the implied com-patibles noted directly from the columns of the statetable (first-level implications) are said to be "primaryimplications" and the set of primary implications isdenoted by Pi.

Definition 6: A compatible Ci excludes a compatibleC, if Ci D C, and Pi < Ej. This definition is a modifiedversion of the definition of Grasselli and Luccio [6] andin fact covers the latter.

II. DELETION THEOREMS AND GENERATIONOF SYMBOLIC COMPATIBLES

In this section, we will develop a new method for theminimization of ISSM's. The conceptual framework isas follows. We first form a set of what are called "primarycompatibles" which is simply the set of all subsets ofMC's. We then investigate systematically whether wewill be able to delete some compatibles from the originalset without in any way affecting our chances of finding aminimal closed cover. By the repeated application ofTheorems 1 and 2, given in the text, we may delete asubstantial number of primary compatibles. The primarycompatibles which remain undeleted are called "basiccompatibles." Thus, we obtain a smaller set of compatiblesto deal with. From among the basic compatibles, some are"excluded" according to Definition 6, making the set stillsmaller. The compatibles remaining in the set after dele-tion and exclusion effected in that sequence are called"symbolic compatibles." Each symbolic compatible servesas a representative for a subset of primary compatibles

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RAO AND BISWAS: MINIMIZATION OF ISSM'S

and hence this name is chosen. The set of symbolic com-patibles is usually much smaller than the set of primarycompatibles. Theorem 3, given in the text, guaranteesthe existence of at least one minimal closed cover whichis a collection of symbolic compatibles only.

Definition 7: Every subset of an MC is a primary com-patible.

Illustration: Consider the machine represented by theflow table given in Table I. The corresponding implicationchart is given in Table II. The set of MC's is easily ob-tained as'

I acef,abd,acd,bdgh,cdh,efg I . (1)

The primary compatibles and their closure class sets arelisted in Table IV. There are 40 primary compatibles forthis machine including 8 single-state compatibles.

Definition 8: A compatible Ci is said to be an "impliedcompatible" (IC) if each compatible pair C, C Ci is asubset of some member of the closure class set of somecompatible contained in the set of compatibles underconsideration.

Definition 9: A compatible Ci is said to be an "unimpliedcompatible" (UC) if none of the subsets of Ci are impliedby any compatible contained in the set of compatiblesunder consideration.

Illustration: Observe in Table I that the compatibleacef is implied by the compatible bdgh. We say that acefis an IC. It will be seen that each of the compatible pairsab, ac, ae, af, ce, cf, dg, ef, eg, fg is implied by at least oneother compatible. These IC's are shown in Table III withan I mark in the corresponding cell. The remaining com-patible pairs ad, bd, bg, bh, cd, ch, dh, gh are UC's indicatedby an U mark in Table III.

Definition 10: A maximal implied/unimplied compatible(MIC/MUC) is an IC/UC which is not a proper subsetof any other IC/UC.

Illustration: From Table III, the MIC's are easilyobtained in the same manner as the MC's. The set ofMIC's is

Jacef,ab,dg,efg}. (2)

Similarly, the set of MUC's is obtained as

{ad,bdh,bgh,cdh}. (3)The MIC's and MUC's serve the purpose of referencingin the deletion of primary compatibles, discussed laterin this section.Lemma 1: If Ci and C, are two UC's such that C, C Ci,

then Ei > E,.Proof: Since none of the compatibles which are sub-

sets of C1 are implied by C, and Ci D Cj, it follows thateach element of Ei must be contained in at least oneelement of Ei. Hence Es 2 Ej.Lemma 2: Let Ci be a UC and Ei be the closure class

1 Although a compatible is a set of states of the machine, the setparentheses are understood and not explicitly shown for compatiblesthroughout this paper. Also the set parentheses are omitted in tableswhere they are understood.

TABLE IFLOW TABLE OF AN ISSM

Next State, OutputPresent state

Inputs0 1 2 3

a g,0 e,1 d,-b a, d,- - -Oc c, -,O - g,ld e,O - a,-e ,1 f,- ,1 ,f -,1 e,- a,j -,g f, -,1 b,h--h c,- - a,O

TABLE IIIMPLICATION CHART OF MACHINE OF TABLE I

b dgc dg Xd ae ae cee fg X a Xf eg X V X v

aeg X af X ef v ab

abh X ae V ce X X cf

aba b c d e f g

set of Ci. Let s be an arbitrary state of the machine notcovered in Ei. Then the closure class set Ej of a compatibleCj C C1 does not cover the state s.

Proof: Since Ci D Cj it follows from Definition 9and Lemma 1 that Et 2 Ej. It is clear that all the statescovered in Ej are covered in Ei also but the converse isnot true. Obviously a state s not covered in Ei cannot becovered by Ej. Hence the lemma. As a consequence ofLemmas 1 and 2, Theorem 1 follows.

Theorem 12 (First Deletion Theorem): The deletion ofall compatibles C1 satisfying the following conditions,from the set of primary compatibles, does not precludefinding at least one minimal closed cover from the remain-ing primary compatibles.

a) Ci is a UC.b) The closure class set Et of C1 covers at least one

state s E Ci.Proof: Suppose the compatible C1 is a member of a

minimal closed cover M. Since Ci is a UC, its presence isnot required to satisfy any closure conditions. It servesonly the purpose of covering the states of the machine.Further, each of the compatibles contained in Et (closureclass set of C1) must be contained in at least one of thecompatibles contained in the set3 (M - Ci) and hence(M - Ci) 2 Ei. Let the compatible Ci be given by5S12 5-hkSk±1 '"S where si through s, are contained inthe state set of the machine. Without loss of generality,assume that S1,S2,..-,Sk are the states not covered in Ei

2 This theorem is much more powerful than Luccio [7, rule 1(a)],which requires that at least all but one states contained in Ci must becovered in Ei. Moreover, implementation of this theorem is trivialitysimple compared to that of Luccio's rule.

3 (M - ) denotes the subset yielded on removing the member Cifrom the set M.

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TABLE IIIIMPLIED (I) AND UNIMPLIED (U) COMPATIBLE PAIRS

b IC Id U U Ue I If I I I9 U I I Ih U U U U

a b c d e f g

and the remaining states sk+1,. ,s-1r are covered in Ei. Itis clear that the set (M -C) Ei covers the statess+l..sr. If Ci is replaced by a compatible Cj comprisingof the states S1,S2,-. - Skin M, it follows from Lemma 1that the yielded set is still closed and covers the machine.In order to conclude that Ci may be deleted from the setof primary compatibles, it is to be shown that Cj remainsundeleted. Recall that C, C Ci and none of the statescomprising Cj, namely 8182,... ,sk, are covered in Ei. Itfollows from Lemma 2 that none of the states containedin Cj will be covered in E,. Hence Cj remains undeleted.Thus Ci can be replaced by C, in any closed cover andhence can be deleted from the set of primary compatiblesand ignored in the process of extracting a minimal closedcover. The only restriction on the number of states con-

tained in Ci and covered in Ei is that it should be nonzero.

As C1 is arbitrarily chosen the theorem is proved.Illustration: For the machine of Table I, the primary

compatibles and their corresponding closure class sets are

listed in Table IV. Let Ci be cdh which is a UC. Et = {ce}.

Ei covers one state c E Ci. The compatible cdh can bedeleted from the list of primary compatibles. In this case

the compatible dh can represent the compatible cdh inany closed cover and the theorem guarantees that thecompatible dh remains undeleted in the current list, whichfact can be verified by noting the closure class set of thecompatible dh.An important and useful corollary to Theorem 1 follows.Corollary: Let Ci be a UC and Ei be its closure class

set. Then Ci is not a member of any minimal closed cover

M if Ei covers every state E Ci.Proof: Suppose Ci is a member of some minimal

closed cover M. Since C? is a UC, it does not satisfy any

closure conditions of (M - C1). Therefore the set(M - Cj) is closed. Further, (M -C) 2 Ei and henceall the states contained in Ci are covered in (M - C).

Hence (M -C) represents a closed cover for the ma-

chine-thus contradicting the hypothesis that M is mini-mal. In other words, Ci may be replaced by a "void com-

patibility class" in any closed cover and hence cannotparticipate in any minimal closed cover.

The above corollary naturally leads to the followingdefinition.

Definition 11: A compatible which can be removedfrom any closed set of compatibles, without affectingthe closure and covering properties of the set, is said tobe a "weak compatible." Otherwise it is a "strong com-

patible."

Illustration: Consider again the machine of Table Iand compatible Ci = ad in Table IV. Ei is the set {ae,fg,ab,dg,ef}. Since Ci is a UC and Es covers the states a andd, the compatible Ci = ad cannot be a member of anyminimal closed cover. Such weak compatibles can beweeded out and ignored even if, for some reason, we haveto find all minimal solutions.

Let us, now, proceed to develop another theorem whichis more powerful and in fact covers Theorem 1. The follow-ing definitions help a precise formulation of Theorem 2.

Definition 12: The "implied part" of a compatible Ci,denoted by I-, is the set of states obtained by the unionof all IC's which are subsets of Ci.

Illustration: Let Ci be bdgh of Table IV. Note also theIC's marked I in Table III. There is only one IC, namelydg, which is a subset of Ci. We then say that Ii = dg.

Definition 13: The "uncovered part" of a compatibleCi, denoted by Ni, is the set of states contained in C1and not covered in Ei.

Illustration: For the compatible Ci = bdgh, the closureclass set Ei = Jacef,ab,efg} from Table IV. The statesd and h contained in Ci are not covered in Ei. We thereforesay that Ni = dh.The following Theorem 2 is different in content and

much easier in implementation than Luccio [7, rule 1].Theorem 2 (Second Deletion Theorem): The deletion of

all compatibles Ci satisfying the condition (li U Nj) C Cifrom the set of primary compatibles does not precludefinding at least one minimal closed cover from the remain-ing primary compatibles.

Proof: Suppose Ci is a member of a minimal closedcover M. Let Cjbe a compatible given by C, = (Ii u Nj) CCi. Obviously Ei > Ej. If Ci is now replaced by Cj in M,then the closure conditions imposed by (M -C) areclearly satisfied by Cj as Cj D Ii. Since (M -C) 2Ei > Ej, it follows that the set yielded on replacing Ciby Cj in M is closed. Further, the states contained inCi and not covered in Ei are now contained in Cj. Hencecoverage also is satisfied. In order to conclude that C1can be deleted from the list of primary compatibles, it isto be shown that Cj does remain undeleted. A little re-flection reveals that I, = Ii and Nj D Ni and hence(Ij U Nj) D (Ii u Nj) = Cj. From the original definitions(Ij U Nj) C Cj. It follows that (Ij u Nj) = Cj. ThereforeC1 remains undeleted. Hence the Theorem.The essence of Theorem 2 is that if at least one state

contained in (Ci- Ii) is covered in Ei, then Ci can bedeleted from the list of primary compatibles.As a matter of fact, Theorem 1 can be treated as a

special case of Theorem 2 when Ii is a null set. However,in practice, it is much easier to implement Theorem 1first and Theorem 2 next. This is the reason for statingthem explicitly. This point is further discussed after thedeletion procedure has been stated.

Illustration: Referring again to Table IV, let C1 = bdgh.Then IX = dg, NT = dh, and TI U NT = dgh C C0. There-fore, the compatible bdgh can be deleted. In fact, the

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RAO AND BISWAS: MINIMIZATION OF ISSM'1

TABLE IVGENERATION OF BASIC COMPATIBLES

Primary Closure class set Deletions underSerial compatible Theorems 1 and 2number Primary impli- Others

cations Iteration numbers1 2 3 4

1 acef efg,dg abS D22 bdgh acef,ab efg D23 ace fg,dg ab,ef D24 aef eg,ae,dg fg,ef,ab D25 aef efg ab,dg D26 cef 0 * * * *7 bdg aef,ab efg D28 bdh ace fg,dg,ab,ef D19 bgh acf,ab eg,ae,dg,fg,ef D110 dgh cef,ab * * * *11 abd ae,dg fg,ef D212 acd ce,ae,dg fg,efab D213 cdh ce D114 efg ab dg V D215 ab dg ef * * * *16 ac dg efab D117 ad ae fg,ab,dg,ef D118 ae fg ab,dg,ef v 4 D119 af eg,ae fg,ab,dg,ef 4 Dl20 bd ae fg,ab,dg,ef D121 bg af eg,ae,fg,ab,dg,ef D122 bh ac dg,ef,ab Dl23 cd ce D124 ce 0 * * $ *25 cf 0 * * * *26 ch 0 * * * *27 dg ef,ab * * * *28 dh ce29 ef 0 * * * *30 eg 031 fg ab dg,ef D 132 gh cf,ab dg,ef D133to a-h 0 (single state compatibles) * * * *40

compatible dgh can represent the compatible bdgh in anyclosed cover and the theorem guarantees that the com-patible dgh remains undeleted in the current list.We will now illustrate in detail the procedure to obtain

a minimal closed cover for the machine of Table I. TableIV gives the list of primary compatibles with their corre-sponding closure class sets. First, it is noted that thecompatibles cef, dgh, ab, ce, cf, ch, dg, dh, eg, eg cannot bedeleted from the list as their corresponding closure classsets do not cover any of the states contained in the respec-tive compatibles. This fact is indicated by stars in thecorresponding rows. For example, for the compatible dgthe closure class set ef,ab} does not cover the statesd, g. The states of a compatible covered in its closure classset are shown in boldface.The deletion of compatibles is achieved recursively as

follows. Consider iteration 1 for which the MIC's andMUC's are given in Table V. All the MIC's and theirsubsets cannot be deleted at this stage and this fact isindicated in Table IV by the check mark. Now Theorem 1is applied to all MUC's and their subsets; as a consequencethe compatibles bdh, bgh, cdh, ad, bd, bg, bh, cd, gh aredeleted, indicated by D1. The remaining compatibles arenow subjected to the test of Theorem 2 and consequently

TABLE VPROGRESSIVE CHANGES OF MIC'S AND MUC'S

Iterationnumber MIC's MUC's

1 acef,ab,dg,efg ad,bdh,bgh,cdh2 ab,ae,cef,dg,efg acd,af,bdh,bgh,cdh3 ab,cef,dg,efg acd,ae,af,bdh,bgh,cdh4 ab,cef,dg acd,ae,af,bdh,bgh,cdh,eg,fg

the compatibles bdgh, bdg, abd, acd are deleted, indicatedby D2. This completes the first iteration.The above deletions of compatibles may result in some

of the IC's becoming UC's. For example, after the dele-tions in the first iteration, the compatibles ac and af, whichwere IC's earlier, now become UC's. The updated MIC'sand MUC's are given in Table V. With the new list, theprocedure is repeated. The procedure terminates when theset of MIC's and hence MUC's remains unaltered. Forthe machine under consideration, the deletion procedureterminates after four iterations.One point is to be noted here. The frequency of iteration

may be doubled by iterating alternately on the applicationof Theorem 1 and Theorem 2. This is the reason for stating

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the two theorems explicitly. The frequency of iterationmay be increased still further by starting a fresh iterationwhenever a primary compatible is deleted, but, however,the returns of such extra work may not be attractive inmany examples.The compatibles remaining after the deletions in Table

IV are called "basic compatibles."Definition 14: A basic compatible is a primary com-

patible which cannot be deleted by the application ofTheorems 1 and 2.

For our machine, starting from 40 primary compatibles,we have so far deleted 22 of them, yielding 18 basic com-patibles listed in Table VI, with their correspondingclosure class sets. Among these, 8 more compatibles arenow excluded in accordance with Definition 6, indicatedin Table VI. Obviously an excluded compatible can berepresented by a compatible excluding it because thisprocess does not impose any extra closure conditions butmay cover some more states of the machine. The com-patibles remaining after the elimination of excluded basiccompatibles are called symbolic compatibles.

Definition 15: A symbolic compatible is a basic com-patible which is not excluded by any other basic com-patible.

For our machine, the symbolic compatibles, 10 in all,are listed in Table VII together with their correspondingclosure class sets. The compatibles represented by eachsymbolic compatible are also listed in Table VII.

It is interesting to note that a primary compatible maybe represented by more than one symbolic compatible. Forexample, the compatible ac may be represented by eitherof the compatibles cef or ch in any closed cover. In thisexample, this situation arose due to the fact that thecompatible ac can be represented by the single statecompatible c which is excluded by two compatibles cefand ch. Another interesting point to be noted is that thecompatibles acd, ad, ae, af, bd, bg, and fg are not repre-sented by any symbolic compatible. These are weakcompatibles, discussed earlier, and as such they cannotparticipate in any minimal closed cover in accordancewith corollary to Theorem 1. Hence they do not requireany representation.Now we need consider only the relatively small set of

symbolic compatibles in the search for a minimal closedcover for the machine. The following theorem guaranteesthat we do not miss the solution.

Theorem 3: There exists a minimal closed cover whichis a collection of symbolic compatibles only.

Proof: Let M be a minimal closed cover and supposethat a compatible Ci, which is not a symbolic compatible,is a member of M. Then C, must be either a basic com-patible or not a basic compatible. If C, is a basic com-patible, then there exists a symbolic compatible C, whichexcludes Ci and Ci can be replaced by C, in M. If Ci isnot a basic compatible, then it must be a primary com-patible which can be deleted under Theorem 1 or Theorem2, in which case there exists a symbolic compatible Ckwhich can represent Ci in M. Hence the theorem.

GENERATION OF SYMBOLICTABLE VICOMPATIBLES FROM BASIC COMPATIBLES

Closure class set ExcludingSerial Basic compatiblesnumber compatible Primary Im- Others

plications

1 cef 02 dgh cef,ab3 ab dg ef4 ce 0 cef5 cf 0 cef6 ch 07 dg ef,ab8 dh ce9 ef 0 cef10 eg 011 a 012 b 013 c 0 cef,ch14 d 015 e 0 cef16 f 0 cef17 g 0 eg18 h 0 ch

TABLE VIISYMBOLIC COMPATIBLES AND THE PRIME CLOSJD SETS GENERATED

BY THEM

Symbolic States Closure Primary com- Primecompatible contained class patibles repre- closed

in Ci set sented by Ci sets

C1 cef 0 acef,ace,acf, C1aef,cef,efg,ac,ce,cf,ef,c,e,f

C2 dgh cef,ab bdgh,dgh C1C2C3C3 ab dg,ef abd,ab C1C2C3;

C1C3C5C4 ch 0 bdh,bgh,ac,bh, C4

ch,gh,c,hC5 dg ef,ab bdg,dg C1C3C5C6 dh ce cdh,dh C1C6C7 eg 0 eg,e,g C7C8 a 0 a C8Co b 0 b C9C10 d 0 cd,d C10

For the symbolic compatibles, prime closed sets areeasily obtained [9], [10]. They are listed in Table VII.Now the problem is only that of coverage and the minimalclosed cover is easily obtained as the set of symboliccompatibles C1, C2, C3 given by

{cef,dgh,ab}. (4)The results of this section are summarized in Procedure 1below.

Procedure 1-Generation of Symbolic Compatibles

Step 1: From the flow table, obtain the implicationchart and find all MC's.

Step 2: Prepare a table listing the primary compatibles(all proper and improper subsets of MC's) and theircorresponding primary implications.

Step S: In the table obtained in Step 2, for each primarycompatible, augment the list of primary implicationswith all implied compatibles obtained by repeated useof the transitivity of implication. The primary irffplica-tions together with the additional compatibles thus ob-

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RAO AND BISWAS: MINIMIZATION OF ISSMS1

tained constitute the closure class set for the correspondingprimary compatible. For this step, no reference need bemade to the flow table.

Step 4: List all the IC pairs and find MIC's. Similarlyfind MUC's.

Step 5: Delete from the list each primary compatibleCi which is a subset of some MVIUC if at least one statecontained in Ci is covered in Ei.

Step 6: For each primary compatible Cj which is not asubset of any MIC or MUC find Ij U Nj and delete Cjfrom the list if Ij uNjC Cj.

Step 7: From the reduced list of primary compatibles,form a fresh list of MIC's and MUC's. If these are identicalwith the earlier lists, go to Step 8, otherwise replace theearlier lists by the current lists and go to Step 5.

Step 8: For each pair of compatibles Ci and Cj suchthat Ci D C, compare Pi and Ej. Exclude (remove) Cjif Pi< E_. The compatibles now remaining in the listare all symbolic compatibles.

III. INITIALIZATION OF THE CARDINALITY OFMINIMAL CLOSED COVER

A prior knowledge of the upper bound on the numberof states in a minimal machine which covers a givenISSM greatly helps in reducing the computational workin the generation of prime closed sets of compatibles. (Aprocedure to generate prime closed sets of symbolic com-patibles, an improvement over that given in [9], [10],involving less computational work, is proposed in SectionIV.) Let the upper bound be denoted by u, obtained byactually having on hand a covering machine with u states.Then, in the process of minimization, we need search fora covering machine with less than u states. As such, allprime closed sets containing u or more compatibles neednot be generated. The computational work, perhaps,increases exponentially with the value of u. Thus, if ucan be lowered at an early stage in the process, our jobis significantly simplified. The presently used bound uis given by (see for instance [13])

u = minimum of (m,n,v) (5)

where

m number of MC'sn number of states of the given machinev cardinality of the minimal cover obtained using

only compatibles with void closure class sets.

A technique to reduce the initial cardinality of 'theminimal closed cover, whenever feasible, is proposed inthis section. The essential idea is to find a minimal closedcover containing only MC's.4 For brevity let us call this"min-max cover." For many machines, particularly whenthe number of MC's is large, the cardinality of the min-max cover may be smaller than u. Let x be the cardinality

4This is analogous to the procedure described by Bennetts etal. [11].

of the min-max cover. The modified upper bound, denotedby u', is given by

u = minimum of (u,x). (6)Thus we need search for a covering machine with u' - 1or less number of states. If, however, the number ofinternal state variables were to be the only criterion forminimizing the given machine, we need search for a cover-ing machine with its number of states being 2k or smallerwhere k is the integral part of log2 (u' - 1).

Illustration: To illustrate the ideas of this section andthe next, let us consider a more difficult example of anincomplete sequential machine represented by TableVIII. This machine is the same one used by Unger [16],Roy and Sheng [15], and Yang [13] but for renamingof the states by alphabet in place of numerals. The set ofMC's, ten in all, and their closure class sets are listed inTable IX. For this machine m = 10, n = 9, and v = 6.Therefore u = 6 and we need search for a covering ma-chine with five or less states as we already have a coveringmachine with six states.Now, consider the set of MC's. We first pick out the

state which is contained in the least number of MC's. Ifthere is a tie, the choice is arbitrary.5 Note that the stated is contained in two MC's M3 and M1o while each of theremaining states is contained in more than two MC's. Itfollows that either M3 or M1o or both must be containedin the min-max cover. So we arbitrarily choose M3 andgenerate the prime closed sets of MC's initiated by M3taking full advantage of the fact that those containingsix or more MC's may be ignored in the process as wealready have a cover with u = 6 compatibles. The primeclosed sets of MC's generated by M3 are

{M1,M3,M4,M5,M61, M1,M3,M4,M6,17},{21,M3,15,M,A8},{1M2,M3,M5,M6,M8}.

It is easily verified that each of the above sets covers allthe states of the machine. So we now have a coveringmachine with five states only. Next, we obtain prime closedsets generated by M1o containing less than five MC's. Itwill be seen that there is no such set. Hence we concludethat x = 5 and the initialized number of states in theminimal closed cover for the machine is given by

u= minimum of (5,6) = 5. (7)Now that we have a covering machine with u' = 5

states, we need search for one with u' - 1 = 4 or lessnumber of states. This enables us to ignore all primeclosed sets of symbolic compatibles, with a cardinality > 4.The process obviously converges to the solution muchfaster with the use of the reduced upper bound and thecost we pay is a little extra computation restricted toMC's only, which is negligible when compared with theamount of work saved.

6 Bennetts et al. [111 have suggested an intuitive choice. However,our objective here is to develop nonintuitive procedures.

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A MORE DIFFICULTTABLE VIII

EXAMPLE OF AN INCOMPLETE SEQUENTIALMACHINE

Present state Next state, Output

Inputs XlX200 01 11 10

a b,- d,- -b f,- i,- --c , --hgd b,- a, f,- e,-e - - f-f a,0 - b,- -,1g e,l b,-h e, - - a,Oi e,- c,- -e-

TABLE IXMCOS AND THEIR CLOSURE CLASS SETS FOR THE MACHINE OF

TABLE VIII

States con-MC Mj tained in Mi Closure class set of Mi

ml bcghi ef,ah,be,acM2 beghi ef,bci,af,ab,bf,di,ac ch,ahM3 deghi be,abc,aef,bf,ch,cf,ah,bg,bi,ciM4 abcf di,bg,ch,be,ef,bi,ci,ahM5 abch bef,di,afM6 abef di,cf,ac,ch,ah,bg,bi,ciM7 abeh bef,di,acf,bg,ch,bi,ciM'3 befi aef,bg,ab,be,di,ac,ch,ahM9 befi aef,ci,bg,ab,cf,di,ac,ch,ahM1o defi abe,ac,bf,cf,ch,af,bg,ah,bi,ci

IV. ADDITIONAL TECHNIQUES TO REDUCECOMPUTATIONAL WORK

Rank of a Compatible

Some more techniques which help in reducing thecomputational work involved in the minimization ofincomplete sequential machines are proposed in thissection. As discussed in Section II, we first obtain the setof symbolic compatibles following Procedure 1. The laststep of Procedure 1 requires exclusion of some basic com-patibles. For this, we have to compare each pair of com-patibles Ci, Cj such that Ci D C, and their implications.The number of comparisons to be made can be appreciablyreduced if we classify the compatibles according to their"order" and "rank" defined below.

Definition 16: The order of a compatible denoted by nis the number of states contained in it. For example theorder of the compatible dgh in Table VI is 3.

Definition 17: The rank of a compatible, denoted by r,is the order of the highest order compatible(s) containedin its closure class set. For example, the rank of the com-patible dgh in Table VI is 3 as it implies the compatiblecef of order 3.The following theorems follow as a consequence of the

above definitions.

Theorem 4: The rank of a compatible cannot exceed itsorder.

Theorem 5: A compatible Ci of rank ri cannot excludea compatible Cj of rank rj < ri.

Illustration: Referring to Table VI, let Ci = dgh andCj = dg. We have

Pi= {cef,ab} and Ej = {ef,ab}.Therefore ri = 3 and rj = 2. The compatible Ci does notexclude the compatible Cj. Even if Ej were to be {ce,cf,ef,ab}, the compatible Ci would not exclude Cj as E5 doesnot contain the third-order compatible cef. We, therefore,conclude that it is futile to make further comparisons, oncewe recognize the fact that ri > rj. In order to exclude acompatible Cj we need compare E' with Pi of anothercompatible Ci if and only if Ci D C5 and ri < rj. Thus,listing the basic compatibles in accordance with theirorder as well as rank will reduce the number of comparisonsto be made.

Generation of Prime Closed Sets of Symbolic Compatiblesand Obtaining a Minimal Closed Cover

Consider the machine of Table VIII and its MC's listedin Table IX. This machine has a total of 97 primary com-patibles (not listed). The reader is advised to solve thisexample completely in order to judge the efficacy andsimplicity of the method proposed in this paper. Following

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RAO AND BISWAS: MINIMIZATION OF ISSM S

TABLE XIC'S Ca AND. THE SYMBOLIC COMPATIBLES CQ SUCH THAT Cc, C Cp

IC C", All symbolic compatibles Cp D C.

abc C2,X3abe C4,C5acf C2aef C4bei C1,Ce,C7bef C4,C,17ab C2,C3,C4,C5,C13,C14ac C2,C3,C13,C15ae C4.C5,C16,C29af C2,C4,C14ah C3,C5,C16bc C1,C2,C3,C6,C7,C13,C30be C4,C5,C8,C9,C17,C18,C19,C32bf C2,C4,C6,C8,C14,C17bg Ci,C7,C9,Cio,C18,C2obi C1,C6,C7,C8,C9,C01,C19tC21cf C2,C6,C22,CS3ch Cl,C3,Cll,C15,C23,C26ci C1,C6,C7,C11,C22,C2,C2,Cdi C12,C2,37ef C4,C8,C12,C17,C38

Procedure 1 of Section II, 12 compatibles are deleted inStep 5 and 13 in Step 6 and 28 compatibles are excludedin Step 8. Thus 53 compatibles in all may be ignored andwe have to deal with the remaining 44 symbolic compati-bles only in the search for a minimal closed cover. Eventhis is a formidable task for hand computation unless wehave a fairly efficient technique which converges fast to asolution. A technique which the authors have used advan-tageously for solving several examples is illustrated below.

For the machine of Table VIII the set of symboliccompatibles, denoted in parentheses by Ci, 1 < i < 44, isgiven below.

{bcghi(Cj) ,abcf(C2) ,abch(C3) ,abef(C4) ,abeh(C5) ,bcfi(C6),bcgi(C7) ,befi(C8) ,begi(C9) ,bghi(Cio) ,cghi(Ci) ,defi(C12),abc (C13) ,abf(CA4) ,ach (Ci5) ,aeh (C16) ,bef(C17) ,beg (C18),bei(C19) ,bgh(C20) ,bhi(C2j) ,cfi(C22) ,cgh(C23) ,cgi(C24),chi(C25) ,dgi(C26) ,egi(C2,) ,ghi(C28) ae(C29) bC(C30),

be (C31) ,bh (C32) ,Cf(C33) ,cg (C34) ,ci(C3s) ,dh (C36) ,di (cG),ef(C3) ,eg (C39) ,ei(C40) ,gh (C41) ,hi(C42) ,a(C43) ,d (C44) }

(8)Observing the implications of the above symbolic

compatibles, we get a set of IC's listed in Table X. InTable X, for each IC, all the symbolic compatibles inwhich the IC is contained are also listed. For example, theIC abc is a subset of abcf(C2), abch(Cs), and abc(C13).The symbolic compatible which occurs most in TableX contains in it the largest number of IC's. If only wecould eliminate such compatibles early enough, TableX would be simplified. Hence we form prime closed sets ofsymbolic compatibles in the order of their frequency ofoccurrence in Table X. The compatible C4 = abef occursmost (nine times). So, we will now find prime closed sets

generated by C4 ignoring in the process all those containingfive or more compatibles since the initialized cardinalityof the minimal closed cover is now five, as discussed inSection III. The closure class set E4 of C4 = abef is easilyobtained as

E4= {di,cf,ac,ch,ah,bg,bi,cij. (9)

Now, we obtain a Boolean expression in product of sumsform, one sum corresponding to each compatible con-tained in E4 together with C4 itself as one sum. For exam-ple, to account for the compatible di, we must have oneof the symbolic compatibles C12, C26, CS7 noted from TableX. Thus the initial expression will be

(C4) (CH2 + C26 + C37) (C2 + C6 + C22 + CS3) (C2 + CS + C1l

+ C15) (Ci + C3 + c11 + C15 + C23 + C25) (C3 + C5 + C15

+ C16) (C1 + C7 + C9 + C10 + C18 + C20) (Cl + C6 + C7

+ C8 + C9 + C1o + Clg + C21) (C1 + C8 + C7 + Cll + C22

+ C24 + C25 + CBS), (10)

where each ci is a binary variable corresponding to thecompatible Ci.

In the above expression C4 is in boldface to indicatethat the closure conditions of C4 have been taken care of.Noting that the first, second, third, sixth, and seventhsums do not contain any common terms, we concludeimmediately that all the prime closed sets generated byC4 contain five or more symbolic compatibles. Hence C4cani be eliminated from the set (8) and Table X. As aconsequence of striking off C4 from Table X, we observethat the compatible aef is no longer contained in anysymbolic compatible. Therefore, all those symbolic com-patibles which imply the compatible aef cannot be mem-bers of any covering machine with less than five states

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and may be eliminated. Thus C6 and C8, each of whichimplies aef, are also eliminated along with C4.

Next,-we generate prime closed sets starting from C2which occurs eight times in Table X.

E2= {di,bg,ch,be,ef,bi,ci,ahj. (11)

The initial Boolean product of sums is obtained as

(C2) (CL2 + C26 + C37) (C1 + C7 + C9 + c10 + c18 + c20)(Ci + C3 + Cl1 + C1i + C23 + C25) (C5 + C9 + C17 + C18

+ C19 + C31) (C12 + C17 + C88) (C1 + C7 + C9 + CIO + C1g

+ C21) (Cl + C7 + Cll + C22 + C24 + C25 + C35) (C3 + C5

+ c1i + c16). (12)In the above expression, we fiist observe that the first,second, third, and last sums do not contain any commonliterals and their product yields all four-literal terms. Aswe want to ignore all terms with five or more literals, wemay ignore in the remaining sums any literal which hasnot already occurred in the above four sums. After remov-ing the redundancies we obtain the expression

(c2) (C12) (Ci + CO) (C3 + CG + C1i + C16) (C5 + C9 + C18)(Ci+ Cs + cii). (13)

Again we observe that the first four sums in (13) yieldall four-literal terms and hence ignore cg, C18 in the fifthsum and the expression simplifies to

(C2) (Ci) (C12) (Ci + C7) (Cl + C3 + C15). (14)

Now, ignoring C3 and ci5 in the last sum we obtainC2CiC5C12. (15)

We now have to take into account the closure conditionsof the literals which are not boldface in the above expres-sion. The expression.obtained after simplification usingthe rules of Boolean algebra and an additional rule x * x = xis

C2CiCSCi2(C8 + C17) (16)in'which each term contains five literals. We thereforeconclude that the compatible C2 cannot generate anyprime closed set with less than five symbolic compatiblesand hence ignore C2. Thus, we eliminate C2 from the set(8) and Table X. As a consequence, the compatible acfwill not, be contained in any symbolic. compatible andhence C5 and C16, each of which implies aCf, are also elimi-nated. Elimination of C5 from Table X in turn resultsin the elimination of C12 in a similar way.By now the efficacy of this method and the utility of

Table X must be clear. Every elimination of a compatiblefrom Table X drastically reduces the computational workinvolved in the generation of prime closed sets for theremaining compatibles. Furthermore, elimination of onecompatible may give rise to further eliminations in achainl without any extra work. For example we have sofar generated prime closed sets for C4 and C2 only, buteliminated not only C2, C4 but C5, C6, C8, C12, C16 alsowithout any extra computation. The key to reduction of

computational work is that we must score out at everystage maximum possible entries in Table X. Anothertechnique also can be used to achieve this purpose. Oncewe establish that a symbolic compatible Ci implies anothersymbolic compatible Cj, we may cross out Ci from everyrow of Table X containing C0. For example, at a laterstage, we find that each of the symbolic compatiblesC, -C10, C18, and C20 implies C7 and we may cross out theformer compatibles from the bg row and bi row of Table X.

After we have eliminated C2 and with it C5, C12, Cle6we proceed to compute the prime closed sets for C1, whichoccurs six times in Table X. It will be seen that each ofthe-prime closed sets generated by Ci contains four sym-bolic compatibles. None of them cover all the states ofthe machine. Hence we eliminate C0 also.Next we examine C3, which occurs six times in Table

X, and find that it generates two prime closed sets, eachcontaining four symbolic compatibles. The working forC3 is shown below.

C3 = abch

E3 = {bef,di,af}

(17)

(18)

(C3) (C8 + C17) (C12 + C26 + C%7) (C14)

= C3C8Ci2C14 + C3C8Cl4C26 + C3C8Cl4C37 + C3C12Cl4Cl7

+ C3Cl4Cl7C26 + C3Cl4Cl7C37. (19)For each term in (19), we now take into account theclosure conditions imposed by all the literals which arenot boldface. The logical expressions thus obtained aresimplified using Boolean algebra. Ignoring terms contain-ing five or more literals, the final expression for C3 isobtained in the next step only as

C3Cl4C17C26 + C3Ci4Cl7C37. (20)

Now we have two prime closed sets in (20). The set givenby

{abch(C3) ,abf(Ci4) ,bef(C07) ,dgi(C26) I (21)

further covers all the states of the given machine andhence constitutes a covering machine with four statesonly. The second prime closed set indicated in (20) doesnot cover the state g and hence it is discarded.Now that we have a covering machine with four states

only, the new value of u' is 4 and for the remaining sym-bolic compatibles, we need generate prime closed setscontaining three or less compatibles only. This drasticallyreduces the computational work. It is easily verified thatno other prime closed set or union of prime closed setscontaining three or less compatibles covers all the statesof the machine. Hence the set (21) is a minimal closedcover for the machine of Table VIII.De Sarkar, Basu, a:nd Choudhury [9], [10] have given

a procedure for generating the prime closed sets. They,however, consider only the primary implications of eachcompatible and successively develop the expressions. Theworking shown in earlier paragraphs clearly demonstratesthat the computational work can be significantly reduced

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RAO AND BISWAS: MINIMIZATION OF ISSM 19

if: 1) the initial Boolean expression takes into accountthe entire closure class set of the compatible; and 2) thecompatibles are ordered in accordance with their frequencyof occurrence in Table X and the prime closed sets are

generated for the compatibles in that order. This point,however, does not apply to the generation of min-maxcover discussed in Section III, in which case we generatethe prime closed sets for only those MC's which containthe state occurring in least number of MC's.The entire procedure for obtaining a minimal closed

cover is summarized below.

Procedure 2: Finding a Minimal Closed Cover

Step 1: Obtain the set of symbolic compatibles followingProcedure 1 and tabulate along with their closure classsets.

Step 2: List all IC's in the table obtained in Step 1and prepare another table showing all symbolic compati-bles containing each IC. For convenience in referencing,let us call this the inclusion table.

Step 3: Let the index u' represent the initialized numberof states in a minimal covering machine found. Find thevalue of u' as discussed in Section III.

Step 4: Count the number of times each symboliccompatible occurs in the inclusion table and performthe following steps selecting compatibles in the order oftheir frequency of occurrence in the inclusion table.

Step 5: Compute prime closed sets generated by thesymbolic compatible selected, ignoring in the process all

those containing u' or, more compatibles. If no more

remain, eliminate the compatible and all compatiblesimplying it recursively. Repeat this step selecting the nextcompatible in order until all symbolic compatibles are

examined.Step 6: Test every prime closed set containing u' - 1

symbolic compatibles and ignore if it does not cover allthe states of the machine. If no more remain, eliminatethe symbolic compatible and all those implying it recur-

sively from the table of Step 1 and the inclusion table.Proceed to generate prime closed sets for the next com-

patible in order beginning from Step 5. If a prime closedset containing u' - 1 symbolic compatibles and coveringall the states of the machine is found, it constitutes thecurrent solution; record it. Decrease the value of u' by1 and repeat this step. Record the remaining prime closedsets if any and go to Step 5, choosing the next compatiblein order until all symbolic compatibles have been examined.

Step 7: Find a minimal cover using the prime closedsets recorded. If this contains less than u' compatibles, itis a minimal closed cover for the machine. Otherwise theinitial solution or the solution found in Step 6 if any,

whichever contains less number of compatibles, representsthe minimal closed cover.

All Minimal Closed Covers

We have, so far, discussed only the problem of finding

one minimal closed cover. If, for some reason, all minimalclosed covers are to be found, we may adopt the followingprocedure.

We may delete only weak compatibles from the list ofprimary compatibles because weak compatibles can neverparticipate in any minimal closed cover according tocorollary to Theorem 1. Further, if an excluded compatiblehas identical closure class set with that of one of theexcluding compatibles, it has to be retained in the list.This, in effect, means that we have to deal with a largernumber of compatibles and it is the price we pay to obtainall minimal closed covers.

Alternatively, we have to generate all prime closedsets of symbolic compatibles containing u' compatiblesalso until we have found a closed cover with less numberof compatibles when the value of u' is updated. Thisprocedure yields all solutions, each of which is a collectionof symbolic compatibles only. Then we have to investigatethe representations of each symbolic compatible. Forinstance, the symbolic compatible dgi(C26) of the machineof Table VIII represents the primary compatible degi.If dgi is now replaced by degi and/or abf is replaced byaf in the minimal closed cover given by the set (21), theyielded sets are still closed and cover the machine. Thuswe obtain four different solutions.

V. CONCLUSIONSThe principal contributions of this paper are: 1) A

simple and efficient method for the minimization of ISSM'sis developed. The central theme of the new method is toweed out such of those compatibles which cannot bemembers of any minimal closed cover and further ignorea substantial number of compatibles, elimination of whichdoes not preclude finding at least one minimal closedcover. The tests proposed to achieve this objective areextremely simple and are easily implemented at the veryfirst stage in the process of minimizing an ISSM. As aconsequence, we have to deal with a relatively small setof compatibles only. 2) A lower upper bound for thecardinality of the minimal closed cover is proposed and atechnique to find it has been presented. 3) Rank of acompatible is defined and it is shown that ordering thecompatibles in accordance with their rank reduces theamount of work involved. 4) An improved technique ofgenerating prime closed sets, which converges faster to thesolution, is proposed.The methods proposed in this paper are simple, efficient

and systematic. From the recursive nature of the pro-cedures developed, it is evident that the methods areprogrammable. These methods have been successfullyused by the authors for minimizing several machines ofmoderate size by hand computation.

REFERENCES[1] M. C. Paull and S. H. Unger, "Minimizing the number of states

in incompletely specified sequential switching functions,"IRE Trans. lectron. Comput., vol. EC-8, pp. 356-367, Sept.1959.

[21 S. H. Unger, "Flow table simplification-Some useful aids,"IEEE Trans. Electron. Comput. (Short Notes), vol. EC-14,pp. 472-475, June 1965.

[31 E. J. McCluskey, "Minimum-state sequential circuits for arestricted class of incompletely specified flow tables," BellSyst. Tech. J., pp. 1759-1768, Nov. 1962. r

[4] D. Pager, "Conditions for the existence of minimal closedcovers composed of maximal compatibles," IEEE Trans.Comput. (Short Notes), vol. C-20, pp. 450-452, Apr. 1971.

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[5] H. D. Ehrich, "A note on state minimization of a special classof incomplete sequential machines," IEEE Trans. Comput.(Short Notes), vol. C-21, pp. 500-502, May 1972.

[6] A. GrasseUi and F. Luccio, "A method for minimizing thenumber of internal states in incompletely specified sequentialnetworks," IEEE Trans. Electron. Comput., vol. EC-14, pp.350-359, June 1965.

[7] F. Luccio, "Extending the definition of prime compatibilityclasses of states in incomplete sequential machine reduction,"'IEEE Trans. Comput., vol. C-18, pp. 537-540, June 1969.

[8] W. S. Meisel, "A note on internal state minimization in in-completely specified sequential networks," IEEE Trans.Electron. Comput. (Short Notes), vol. EC-16, pp. 508-509,Aug. 1967.

[9] S. C. DeSarkar, A. K. Basu, and A. K. Choudhury, "Simpli-fication of incompletely specified flow tables with the helpof prime closed sets," IEEE Trans. Comput. (Short Notes),vol. C-18, pp. 953-956, Oct. 1969.

[10] -,"On the determination of irredundant prime closed sets,"IEEE Trans. Comput. (Short Notes), vol. C-20, pp. 933-938,Aug. 1971.

[11] R. G. Bennetts, J. L. Washington, and D. W. Lewin, "A com-puter algorithm for state table reductions," IERE Radio andElectron. Eng., vol. 42, pp. 513-520, Nov. 1972.

[12] J. Kela, "State minimization of incompletely specified se-quential machines,"' IEEE Trans. Comput., vol. C-19, pp.342-348, Apr. 1970.

[13] C.-C. Yang, "Closure partition method for minimizing in-complete sequential machines," IEEE Trans. Coomput., vol.C-22, pp. 1109-1122, Dec. 1973.

[14] N. N. Biswas, "State minimization of incompletely specifiedsequential machines," IEEE Trans. Comput., vol. C-23, pp.80-84, Jan. 1974.

[15] P. K. S. Roy and C. L. Sheng, "A decomposition method ofdetermining maximum compatibles," IEEE Trans. Comput.(Short Notes), vol. C-21, pp. 309-312, Mar. 1972.

[16] S. H. Unger, Asynchronous Sequential Circuits. New York:Wiley-Interscience, 1969.

117] Z. Kohavi, Switching and Finite Automata Theory. New York:McGraw-Hill, 1970.

C. V. S. Rao was born in ChiUakallu, AndhraPradesh, India, on October 19, 1934. Hereceived the B.Sc. degree in physics from theAndhra University, Waltair, Andhra Pradesh,India, in 1952, the D.M.I.T. degree in elec-tronics from the Madras Institute of Tech-nology, Madras, India, in 1955, and theM.Tech. degree in electrical engineeringfrom the Indian Institute of Technology,Kanpur, India, in 1966. He is currentlyworking for the Ph.D. degree at the School

of Automation, Indian Institute of Science, Bangalore, India.

From 1955 to 1956 he served the Andhra Electricity Departmentas a Junior Engineer in the charge of power-line carrier telephony.From 1956 to 1961 he was a Research Assistant in the InternationalGeophysical Year scheme under the Council of Scientific and In-dustrial Research (CSIR) and took active part in the research pro-gram, including the establishment of the Ionospheric SoundingStation at Trivandrum. In 1961, he became a Lecturer in the De-partment of Electronics and Communication Engineering, OsmaniaUniversity Engineering College, Hyderabad, India, where he hasbeen a Reader since 1966. He is presently on deputation at theSchool of Automation, Indian Institute of Science. His currentresearch interests include switching theory and fault-tolerant com-puting.

Nripendra N. Biswas (SM'65) was bornin Hili, West Bengal, India. He receivedthe B.Sc. (Hons.) degree in physics from theCalcutta University, Calcutta, India, in1948, the D.I.I.Sc. and A.I.I.Sc. degrees inelectrical communication engineering fromthe Indian Institute of Science, Bangalore,India, in 1952 and 1954, respectively,. andthe Ph.D. degree from the Indian Institute ofTechnology, Kharagpur, India, in 1960.In 1959-1960, he visited the United States

on a fellowship sponsored by the Governments of the United Statesand India, and attended Texas A & M University, College Station,and the University of Wisconsin, Madison.

In 1955, he joined the Department of Electronics and Communica-tion Engineering, University of Roorkee, Roorkee, India as a Lec-turer. He was made a Reader in 1958, an Associate Professor in1963, and a Professor in 1965. In 1961 he received the Khosla Re-search Award from the University of Roorkee for doing meritoriousresearch there. In 1967, he went to the United States as a Professorof Electrical Engineering with St. Louis University, St. Louis,Mo. In 1970, he returned to India to take up his current assignmentas a Professor of Computer Science with the Department of Elec-trical Communication Engineering, Indian Institute of Science,Bangalore. Author of a number of papers, he is also the author ofPrinciples of Telegraphy (listed in the bibliography of McGraw-Hill'sEncyclopedia of Science and Technology), Principles of Telephony, andPrinciples of Carrier Communication (Asia Publishing House, Bom-bay, India), and Essentials of Computer Programming in Fortran IV(Radiant Books, Bangalore, India).

Dr. Biswas is a member of the Institution of Electronics and Tele-communication Engineers and the Institution of Engineers (India).

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Minimization of Incompletely Specified Sequential Machines