a tabular minimization procedure for ternary switching functions

578 IEEE TRANSACTIONS ON ELECTRONIC COMPUTERS, VOL. EC-15, NO. 4, AUGUST, 1966

A Tabular Minimization Procedure forTernary Switching Functions

ROY D. MERRILL, JR., MEMBER, IEEE

Abstract-A tabular minimization procedure for temary switching binary variables of the functions, and all variables offunctions is developed. The theory is analogous to that used in the which the functions are independent. The results ob-McCluskey simplification method for Boolean functions. Using theternary function truth table, the procedure provides a systematic tamed f te pu nerally usefu for anmethod of applying a limited set of reduction rules in a converging system of ternary gating functions which might beprocess for obtaining a minimal irredundant form of the function. considered. This is illustrated with a system composedIt is shown how the procedure can be used to derive simplified of one set of functionally complete threshold gatingexpressions for arbitrary ternary functions in terms of a particularly functions.attractive system of threshold gating functions. The procedure has Several authors have considered the problem of sim-more general applications in providing a simple method of finding,for any given function, all binary variables of the function and all plification with respect to some chosen system of gatingvariables of which the function is independent. functions [8-10, and 14]. The procedure developed

in this paper considers switching function simplificationI. INTRODUCTION from a somewhat more general viewpoint so that the

ALTHOUGH the machine states as well as the results should be applicable in any logic system used fornumber representation of virtually every digital design.computer are implemented with binary logic, I I. DEFINITIONS

there are other systems such as the ternary and quinary Let the vector space Vn be the set of n-tuples Vi v

that also could be used. With the advancing electronictechnology, there are indications that ternary logic, par-

as. o)i= 1,u2, valresentingall possbe

ticularly, can be reliably and economically implemented. assignments of,the truth valuesa-1,s0,tandn+1ftoctheMoreover, there have been several instances reported f(x1, x2, , xn,), or simply, f(X), defines a partition ofrecently where ternary logic and switching networks Vn into the three truth sets f, f0, and f+ which arehave found practical application. For example, the mapped by f into -1, 0, and +1, respectively. Any twoSET'UN, a Russian stored-program computer, incor-porteth tenr nube syte inisaihmtcui of these three sets uniquely representf. Such a represen-

tation is the truth table form of f where each vector ofto take advantage of the relative ease of representing the two sets chosen constitutes a row of the table. Thenegative and positive numbers and the simplicity of form used in this discussion will beperforming round-off in the ternary representation 1 ],[2]. Also many proposed content addressable memories XIX2 . . . Xn f(X)utilize tristable switching and memory elements to ac- I+1complish masking and associated operations [3]. Ter-nary switching theory has been proposed as a useful +1means of designing hazard-free binary gating, and se-ries-parallel contact networks [4-5]. NASA has activelysupported the development of tristable fluid logic de-vices for hydraulic switching systems [6]. Still another _+area is the use of ternary logic redundancy in binary |networks to improve operating reliability [7]. _

This paper develops a procedure for simplifying ter-nary switching functions represented in truth table [form. The procedure is shown to be a simple systematicmethod of determining for the given function and all Later developments in this paper make extensive useits subfunctions (made up of subsets of the rows of the of the 1-variable ternary switching functions. Theseoriginal truth table) which have simplifications, all functions, called transforms and denoted by i,t'(x), i

=0O +1, * ±,+13, are given in TablelI.Manuscript received September 15, 1965; revised May 2, 1966.

The research reported in this paper was sponsored in part by the Air III THE SIMPLIFICATION PROBLEMForce, under Contract AF 33(657)-8777, and the Lockheed Inde-Xpendent Research Program.* *

Teathor is wit th Elcroi Scene Laoaoy Lochee Regardless of the syzstem of gating functions whichMissiles and Space CoDmpany15, Palo Alto, Calif. might be used in logical design, it is desirable to first

MERRILL: TABULAR MINIMIZATION PROCEDURES 579

TABLE ITRANSFORMS OF X

i~~~~ ~ ~ ~ ~~~~~~iX-13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13

x\

_ -0 + 0 + -0 + -0 + -0 + - 0 + -0 + - 0 + -0 +0 -0- 0 0 + + + -0 0 + + + ---0 0 0 + + +

+-0 0 0 000 0 0 0 +++++ + + + +

reduce the switching function to its simplest form. In Clearly, every switching function has a unique repre-this development the criteria for simplicity are the sentation in the set of functionals created using (1) andfollowing: consider all decompositions of the switching (2) and containing one functional for each row in thefunction whose - and + truth sets are included in the truth table form of the function. By using this represen-- and + truth sets, respectively, of the original func- tation we perform reduction operations similar to thetion.' (These subfunctions are generated, in effect, by Quine-McCluskey reduction technique used for binaryconsidering all combinations of the rows of the truth functions [11].table for the function.) For each decomposition, or sub- The functional representation off of Table IT is givenfunction, determine all input variables for which the in Table 111(a). For convenience only, the transformfunction is independent. Further, determine all binary subscripts are used in representing the functionals.input variables of the subfunction; that is, variables for Further, since the transforms of a functional uniquelywhich two of the three truth states are indistinguishable identify the truth set to which it is associated, we mayby the function. dispense with the truth table column designatingf and,The following example will illustrate the general ap- if desired, the dotted line separating the truth sets.

proach taken in the simplification procedure. Consider The corresponding representation for Table 11(b) isthe function in Table 11(a). Note thatf is independent given in Table III(b) where, since f is independent ofof xi [denoted by the asterisks in Table II(b) and (c)] xl, the transforms in the functionals forf are constantand dependent only on x2 being 0, and + 1 or -1 [de- transforms. The fact thatf does not distinguish betweennoted by the grouping of + 1 and -1 in parentheses in x2 having the truth values -1 and +1 is reflected byTable II(c)]. {110(x2) which maps + 1 into +1 and 0 into 0.

In order to reduce confusion where there may be many Still another form of simplification is possible as sug-subfunctions with simplifications, we express the ele- gested in Table 111(d). Specifically, it will be noted thatments of the rows in the truth table for f in terms of the functionalthe transforms {'i(x) such that for every ViEf+ or f [,6(V.) ,67(v-2), +s(v 3)]form afunctional (an n-tuple of transforms): (V= (0, ±1, +1) for all Vi8f

gi(X) = [gi&(xl), gi2(x2), . . ., gin(xn)] = (0, -1, -1) for all Vief-(where, for VjCf+, ^ (0, +1, +1) nor (0, -1, -1) otherwise.

(xi) v ii = -1 Thus, this single functional uniquely represents thegij(Xj) 3(X)X vij = 0 , allj (1) original switching function. It will be shown later that,

I1 (X)X ij = +1in one system of gating functions at least, this simplifiedform can result in a most economical network realiza-

and where, for ViE ft, tion. The important point to note, however, is the con-

Vi(xj),~~ vj = - 1ciseness of representing the simplified form.If_l(xj) vij =-1It is easy to verify that, in general, a functional gi(X)

gi1(xj) = 41i3(x) '~z vi = 0 ,all j. (2) uniquely defines a partition of Vn into the three sets

L1&s(X;)X ij = + 1 gj+, gj, and gil. Each VkCgi+ and gj- is mapped by gjinto an n-tuple, all of whose transforms not identically

It may be noted that transforms t'(x) = ±1^ +3 ±9, zero are +1 and -1, respectively. For each VkCgi°,are two-valued. either at least two transforms of gj( Vk) not identically

zero will be opposite in sign, or at least, one transform1 Any two of the three truth sets can be considered as the truth of g~( Vk) not identically zero will take the truth value 0.

table representation for a function. Throughout this discussion the Equation (3) illustrates this functional partitioning- and + truth sets are arbitrarily chosen as this representation. Itshould be clear in the development to follow that the results obtained property for the example in Table III. There are severalare equally applicable to any of the three truth table forms represent- additional properties of both functionals and switchinging either of the functions in each of the pairs: [f, +_'8(f)], [+6(f), fntoswihrqiecnieain^IK6(f)], and [+2(f), +_2(f)I.fntoswhc eur cnleazn

580 IEEE TRANSACTIONS ON ELECTRONIC COMPUTERS AUGUST

TABLE IISIMPLIFICATION OF f

- + -I+0+-i+1++- I+±I_+- I~+ I

0 + + F((±)I _ o 0

-L+ ---'-- L * 0 (c)o 0 ±+ (b)

(a)

TABLE III only if (gj, hj) is one of the pairs in Table IV.SIMPLIFIED FUNCTIONAL REPRESENTATION FORf The proof of Theorem 1 in the Appendix provides the

3 9 1 following method of finding l(X): select li=gi for every9 9 1 i5zj and select Ij as a function of (gj, hj) from Table IV.F

9 1It will be clear later that only the combinations of Table

1 1 IV(a) need be considered in carrying out the simplifica-1 1 1 F 13 9 1

1tion procedure. The simplifications derived from Table

3 1 131 F 131011 IV(b) will always be realized by two iterative applica--13 -3 -9 L -1-3 _-_3_-_9 tions using Table IV(a), together with Theorem 2. Also,

-3 -3 -9 b)-c)1--3- 9- 3 -3 - all combinations in Table IV(c) are degenerate in that-1 -3 -9 [O 7 -8] every case either g < h or h< g.

-9 -3 -9 (d) Theorem 2(a) Let g(X) and h(X) be two functionals such that gi

z, hi for all i. There exists a functional 1(X) such thatIV. COVERING RELATION AND 1+ g+Uh+ and -=g-Uh- if and only if each (gj, hi) is

REDUCTION THEOREMS one of the pairs of transforms in Table V.

Let g, h, and I each be ternary functionals or switching The Theorem 2 proof in the Appendix provides thefunctions with -, +, and 0 truth sets: g-, h-, 1-; g+, h+, following method of finding l(X): select each 1i as al+; and g°, h0, 10, respectively. If for all ViC Vn, h+Dg+ function of (gi, hi), for all i, from Table V.and h-Dg-, then h is said to cover g, denoted by h>g. V. SIMPLIFICATION PROCEDUREThis covering relation is illustrated in Table III where,for example, Using Theorems 1 and 2, a simplification procedure

for ternary switching functions can be systematized in13(x1), 41o(x2), 4'1(x3)j > [{3i1(X1), i1(x2), #1(x3)] a manner analogous to the Quine-McCluskey procedure

Clearly, by this definition, if l> g and l> h, then g+Qh- [II ]. First, a set of functionals, one for each Vief+ and=d/, the null set. Also, g-fh+=4. The covering relation f-, is formed according to (1) and (2). These are calledis transitive, that is, if I > h and h > g, then I > g. More- original functionals.over, it is easily shown that if hI> g and g> h, then h+ Next, the original functionals are grouped according= g+, h- = g-, and h° = g°. This property is written h'-g. to their corresponding truth sets and compared by pairsIn Table III, note that f-[\o0(xi), +t7(x2), 4'8(x3)]. for reductions possible under Theorem 1. The new

It is now possible to formally present the reduction covering functionals generated by this process are againrules and the simplification procedure illustrated in grouped according to the truth set being covered andTable III. Theorem 1 is again applied. This procedure is repeated

Theorem ~~~~~~~~~~~iteratively until no new functionals can be created. ThisTheorem ] ~~~~~~~~~stepis greatly simplified by noting from the Theorem 1

Let g(X) = [g1(x2), g2(x2), , gn(xn) ] and h(X) proof that only the first six combinations in Table JV(a)- [hi(xD), h2(x2), * , hn (xn) ] be two functionals such arise for ± truth set original functionals and their subse-that g, hs for every i except i =j. There exists a func- quently created covering functionals. Similarly, only thetional t(X) such that l+=g+Uh+ and l-=g-Uh- if and second group of six combinations in Table IV(a) arise

1966 MERRILL: TABULAR MINIMIZATION PROCEDURES 581

TABLE IV TABLE VTHEOREM 1 REDUCTION RULES THEOREM 2 REDUCTION RULE

(gj, hi) (g1, hi) (gi, hi) 1; (gi, hi) 1T (gi, hi) Uior 1, or

(hij, gj) (hj, gj) VI1 4-3 41-2 43 VI-9 V-6 VI9 41-4 45.I 'P-9 4-8 'P3 'P-10 'P-7 ;'4 'P-9 '-5

44 V13 414 #3 V2 V-6 4-7 4'1 4'-12 4-11 49 41-1 V8 V110 V-3 474441 9 4,10 432 418 411 V13 41-1 V12 49 IP-3 46 4112 VI-1 4111'P3 4P9 4112 'P6 VI8 'P5 I113 t'-13 VI04'4 '10 4P13 '6 'P-2 '7

'P4 P12 'P13 '1-6 '-8s P-5

'P10 4'12 V13 '-8 1-2 '-il

#2 VI -3'4 4 #4 'I5 '-5 'P0 for - truth set original functionals and their subse-'P-1 4'_9 '-10 '7 'P-7 'P04'-3 'P- 'P-12 p11 -11 40 quently created covering functionals. The example in'P-4 '-10i -13 'P13 'P-13 'Ps Table VI illustrates this step. Column 1 contains theVI-4 4-12 VI-13'_10 P-1i2 '-13 original functionals and columns 2, 3, and 4 contain

covering functionals created using Theorem 1, one for(a) each iteration. (In the simplification tableau, e.g., Table

VI, covered functionals are checked and redundantfunctionals are struck-out.)

In the third step, each functional of the + truth set(g2, hi) (gi, hi) (gi, hj) is compared with every functional of the - truth set

or IU or Ij or IU for reductions possible under Theorem 2. The func-(hi, gi) (hj, gj) (hi, gi) tionals created under this step are grouped separately

{'P ' 'P7 'P3 'P-1 'P2 'P-i 'P12 'Pll from those of Steps 1 and 2 in carrying out the next step.'Pi '112 'P13 '3 'P-8 'P-5 #'-2 'P9 '7

1'P 'P-3 'P-2 '3 'P-9 'P-6 'P-2 'P-9s '-11 Columns 5, 6, 7, and 8 of Table VI contain this grouping'I 1 '-6 -5 'P3 'P-10 'P-7 '-3 'P-8 '-11 for the example as created from columns 1, 2, 3, and 4,44 V;9 V-8 V14 /-9 ; -5 41-3 +-10 4'-13'P1 'P-12 'P-11 '-1 'P-6 -7 '-3 '1Pi -2 respectively.'P2 1P-9 'P-7 Q'P- 1'-12 'P-13 'P-3 I8 'P5 In the fourth step, all functionals created in Step 3'P2 'P9 '11 'P-1i 3 'P2 '-3 149 'P6'P3 'P8 'P11 'P-1 'P6 'P5 '-3 '1s0 '7 are compared by pairs using Theorem 1. This step is'P3 'P 'P113 'P-1 'P9 3'P 'P-4 '1P9 'P greatly simplified by noting that in the first iteration

(b) only transforms in the third group of six combinations of(b) Table IV(a) arise. In the second and subsequent itera-

tion in this step only transforms in the third and fourth,or final, group of combinations in Table IV(a) arise. Itis easily verified by enumeration that at the completion

(gi, hi) (gi, hi) (gi, hi) of Step 4, the set of all unchecked functionals are ex-or or Ij or

(hi, gj) (hj, gi) (hi, g3) actly those obtained if comparisons are made which re-

'Ps 'P-13 'Ps 'Ps 'P5 'Pi 'Pl 'P12 'P12 quire using Table IV(b). However, sometimes the cover-40 4;-13 V'V0 s8 5 V/5 1 -3 4 -12 4+-12 . . . .. .

'P0 '1-12 V0' '8 'Pu11 '1P 1 -3 '1-13 'P13 ings given in Table IV(c) will greatly simplify the reduc-'P9 ';5 'P5 'P-3 'P5 |'5 tion procedure. For example, in Table VI, each func-'P9 'Ps6 'Ps6 -3 'P6 '16'P9 ' 7 '17 'P-3 'P7 'P7 tional with one or more i/o covers all functionals which

'P0 4'+13 'P0 'P9 'P8 'P8 'P-4 '1-13 'P-13 have the same corresponding transforms except in the'P 'P14 '14 'P9 'P9 'P+9 -4 '1P5 '5'PI 'P7 P7 'P9 '1P0o 'P0 'P-6 '5 '1P5 position of the iPo. Hence, we designate these latter'P 'Pis 10's 'P9 'P11 'P11 '-6 '-7 'P-7 functionals with small crosses as in columns 5, 6, and'P1 '1P3 '113 'P9 '112 '112 'P-8 'P5 'P-.'Pl '1-2 'P-2 ',9 'P13 'P13 'P-8 1P-11 '11 7, since they need not be considered further as a group.'P 'P-5 'P-5 '1P10 '17 'P7 4'-9 'P-i5 'P- Individually, they must be compared with the remaining'P 'P-8 'P-8 '1P10 V 13 'P13 'P-9 'P-6 'P-6'P 'P- VPa P112 1P11 '1P11 '1-9 1-7 'P-7 uncrossed functionals, however, since additional simpli-'P3 'Pl 'Pl1'12 '113 1'13 '1_9 'P-8 'P-8 fication may be possible under Theorem 1.'P3 1'12 '112 '1-1 1P-4 1P-4 'P_9 1P_9 'P-9

1'3 '113 '113 'P1-1 1-7 'P-7 '1_9 'P_10 'P-10 Following the exhaustive application of Theorem 1'P^63 'P-i '-1 'P-io 'P-is '1-9 ' -u11 -1u in Step 4, it should be clear that the unchecked and'P3 'PQ-6 'PA-s 'P1-s 1'-13 1'P-i3 '1P-9 1'P-12 1'P-i21''PQ3 1'P-7 1'P-7 'P1-u 'P2 'PQ2 P- 'P113 'PA-aa uncrossed functionals are pri,me functionals; that iS,'Ps4 'P+13 'P{13 'PI-i 'P'PS'P5-is 'P-7 1'P-v functionals which are covered only by the original'P,4 'P- 'P-i 'P_51-i 'P8 'P 'P1-is 1'P-13 'PA-ia'P+6 'P 'Pi 'P51-i 'P1 'Pu1'P112 'PQ-ai 'Pui1 switching function itself. Further, the set of prime'P 'P17 'P+7 'P1-a 'P+-u 'P+-u1 '-12 'P;-ia 'P-13 functionals together cover the original functionals

(c) ~~~~~~created using (1) and (2). In Table VI, [(13, 3, 1), (-2,_________________________ 5, -5), and (7, 0, -8)] is the set of all prime func-


TABLE VISIMPLIFICATION PROCEDURE EXAMPLE

1 2 3 4

V9 3 1 V9 4 1 9 13 1 V10 13 1v9 1 1 A / 9 12 1V 9 9 1 V10 3 1 v10 4 1V 1 9 1 12 3 1 z - =z LV 1 3 1 v 9 10 1 V10 12 1v 1 1 1 v10 1 1V 1 9 3 V 10 9 1V 3 3 1 V 1 12 1 13 3 1

v 1 10 1V 4 3 1V 1 9 4 v10 10 1v 1 4 1

V 1 13 1

V-3 -3 -9 \/-3 -4 -9 V/-3 -13 -9A/-3 -1 -9 V/-3 -12 -9A/-3 -9 -9 V/-3 -10 -9

5 6 7 8

X 6 2 -8 X 6 -5 -8 v6 0 -8 7 0 -8X 6 -6 -8 X 6 11 -8 X 7 -5 -8X 6 -7 -8 X 7 2 -8 X 7 11 -8X 6 -2 -8 X 7 -6 -8 X 7 7 -8X 6 -8 -8 X 7 -7 -8 V-2 0 -8X 6 -11 -8 X 6 7 -8X 6 6 -8 X 7 -2 -8X 6 8 -8 X 7 -8 -8X 6 5 -8 X 7 6 -8X-2 6 -8 X 7 8 -8X-2 8 -8 X-2 11 -8X-2 5 -8 X-2 7 -8X-2 2 -8 V-2 6 -5X-2 -6 -8 V-2 8 -5X-2 -7 -8 -2 5 -5X-2 -2 -8 X-2 -5 -8X-2 -8 -8V-2 6 -6V-2 8 -6V-2 5 -6

tionals. Moreover, all three are essential; that is, foreach of the prime functionals there exists at least oneoriginal functional which is not covered by any otherprime functional. This can be seen in carrying out thefifth, and final, step of the simplification procedure: con- TABLE VIIstruction of the prime functional covering table as in PRIME FUNCTIONAL COVERING TABLETable VII. Generally, all prime functionals will not be FOR EXAMPLE OF TABLE VIessential, hence the covering table is used to select anirredundant set of prime functionals. Such a set has the PrimeFunctionalsproperty that removal of any one of the functionalsfrom the set results in at least one of the original func- iotionals not being covered.

Since Theorems I and 2 provide all possible ways of e 0generating a new covering functional given two func- Itionals, the simplification procedure produces all possible 9 3 1 x x

functionals each of rhich covers a subset of the original I 1 xfunctionals. Moreover, by checking a functional only if 9 1 xxat least one other functional covers it, we are assured .+ 1 3 1 x xthat by exhaustive application of Theorems 1 and 2, the 9 3 x|set of unchecked functionals will collectively cover allX 3 3 1 xthe original functionals. It follows that by the procedure : 3 -3 -9 x xgiven, one obtains the set of all prime functionals. Using o -3 -1 -9 x xthe covering table then provides the means of selecting -3 -9 -9Xany of the possible irredundant sets of prime functionals._________________________


It should also be recognized that one must consider functionally complete. Hanson has also shown func-the three combinations of the three truth sets taken two tional completeness by deriving a general expansionat a time and apply the simplification procedure to each theorem for the group [9]. The general representationbefore it is possible to select the irredundant set(s) hav- for f(X) developed by Hanson [9] is equivalent to (5).ing a minimal number of prime functionals.1 It is important to note that the set of functions ob-The irredundant set chosen to represent the switching tained by (4) for all original functions of f(X) has the

function in logical design will depend on the group of basic property that if for some VkC Vngating functions used. In the next section, a particular Fi(Vk) F. 0, 1 < i < r.group of gating functions is considered and it is shownhow the truth table simplification procedure can be Then, eitherutilized in obtaining efficient realizations. Fj(Vk) = Fi(Vk) or Fj(Vk) = 0, j = 1, 2, . . , r.

VI. REALIZING TERNARY SWITCHING Further, this property holds for any set of functionalsFUNCTIONS WITH THRESHOLD LOGIC generated for f(X) in the simplification procedure. To

Threshold switching functions appear promising as a see this, note that each functional covers some subset ofsource of gating functions for logical design because of the original functionals. But, from the definition oftheir algebraic and switching theoretic properties, as covering, if the functionals d, g, and h have the relation-well as the relative simplicity of their circuit mechaniza- ship d < h and g < h, then ViCd+ (or g+) implies VjCh+tions for reasonable numbers of inputs [9], [12], and and VjCd- (or g-) implies ViCh-. Hence, if Vich+,[13]. The group of threshold gating functions2 then either ViC-d+ (or g+) or ViGd0 (or g°). The anal-

ogous situation also holds for ViEh-. Thus, in (4) theTY1Y2* ,YnY;2, Yn-n functions for d and g, D(X), and G(X) have the above

property; specifically, for some VkC Vn if D(Vk) 0,and the transform3ofs(y) will be used as one example then either G(Vk) =D(Vk) or G(Vk) =0.for the application of the truth table simplification pro- It follows, therefore, that if (gl, g2, . g,) is a set ofcedure. irredundant prime functionals for f(X), then

Every ternary function f(X) can be expressed interms of the chosen gating functions as follows: Let f(X) TGl(X),.G2(X ) )fl, f2, * * , f, be the original functionals of f. From the 1( ) = l l 1; /-(definitions of (1) and (2), it is easily verified that the where the Gi are obtained from (4) given the gi. Fortruth sets of fi are identically the truth sets of example, the function of Table II which is covered by

g= [{0(xl), {7(x2), 41s(x3)], has the threshold realizationFi(X) - Tf'?:j') (4) G 7 (X2) X3

where the transforms fij(xj) have the realizations of Ti;i1_i = CTable VIII which are expressed in terms of the chosen Similarly, the function in Table VI has the realizationgating functions.

G1,G2,G3f(X) then has the following canonical representation: Ti,i i; /-

whereF1(X),F2(X), -* *,F,(X)f(X) = T1,1,.. *,1;1/-1 .(5)¢/3 (x2) 411 (X3)

It is easily verified that the canonical expression (5) G1 = Tl,1;2/-2is unique and the selected group of gating functions is G T-2(T1),1 (X3)P-(3

2 The author's definition for threshold functions [13] is used:f(X) andis a threshold function if there exists weights (wl, w2, * wn) andthresholds (t+, tL) such that G (X1),X3

57n C3 = Tl,1; 2/-2X+1 X4 xwiwi > t

For the examples of Tables II and VI, the primef(X) = n-1Ez < functionals are all essential. In general, this will not bexii the case. The function of Table IX illustrates the more

l0 otherwise general situation where not all prime functionals arewhich is written essential: columns of the covering table with circled

f(X) - /tlX,2w*z t+/t_ entries correspond to essential prime functionals.43In most threshold function mechanizations /'8(x) can he real- There are five distinct irredundant prime functional sets

ized directly, for example, in Table IX. The set { (3, 9, 10), (2, 6, -5), (-2, -6, 7),

and :IV :'v2 , The simplification procedure tableau for this example can bef_s(T'.l>i.;'+jus ) = T_'Ij1,_ ,- ,_;---LIt+ found on p. B-227 of [12].


TABLE VIII TABLE XNONTRIVIAL TRANSFORM THRESHOLD REALIZATIONS* AN ADDITIONAL SIMPLIFICATION FOR THRESHOLD REALIZATIONS

4,1(x) = _/Txl ;12/-2] 7(x) = T1,2,4;_ i Irredundant Sets of Functionals

(I'2(X) = x withoit (-+ 1,0, -1) with (+ 1, 0,-i1)

t'3(X) = Ti,;111 I 1)9g(x) =Q_jT1,1;2/-2 I414(X) T 4,jo(x) = Tx,_1(x) C\

l6(x) TT1,'2,l_ 4,11(x) = Tl2' _ | n o o |x

4q(x) = V)12(X) = Tl ;ii_ 3 9_ 1 x --x_________________________________3 9 1 x x

3 9 3 x xIn general, 4ti(x) =ts8[1,j(x)]; hence only one-half the nontrivial . 3 99 x x1-place functions need be given. 9 9 1 x x

9 1 1 x xTABLE IX Cd (9 3 1) x

-9 -3 -9 x xCOVERING TABLE FOR THE THIRD EXAMPLE . -9 -3 -3 x x

l (-9 -3 -1) x

Prime Functionals

-I . -

demonstrates one approach of doing this to simplify therepresentation (6) for the switching function. Note thatby utilizing the 0 truth set vector (+1, O, -1), it is

~ necessary to form two functionals; one by (1) and theX - other by (2). The simplification procedure then is carried

out as before, however, in using the covering table to1,3 1, 1 0 x xX obtain the best irredundant set of functionals for form-

9, 1,I ing (6), it is necessary to include as many functionals1, 3, 1 x X ® 1 covering (9, 3, 1) as there are covering (-9, -3, -1).3, 3, 1 0 In this way we are assured that (6) evaluated for9, 3, 1 x X=(+1, O, -1) will be 0. However, further research

c 1, 9, 1 will be required before a unified method can be de-r.

, , veloped.9)9,1 I

C| 1, 9' 3 0 VII. CONCLUSIONS1, 3,1 90

. 3, 9, 9 0 In this paper, a truth table simplification procedureo -3, -3, -3 X was developed by which irredundant sets of prime func-

-3, -9, -3 x x x (0 tionals can be obtained having minimal numbers of

-3, -3 93 (i) prime functionals. Further, it was shown how the pro--9, -9, -9 0 cedure can be used to derive simplified expressions for

_______________ _____________________________- an arbitrary ternary switching function in terms of aparticular group of threshold gating functions. The

(-5, -5, -11), (13, 13, 1), and (7, 7, -11)} has the procedure has more general application, also, in provid-minimum cost function: the number of nonconstant ing a simple, systematic method of finding all binaryarguments in the functionals plus the number of func- variables in the given function and all subfunctionstionals with more than one argument. This cost function (which can be found using subsets of the rows of therepresents the number of threshold gate inputs in (6) for original truth table) as well as all variables of whicha switching function without considering the transform these functions are individually independent.realizations. If the latter realizations merit inclusion,still other cost functions might be used. APNI

THEOREM PROOFSAn Additional Consideration. It is known that through Proof of Theorem 1the simplification procedure alone, there is no assuranceofotiig a thehl relzto wit a miia cos To prove an I exists if (g1, h1) occurs in Table IV we

function of the above form. For example, in the simpli- consider two cases.fication procedure rather than limit the original func- Case 1: If either gj or hJz equals ,60 then g or h, respec-tionals to those generated from the truth table by (1) tively, is independent of x1. Suppose gj =f'o, then it fol-and (2), suppose we also consider functionals generated lows that g>h. Hence by selecting l=g the theorem isfrom vectors in the 0) truth set. The example in Table X satisfied. Similarly, if h>g select I =/h.


Case 2: If neither gj nor hj equals 41o, then from Table Case VkCg+ Vocg- Vkeh+ VkCh-IV it is known that every Vk for which gj(vkj) # Oimplies 1that either gj(Vkj) = h3(Vki) or h1(vkj) =0. If gj(vkJ) =0, b 0 o 0 0then there are no restrictions on h1(vkj). Therefore, c 0 0 0 1

Vk(Eg° implies either g(Vk) =Ih(Vk) or VkEEh. Further,VkCg0 implies either g(VkI) =h(Vk) or VkEIh°. Hence by where 1_yesand 04 noselecting li=gi for all i #j, and selecting Ij such that For case (a), VkCg+and VkEPh. Thusgi(vki) = + 1 forallgJ(vki) #0 implies IJ(VkJ)=g9(Vkj) and gj(vkj)=O implies i. But by the choice of i given in Table V, VkCg+implieslj(Vkj) =hj(vkj), the resulting I satisfies the "if" part of that li(Vki) =+1 for all i ecxept where gi=C13 in whichthe theorem. The (gj, hj) enumerated in Table IV case li(Vki) O. Hence, VkCl+ for those Vk giving rise tocovers only combinations satisfying conditions in each case (b), and conversely.of these cases. For case (b), VkCgO, h°. This implies there exist a gpTo prove an I exists only if (gj, hj) occurs in Table IV, and an hq such that g,(Vk,)#+1 and hq(Vkq)#-. By

we first note that the existence of an l(X) such that the choice of Ii 1,(Vk,)X+1 and 1,(Vk)X-1 Hence+ =g+AJh+ and 1- =g-Uh- implies g+nGh- = and Vke 1Ifor all Vkof case (b). Conversely, when VkCIOthereg2\h± =q$. Hence the criterion under which two trans- exist at least two transforms I, and Iq not identicallyforms may appear as gj and hj is that either gj(x) = hj(x) zero such that -2 <lp(vkp) +l(Vkq) < +2. Therefore, by

the choice of the Ii, Vkegoy ho.or gj(x) =0, or h1(x) =0 for all truth values of x. The For case (c), Vk1go and VkCh- which implies hi(Vki)following itemized lists exhaust all possibilities under . for thosethis criterion as can be verified from Table I. V .Igoand imSiial, if Vs underthe cnto of

Vkeg° and h-. Similarly, if VkEl- under the conditions of1) 1/i and l/-12, 41-11, 1-9, -8, 41-6, l/5, 41/-3, 1/-2, 410, this case hi(Vki) = -1 and there exist at least two trans-

13/, 14, 1/6,1 7, /19, 1/,10i2, or 113- forms gp and gq such that -2 <gp(Vkp) +gq(Vkq) <+2.2) 1/2 and 41-lo, V_9, '6, 1-1, 410, 13, 18, 19,/',, or 4112. Hence, VkCl- implies VkEh- and g.0 Since these three3) 13 and 41-1i, 9-), 1-8, 1/7', L6, 1-51, +1-, 0, 411, 1/2, cases are exhaustive on Vn, the proof is complete.

1/14, 41/8, 41/9, 1,t'i0, 1/'ii,1,1/12, or 1/113. ACKNOWLEDGMENT4) {4 and AI-9, 1-8, 4-6, 4-5, VIO 411, /31 V/99 +10X 4112, or

V/13. The author gratefully acknowledges the assistance of5) {5and 4, -3,1-1,10, 6,18,or /9. Dr. C. E. Duncan, Senior Member of the Lockheed6 /6 and 41/4, 41/3, l11, 1/1, \1, 5

Missiles and Space Company, Electronic Sciences6) V17 and./1, /2, 1i, 11, 1i, 19, 110o Laboratory, and the helpful advice and consultation of7) 1/7 andt 1/13, l/12, 1/10,/1l,1/16,1/19, or AoOFr8) V18 and 1/-4, 1/-3, /-14, /01, 1/'3, 5, /16, 1/9, /11i, or 12-

0. Firschein.

9) 41/9 and 1/'4, 1-3, 1/-2, 1-1/,;1,/,;1/,;13, 1//4, 415, 1/6, 1/7X REFERENCES18, it'10 311, i12, or 113- [1] Y. A. Zhogolev, "The order code and an interpretative system

for the SET'UN computer," USSR Comp. Math. and Math.10) 1/io, and 1/-3, 1/-2, 1/0, 11, 13, 4/4, 1, /7, /9, 1CO, 1/12, or Phys., vol. 3, Oxford: Pergamon, 1962, pp. 563-578.

[2] R. D. Merrill, "Ternary logic in digital computers," presented atthe 1965 SHARE Design Automation Workshop, Atlantic City,

11) VI1, and 1/,, 10o, /2, V13, 18/, V19, or 112- N. J., to be published.[3] M. H. Lewin, "Retrieval of ordered lists from a content-address-

12) C12 and /-1, 1/O, V1l, /2, 13, 1/4, 18, /9, /10, 41/1, or 1/13- able memory," RCA Rev., vol. 23, pp. 215-229, June 1962.[4] M. Yoeli and S. Rinen, "Application of ternary algebra to the

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obtained from the above tabulations by multiplying all [5] E. B. Eichelberger, "Hazard detection in combinational andtransform subscripts by sequential switching circuits," IBM J. Res. and Dev., vol. 9, pp.

90-99, March 1965.A comparison between Table IV and the above lists [6] T. D. Reader and R. J. Quigley, "Research and development in

fluid logic elements," Univac Monthly Progress Repts., nos. 1-8,will verify that all allowable combinations of gj and hj NAS8-11021, July 1, 1963-February 29, 1964.have been included. Again, I is selected as specified in [7] V. I. Varshavskii, "Ternary majority logic," Avtomatika i

Telemekhanika, vol. 25, pp. 673-684, May 1964.Cases 1 and 2. Since this proves the "only if " part of the [8] J. Santos and H. Arange, "On the analysis and synthesis of three-

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The exitencef an 1X) suchthat 1 = g~Yh~ and circuits," IEEE Trans. on Electronic Computers, vol. EC-14, pp.The exstenceof an (X) suh thatl+=g+U+ and 19-29, February 1965.1-= g-J It implies that g+QhI- = j and g-Q'h+t = ct. [11] E. J. McCluskey, "Minimization of Boolean functions," Bell

. , . ,. . ~~~~~~~~~~~Sys.Tech. J., pp. 1417-1443, November 1956.Further, 81/', 1/', 1/'4 1/1s, 1/'i, +1/' iS the set of all trans- [12] Y. A. Keir, R. D. Merrill, and C. L. Thornton, "Ternary logic,"forms for which 1/'(x) >0O for all truth values of x, and in Research on Automatic Computer Electronics, vol. 2, Lockheed

, , . ' ~~~~~~~~~~~~~~Missilesand Space Co., Palo Alto, Calif., RTD-TDR-63-4173,l 24l 1-34-4+-9 -1, u-12 uJi3 s the set of all trans- pp. B-151-B-231, October 31, 1963.

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only one of the following three cases can arise for each [141 J. Santos, H. Arango, and F. Lorenzo, "Threshold synthesis ofTr Trn ~~~~~~~~~~~~~~~ternarydigital systems," IEEE Trans. on Electronic Computersvk; V". (Short Notes), vol. EC-is, pp. 105-107, February 1966.

a tabular minimization procedure for ternary switching functions

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