IEEE TRANSACTIONS ON COMPUTERS, VOL. c-35, NO. 2, FEBRUARY 1986

Regular Ternary Logic Functions- Ternary LogicFunctions Suitable for Treating Ambiguity

MASAO MUKAIDONO

Abstract-A special group of ternary functions, called regular ternarylogic functions, are defined. These functions are useful in switching theory,programming languages, algorithm theory, and many other fields -if weare concerned with the indefinite state in such fields. This correspondencedescribes the fundamental properties and representations of the regularternary logic functions.

Index Terms -Ambiguity, canonical disjunctive form, detecting haz-ards, fail-safe logic, functional completeness, indefinite state, Kleene alge-bra, regular ternary logic function, ternary function.

I. INTRODUCTIONLogics and algorithms are generally based on the two-valued

principle, that is, true or false, or yes or no. However, in somecases, we experience a state in which it is impossible or unnecessaryto decide true or false. For example, each value of a signal in a logiccircuit, which takes 0 or 1 in a steady state, changes from 0 to 1 orfrom 1 to 0 in a transient state; that is, it is impossible to decidewhether the value is 0 or 1. The initial states of sequential circuitsis another example where it is difficult to know whether the valueis 0 or 1 in many cases. Furthermore, it may be said that an algorithmdoes not stop for a given data, or that some data are not applicableto the algorithm. In the cases mentioned above, we may use ternarylogic (three-valued logic), instead of binary logic (two-valued logic),in which the third truth value is introduced to represent an ambigu-ous state apart from true and false.On the other hand, ternary functions have been studied for some

time from the standpoint of their functional completeness or repre-sentation. When applying ternary functions to various fields ofengineering, we seldom use all the ternary functions; instead, weemploy only some subsets, which have special properties or mean-ings. In fact, Mukaidono [1] has introduced some special subsets ofternary functions called regular, normal, and uniform, respectively,which have important and useful properties.

This correspondence discusses in detail a special group of ternaryfunctions, called regular ternary logic functions and introducedfirstly in Mukaidono [1], which are significant if the third truthvalue is considered to represent an ambiguous state. That is, regularternary logic functions, which will be studied in this correspondence,are suitable for treating ambiguity. In Section II, we shall introduceregular ternary logic functions from three different standpoints andshow that they are all the same definitions. A representation of regu-lar ternary logic functions is discussed in Section III, and theiraxioms and functional completeness are explained in Section IV.Finally, in Section V, the canonical form, which is determineduniquely for any given regular ternary logic function, is studied.

II. REGULAR TERNARY LoGic FUNCTIONSA ternary function is defined as follows, using the symbol 1/2 as

the third truth value in contrast to 0 (false) and 1 (true): LettingV = {0, 1/2, 1}, a n variable ternary function F is defined to be amapping from Vn to V:

F: Vns- V.

Here, we will interpret the truth value 1/2 as "uncertain 0 or 1," thatis, "ambiguous." Then, we can define the truth tables of logic

Manuscript received June 28, 1985; revised September 20, 1985.The author is with the Departement of Electronics and Communication,

Meiji University, Tama-ku, Kawasaki-shi 214 Japan.IEEE Log Number 8406762.

TABLE ITRUrH TABLES OF TERNARY AND, OR, AND NOT

0. 112 _ A 0 112 1 0 112L

_ 1 _ 1 .O12 12 LJJ142

E NOT : Aii12J

AND: A-B OR: A+B

connectives AND('), OR(+), and NOT(-) as in Table I. Also, let usconsider the ternary functions defined by the following condition:

Condition 1: the ternary functions which can be represented bywell-formed logic formulas consisting of variables x1," ,xn, con-stants 0, 1/2, 1, and logic connectives AND(-), OR(+), and NOT(-)defined in Table I.

Hereafter, we call a ternary function satisfying the above condi-tion a ternary function representable by a logic formula.

Note 1: The truth tables of Table I are called Kleen9's ternarylogic system [2]. The same truth tables as Table I were used indepen-dently by Goto [3] to analyze indefinite behaviors of relay circuits.

Here, let us define a partial ordered relation "oc" concerningambiguity on V = {0, 1/2, 1} and Vn as follows:

Definition 1: 0 ax 1/2, 1 x 1/2, i cc i, i E V. In the relation cc,0 and 1 are not comparable to each other. The relation can beextended among Vn as follows: For two elements A (a,, ,an)andA' = (a', ,a') ofV, A' oc A if and only if a' cc ai for allvalues of i. If A' xc A, then A' is said to be less ambiguous than orequal to A.Example 1: Suppose Al = (0, 1/2,1/2), A2 = (1, 1/2, 0), and

A3 = (1/2, 1/2, 1/2); then A1 oc A3,A2 cc A3 where A1 and A2 arenot comparable to each other.As a condition for a ternary function F to be significant when the

truth value 1/2 is assumed to be represent an ambiguous state, it willbe postulated that if the value of F(A ) is definite, that is, 0 or 1, thenF(A') takes an equal value for every element A' which is lessambiguous than or equal to A; that is,

Condition 2: Regularity: if F(A) E B = {O, 1}; then F(A') =F(A) for every A' such as A' caA.

Definition 2: A ternary function F is called a regular ternarylogic function if and only if F satisfies the regularity Condition 2.Example 2: Let the two-variable ternary functions F1 and F2 be

given by Table II. Then F1 is a regular ternary logic function whileF2 is not. In fact, (1,1) cc (1/2, 1), but F2(1, 1) = 1 A 0 =F2(1/2, 1).

Note 2: The condition of regularity defined above is an exten-sion of Kleene's definition to n variable ternary functions whereKleene's original definition [2] of regularity for a truth table is asfollows: The truth table never takes 0 or 1 as entry in the "1/2 row(or column)" unless this entry 0 or 1 occurs uniformally throughoutits entire column (or row, respectively).

Next, a ternary function which satisfies the following condition.Condition 3: Monotonicity for ambiguity: If A' Xc A, then

F(A') cc F(A), is called anA ternary logic function. It is known [4]thatA ternary logic functions can be applied to design fail-safe logiccircuits by letting 1/2 correspond to a failure state.

Note 3: A ternary function F which satisfies the above condi-tion and, also, the condition of normality [1], that is, ifA E Bn ={O, If, then F(A) E B is called a B ternary logic function [5] andis applied to detecting hazards [6], [7] and fail-safe logic [8].

Thus far, three different conditions (1, 2, and 3) have been de-fined for ternary functions. In the following, we will prove thatthese three conditions are equivalent to each other.

0018-9340/86/0200-0179$01.00 © 1986 IEEE

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TABLE II

EXAMPLE 2

1 12_ 1T27 1

O T 112 1 112

11/21 1 112 112 112 112 112 1

JLI 1171112 _ O_-1

F --Regular ternary F2--Non-regular ternary

logic function function

Theorem 1: F is a regular ternary logic function if and only if Fis a A ternary logic function.

Proof: Let us suppose that if F(A) E B, then F(A) = F(A')fot everyA' such thatA' x A. If F(A) = 1/2, then it is evidentthatt(A') cx F(A) = 1/2 holds for every A'. If F(A) E B, thenF(A') cx F(A) holds for every A' such that A' cx A by the sup-position. That is, it is always valid that if A' oc A then F(A') cx

F(A). Conversely, let us suppose that if A' oc A, then F(A') c: F(A).If F(A) E B, then F(A') oc F(A) implies F(A') = F(A).

Q.E.D.Theorem 2: If F is a ternary function representable by a logic

formula, then F is a regular ternary logic function.Proof: It will bb shown by induction concerning the number of

logic connectives. It is evident that the constants 0, 1/2, and 1, andeach variable xi, , x,' satisfy Condition 3. Suppose that all ter-nary functions representable by logic formulas in which the numberof logic connectives is smaller than or equal to X, satisfyCondition 3. Next, let us suppose that F is a ternary function repre-sentable by a logic formula in which the number of logic connectivesis, n + 1. Hereafter, for simplicity, we will identify a logic formulawith the ternary function represented by the formula. F is one ofF,,F, * F2 and F1 + F2. F, satisfies Condition 3 because of the fact thatFi(A') oc FI(A) is equal'to Fi(A') oc Fi(A). Suppose that A' x Aand (F, * F2) (A') $ (F, * F2) (A). Then, this fact leads to one of1) (F1i F2) (A) = 0 and (F, * F2) (A') 0, 2) (F1 * F2) (A) = Iand (F, * F2) (A') 1. Either case does not hold as shown below. If(F1 * F2) (A) = 0, then F,(A) = or F2(A) = O. By the assumptionof'deduction, we can obtain that F1(A') = 0 or F2(A') = 0, that is(F1 I F2) (A') = 0. This contradicts the assumption. It is similar inthe case of 2). Therefore, F, F2 satisfies Condition 3. Next, sup-pose that A' c A afid (F, + F2) (A') (F, + F2) (A). In a similarmanner, we can show that F1 + F2 satisfies Condition 3, because(F. + F2) (A) = 0 and (F, + F2) (A) = 1 lead to a contradiction.From the above, it has been shown that all ternary functions repre-sentable by logic formulas satisfy Condition 3. Q.E.D.The converse of Theorem ?, that is, every regular terhary logic

fun'ction can be represented by a logic formula, will be shown in thenext section.

III. REPRESENTATION OF A REGULAR TERNARY LOGIC FUNCTION

A literal is a variable xi or xi, the negation of xi. A conjunctionof one or more literals is called a simple phrase if it does not containa literal and its negation xi * xi simultaneously for at least one vari-able xi, and is called a complementary phrase otherwise. A dis-

junction of one or, more literals is called a simple clause if it doesnot contain a literal and its negation xi + xi simultaneously for atleast one variable'x,, and is called a complementary clause other-wise. In the above definitions, it is assumed that any repeatedliterals are removed.

Note 4: As evident from Table I, xi * xi = 0 and xi + xi = 1

whenx 1/2 does not hold in Kleene's system. Therefore, we can

not ignore conjunctions and disjunctions containing a literal and its

negation simultaneously.Definition 3: Let A = (a,,.*., an) be an element of Vn. Then

A gnd a simple phrase a = x n1* xan (simple clause 13 = xB +

+ xn) correspond to each other if the following conditions

hold: If ai = 0, then x7i = xT (xai = xi); if ai = 1, then x?i = xi(x?i = xi); and if a, = 1/2, then there is no variable xi in aQ(,).Example 3: LetA = (1, 1/2,0). Then, the simple phrase a cor-

responding toA is a = xl X3, and the simple clause 13 correspond-ing to A is /8 = XI + X3.

Definition 4: Let A = (a,, ,an) and A' = (a...* ,a') beany two elements of Vn. Then, it is said that A and A' are disjointto each other and written as A n A' = 0 if there is i in {1, * * *, n}such that ai is 0 or 1 and ai = a''.Lemma 1: Let A be any element of V" and a, /3 be the corre-

sponding simple phrase and simple clause, respectively. Then,1) A' oc A iffa(A') = 1;2) A' nA = 0 iff a(A') =0;3) A' $A andA' nfA # 0 iff a(A') = 1/2;4) A' c A iffB3(A') = 0;5) A' n A = 0 iff ,(A') = 1, and6)A' A andA', nA 0 iff 8(A') = 1/2.Proof: Let A = (al,,< ,a,) anid a = xai1._xik where

a,j(j = 1, ,k) is 0 or 1 and other elements of A are 1/2. For an

element A' (a 1, ,a'), a(A') = 1 if and only if the value ofx`ij [that is, (af)aij] is 1 for allj's (1 < j c k). This means that ifai is O orl, then a, = aij, thatis,A' oc A. Therefore, 1) isjustified.Similarly, a(A') = 0 if and only if there is at least onej such thataij = ai4j that is, A,' n A = 0. Thus, we arrive at 2). Also, 3) isderived directly from 1) and 2). In a similar manner, we can show4), 5), and 6). Q.E.D.

Theorem 3: Let F be a regular ternary logic function and A be an

element of Vn. Then,1) if F(A) = 1, thenF(A') = 1 for everyA' such thatA' oc A;2) if F(A) = 0, then F(A') = Ofor everyA' such thatA' oc A;

and3) if F(A) = 1/2, then F(A') = 1/2 for every A' such that

A cx A'.Proof: These are evident from the condition of regularity

(Condition 2) and monotonicity for ambiguity (Condition 3).Q.E.D.

Let F be an n variable regular ternary logic function. Then,F-'(1), F'(0), and F 1(1/2) represent the subsets of VI mapped to

1, 0, and 1/2, and are called the 1 set, 0 set, and 1/2 set, respec-tively. Theorem 3 indicates that F-'(1), F'(0) and F -'(1/2) are

partial ordered sets in regard to the relation c: and that the setsF '(1) and F '(0) are determined uniquely by their maximal ele-ments while F -'(1/2) is determined uniquely by its minimal ele-ments(Fig. 1). Hereofcourse,F-'(1) U F-1(0) U F'(1/2) Vholds. In Fig. 1, the symbol * indicates the maximal elements ofthe 1 set, the symbol # the maximal elements of the 0 set, and thesymbol x the minimal elements of the 1/2 set.Theorem 4: Any regular ternary logic function F can be repre-

sented by the logic formula

F = F1 + (1/2) * F°where F1 is the disjunction of simple phrases corresponding to allthe maximal elements of the 1 set ofF and where F0 is the conjunc-tion of simple clauses corresponding to all the maximal elements ofthe 0 set of F.

Proof: Let A' be any element of Vn and F(A') = 1. Then,there is a maximal element A in 1 set ofF such that A' oc A. Hence,there is a simple phrase a corresponding to A in F' wherea(A') = 1 [Lemma 1i1]. Therefore, F1(A') = 1, that is, (F1 +

(1/2) - F°) (A') 1. Next, suppose F(A') = 1/2. Then, A' doesnot belong to either the 1 set or the 0 set of F. Hence, there is no

simple phrase in F' and no simple clause in F° corresponding to Asuch that A' x A. As a result, F'(A') # 1 and F0(A') # 0

[Lemma 1-1 and 4]. F0(A') # 0 means that F°(A') = 1 or

F°(A') = 1/2. Thus, we can show that (F' + (1/2) 'F°)(A')1/2. Finally, suppose that F(A') = 0. Then, there is a simpleclause corresponding to A such that A' ax A in F°. Therefore,F°(A') = 0 holds Lemma 1-4. On the other hand, A,' n A = 0 isvalid for every element A of the 1 set of F because A' belongs to the

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TABLE IIITRUTH TABLES OF TERNARY NOR AND NAND

01/21 011/21-O _ 112 X -O El- l l

1/21112- 1/2 1 l/121 X 112 11/2

[1 _ 11 1LJ

NOR: AtB NAND: A4,B

\/ F (0): O-set\

Fig. 1. VW = F-1(1) U F-'(0) U F-1(1/2).

O set. That is, FP(A') = 0 is justified by Lemma 1-2. From theabove, we have (Fl + (1/2) * F°) (A') -= 0. Q. E.D.As we have seen, the three conditions (1, 2, and 3) described in

the preceding section are equivalent to each other. It is apparentfrom the above proof that Fl and F° are determined uniquely (ignor-ing the order of phrases or clauses) for any given regular ternarylogic function F. Therefore, the logic formula described inTheorem 4 can be used as a canonical form of regular ternary logicfunctions. In Section V, we will consider another canonical formcalled the canonical disjunctive form.Example 4: Let us represent F1 of Table II in Example 2 by a

logic formula based on the above theorem. The set of maximalelements of the 1 set is {(0, 1/2)} and that of the 0 set is {(1,-0)}.Therefore, we have F, = xI + (1/2) * (xI + x2).

IV. AxIoMS AND FUNCTIONAL COMPLETENESSOF REGULAR TERNARY LOGIC FUNCTIONS

Any regular ternary logic function can be represented by a logicformula composed of constants 0,1 /2, and 1, and logic connectivesAND('), oR(+) and NOT(-) defined by Table I. As an algebraicsystem, the set of regular ternary logic functions satisfies the follow-ing equalities which also hold in Boolean algebra:

1) the commutative laws A + B = B + A, A * B = B A;2) the associative laws A + (B + C) = (A + B) + C, A

(B - C) = (A B) * C;3) the absorption laws A + (A * B) = A, A (A + B) = A,4) the distributive laws A * (B + C) = (A B) + (A * C),

A + (B * C) = (A + B) * (A + C);5) the idempotent laws A + A = A, A * A = A;6) DI Morgan's laws (A + B) =A *B, (A* B) =A + B;7) the double negation law A = A;8) the least element 0 + A = A, 0 * A = 0;9) the greatest element 1 + A = 1, 1 A = A;10) Kleene's laws (A A) + B + B = B + B, A A

(B + B) = A *A; and11) center 1/2 = T12.Except (the complementary laws) A + A = 1, A A = 0.

The equalities 1-10 except 11 are equivalent to axioms of Kleenealgebra' or fuzzy algebra and have been studied in detail in [9]. Theelement satisfying 11 is called a center. The regular ternary logicfunctions satisfy the axioms of Kleene (or fuzzy) algebra witha center.

Next, we will examine these regular ternary logic functions froma standpoint of functional completeness. The set of logic connec-tives {0, 1/2, 1, +, ,,} (we consider constants as 0 variable logicconnectives) cannot represent all ternary functions; that is, it is notfunctionally complete for ternary functions, but as rnentionedabove, it is functionally complete in a strong sense [1] for regularternary logic functions. That is, any regular ternary logic functioncan be represented by {O, 1/2, 1, +, , }; and conversely, a ternaryfunction represented by {0, 1/2, 1, + , -} is always regular.

TABLE IVNONREGULAR ONE-VARIABLE TERNARY FUNCrIONS

x 1/2 I

u1(x) 0O 01/

u2(x) 1/2 0 0

u3(x) 1/2 0 1/2

u 4(x) 1/2 1 1/2

u5(x) 1/2 1 1

u6(x) 1 1 1/2

Let us define ternary NOR( 1) and NAND( 4.) as in Table III.Theorem 5: The set of logic connectives {0, 1/2, 4 } is func-

tionally cornplete for regular ternary logic functions.Proof: ItisshownbyA A T A,1 0=, A +B =A f B

andA B A t . QED.Theorem 6: The set of logic connectives {0, 1/2, 4 } is func-

tionally complete for regular ternary logic functions.Proof: It is shown byA =A A, 1 =QA +B=BA t

andA B =A B. Q.E.D.Note 5: The following problem arises: if a nonregular ternary

logic function is added to the set of regular ternary logic functions,is the new set always functionally complete for ternary functions?That is, are regular ternary logic functions maximal? The answeris negative. In fact, one-variable ternary functions ul,, , u6 ofTable IV are nonregular, and even if one of them is added to thefamily of regular ternary logic functions', they are not functionallycomplete for ternary functions. But it can be proved that if anynonregular one-variable ternary function except those of Table IV isadded to the set of regular ternary logic functions, then they arefunctionally complete for ternary functions.

V. CANONICAL FORM OF REGULAR TERNARY LOGIC FUNCTIONSIn this section, we shall introduce a canonical form for regular

ternary logic functions, which is different from that of Theorem 4.We shall also discuss the methods to obtain such a canonical form.Any logic formula representing a regular ternary logic function Fcan be expanded into a disjunctive form

F =yl + +ym

where yi(i = 1, - - *, m) is a product term, because the distributive,absorption, De Morgan's, idempotent, and other laws stand valid asstated in the preceding section. Here, each product term yi is one ofthe following three types:

type 1: a

type 2': (1/2) * a

type 3': ..,B

where a is a simple phrase and , is a complementary phrase asdescribed in Section III. If a product term (1/2) ,3 (, is a com-plementary phrase) exists, then we can omit 1/2 and it is equal totype 3' because xi xi - 1/2 stands always true. If a variable xi

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1EEE TRANSACTIONS ON COMPUTERS, VOL. C-35, NO. 2, FEBRUARY 1986

does not exist in a product term (1/2) aa of type 2', then thefollowing relation holds:

(1/2) * a = (1/2) * (xi + xi) a = (1/2) a- * xi + (1/2) * a * xi

as xi + xi7 1/2 is always valid. In a similar manner, if a variablexi does not exist in a complementary phrase E of type 3', then

P = fi' (xi + i) = p xi + p * Xi

holds, since there is a factor xj *x in 83 for a variable xj whereXJ * Xi 1/2 S xi + xi always holds. From the above, we can ex-pand a of type 2' and A3 of type 3' into disjunctions of product termsin which all variables exist, respectively. A simple phrase and com-plementary phrase in which all variables exist are called a mintermand complementary minterm, respectively.

Consequently, any regular ternary logic function can always beexpanded into the disjunction of the following three types of productterms:

type 1: a = x? il . x, ik ... simple phrase

type' 2: (1/2) -a = (1/2) * x l ..*xvn ... a' is a minterm

type 3: p = Xl _x,n * XI-ail ... XI-aik ...

complementary minterm

where a1 or aij is 0 or 1.Next, let us examine the relations of each type of product term.

Here, for two product term 'y and y', if all literals of y exist iny' as well, then it is written as y D y'. In this case, y + y' = yis true; that is, y' is absorbed by y in accordance with the absorp-tion law.

Definition 5: Let A = (an,---,an) be an element of vn. Then,the element A corresponds to a product term of type 2 or type 3 ifthe following relations holds:

ifa1 = 0, then xi =xi

if ai = 1, then x i = xi

if ai = 1/2, then xi =xixi

Example 5: (0, 0, 1) corresponds to a product term of type 2,(1/2) * X1 * X2 X3, and (0, 1/2, 1) corresponds to that of type 3,X* X2--X2 * X3. Product terms of type 2 correspond to elements ofB", and product terms of type 3 to those of V" - B" whereBn = {O, 1}".Lemma 2: Let a be a product term of type 1, a' be that of type 2

or type 3, and A and A' be elements corresponding to a and a',respectively. Then,

1) if a(A') = 1, then a D a' and2) a'(A) = 1/2 if and only ifA' cxA.

Proof: It is shown by the definitions of type 1, type 2, type 3and Definition 5. Q.E.D.

Definition 6: If a regular ternary logic function F is representedby a logic formula

F =7i + +Ym

then it is said that F is in the canonical disjunctive form whereyi(i = 1, * , m) is one of type 1, type 2 or type 3 and yi X j foralli,j (i #j).Theorem 7: Any regular ternary logic function can be repre-

sented uniquely (ignoring the order of the product terms) by thecanonical disjunctive form.

Proof: Let us suppose F1 = Yi + + y, and F2=Yi +.+ y,' are two different canonical disjunctive forms of aregular ternary logic function F. (It is evident from the above dis-cussion that there is at least one canonical disjunctive form of F).Now, we can suppose that a product term y exists in F1 but not inF2 without loss of generality. First, assume y is a product term of

type 1, that is, a simple phrase. If A is an element corresponding toy, then F1(A) = 1 because y(A) = 1. Then, it should beFd(A) = F2(A) = 1. Therefore, there is a product term 7' of type 1corresponding to A' such that A cc A' in F2 [Lemma 1-1 ] where bythe assumption y # y', A # A' holds. Here, y'(A') = 1 leads toF2(A') = 1 which is equal to F1(A') = 1. Therefore, in a similarmanner, there is a product term y" of type 1 corresponding to A"such that A' x A" in F1. Then, y can be absorbed by 7" becauseA x A" and A * A", that is, y C y". Hence, this is contradictory tothe assumption that F1 is the canonical disjunctive form. Second,assume y is a product term of type 2 or type 3. Letting A be anelement corresponding to y, y(A) = 1/2 leads to F1(A) = 1/2,because if we assume that FI(A) = 1, then the following con-tradiction arises: there should exist a simple phrase y' such thaty'(A) = 1 in F, and y is absorbed by y' [Lemma 2-1]. Hence,F2(A) = 1/2 holds. This means that there is a product term y' oftype 2 or type 3 corresponding to A' such that A' cx A [Lemma 2-2]or that there is a simple phrase corresponding to A' such thatA n A' # 0 [Lemma 1-2]. Here, the latter does not hold, since ifso, then F2(A') = F1(A') = 1 dictates that there is a simple phrasey" corresponding to A" such that A' x A" in F1 and y is absorbed byy". Therefore, only the former stands valid. Where, by the assump-tion y # y', A # A'. Similarly, from y'(A') = 1/2, we can showthat there is a product term y" of type 2 or type 3 corresponding toA" such that A" cx A' in F1. Then, y is absorbed by y" becauseA" cx A and A" # A. This is contradictory to the assumption thatF1 is a canonical disjunctive form. Therefore, any product termwhich exists in F, also exists in F2. From the above, we haveshown that the canonical disjunctive form of F is determineduniquely. Q.E.D.The following is an algorithm to obtain the canonical disjunctive

form of any given regular ternary logic function:1) expand the given logic formula into a disjunctive form (a dis-

junction of product terms),2) expand product terms of type 2' and type 3' into the disjunc-

tions of product terms of type 2 and type 3, respectively,3) based on the absorption law, omit, if any, product terms which

are included by other product terms,4) the logic formula obtained finally is a canonical disjunc-

tive form.Example 6: The canonical disjunctive form of the regular ter-

nary logic function of Example 4 is obtained as follows:

F = I + (1/2) - x1 + (1/2) * X2

= xl + (1/2) * xx2 + (1/2) * x12 + (1/2) 'xI *x2

= xl + (1/2) * x x2

This correspondence has concentrated on the canonical dis-junctive form, but the canonical conjunctive form can also be treatedin a similar fashion.

VI. CONCLUSIONWe have defined regular ternary logic functions as a significant

and useful family of ternary functions and have discussed the funda-mental properties of these functions. In particular, we have con-sidered their representations and canonical forms. Recently,Yamamoto [10] has introduced three-valued majority functions as afamily of significant ternary logic functions. The three-valued ma-jority functions are a special example of regular ternary logic func-tions described in this paper.

ACKNOWLEDGMENTThe author would like to thank honorary Prof. M. Goto of Meiji

University for his continued encouragement. He also wishes tothank Prof. J. Berman of the University of Illinois at Chicago Circleand Y. Ohiwa for advising him to refine this correspondence.

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REFERENCES

[1] M. Mukaidono, "Some kinds of functional completeness of ternary logicfunctions," in Proc. IEEE 10th Int. Symp. Multiple-Valued Logic, 1980,pp. 81-87.

[2] S. C. Kleene, Introduction to Metamathematics. Amsterdam, TheNetherlands: North-Holland, 1952, pp. 332-340.

[3] M. Goto, "Application of logical mathematics to the theory of relaynetworks," J. Inst. Elec. Eng. (Tokyo), vol. 69, pp. 125-132, 1949.

[4] M. Mukaidono, "Special kinds of ternary logic functions and their appli-cations to fail safe logic circuits," preprint.

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Design of a Multiple-Valued Systolic System for theComputation of the Chrestenson Spectrum

CLAUDIO MORAGA

Abstract -This correspondence deals with the computation of theChrestenson spectrum of an n-ary, n-place, p-valued function by means ofa systolic system. The design of a systolic system in a multiple-valuedenvironment is discussed in details and some aspects are compared tobinary realizations.The correspondence aims to motivate further research in computer

architecture in the context of multiple-valued logic.

Index Terms- Cellular algorithms, Chrestenson transform, multiple-valued logic design, systolic systems.

INTRODUCTIONIn the last decade we have seen an increasing interest in the

applications of Chrestenson transforms both in logic design and inimage processing [1]-[10]. In this period of time we have also seenthe development of new forms of architecture, which have con-venient VLSI realizations. Among these, systolic arrays [11], [12]have shown great efficiency in the realization of (some) matrixoperations.

Observing these two developments, it becomes a natural questionwhether the computation of the Chrestenson transform of a vectormultiple-valued function, (a problem which may be given a matrixexpression) may be conveniently realized with a systolic array.One of the first difficulties in this attempt is, that the Chrestenson

functions are complex valued (except in the binary case when theyreduce to the Walsh functions) and that the processing elements of

Manuscript received June 20, 1985; revised October 15, 1985.The author is with FB Mathematik und Informatik, Universitaet Bremen,

2800 Bremen 33, West Germany.IEEE Log Number 8406761.

systolic arrays work with real or integer arithmetic. (Some devel-opments using floating point operations may be found in [13].)

In this correspondence we use a polynomial representation ofcomplex numbers, multiple-valued circuitry and a systolic systemwith programmable elementary processors within an environment ofmultiple-valued logic (MVL) to offer a novel solution to the pro-posed problem and, in this way, to motivate further work related tonew architectures in MVL.

DEFINITIONS

Definition 1: Chrestenson Functions [141The Chrestenson functions may be given a discrete representation

as generalized Hadamard matrices over the set of p roots of unity,as follows:

X(p, 1) := [x,j, of dimension p' x p'

with Xij:= x(i,j) = exp(2nrijV7/p), 0 s i,j < p

X(p, n) = X(p, 1) 0 X(p, n - 1) (1)where 0 denotes the Kronecker product and X(p, n) is of dimensionn X pn.The Chrestenson functions are known to be orthogonal and sym-

metric [1], [14], which leads to

X(p, n) * X*(p, n) = X*(p, n) * X(p, n) = pn * I(p, n)

where X*(p, n) is the complex conjugate of X(p, n) and I(p, n) isthe pn X pn identity matrix.

Moreover the Chrestenson functions are closed with respect to theproduct and form an abelian group [1]. Vi,j, k such that 0 S i,j, k < pn: x(i,j) * x(i, k) = x(i, j PD k) where j P k is the digitwisemodulo p addition of the respective p-ary expansions of j and k.

n-I]= EJip;

i=O

n-1

k = E Kip';i=O

Ji, K E Zp . (2)

Definition 2: Chrestenson Spectrum [2]Let F: Z' -* 7Zp Moreover define h: 7Z UpJ (where U,p is

the set of pth roots of unity) such that Vv E 7ph(v) =

exp(2irvA/TT/p) holds.Let z and w be the coding of Zn-1Zn-2..ZIZO and W"-lI

W1 Wo, respectively, as stated in (2) and 0 '- y < m. Then the indi-vidual components of F have the following spectral expansion:

Fy(z) = h-1(p-nES(w) x(w,z)) (3)

with

S,(w) = Z h(F,(z)) * x*(w, z). (4)

Let H be an extension of h to matrices. If A = [aij] thenH(A) := [h(aij)]. Define F := [fij], 0 - j < m; 0 - i < p',with fij = Fj(i) and similarly, S : = [sij] with sij = Sj(i).Equations (3) and (4) may then be given the following expressions:

F = H-'(p-" *XS)

S = X* * H(F)

and letting H(F) = F',

F' =p-n XS

S = X*F'.

(5)

(6)

(7)

(8)

0018-9340/86/0200-0183$01.00 © 1986 IEEE

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