645 SHORT NOTES On Identification of Totally Symmetric Boolean Functions NRIPENDRA N. BISWAS, SENIOR MEMBER, IEEE Abstract-This paper shows how a "unity-ratio" totally sym- metric function can be identified without any further decomposition. The identification is carried out by writing the given function in its ordered partitioned tabular form. The method is general, straight- forward, and programmable on a digital computer. Index Term-Symmetric function. INTRODUCTION The identification of a totally symmetric function is com- pleted when 1) its a-numbers, and 2) its variables of symme- try are determined. It is well known that when the disjunc- tive canonical form of a totally symmetric Boolean function is written in its tabular form, the number of O's and l's of one column is the same as that of another [1]. By finding out this ratio of number of O's to the number of l's, the a- numbers and the variables of symmetry of the function can be determined [1]- [3]. However, when this ratio turns out to be unity, the variables of symmetry are found, according to the existing methods, by further decomposition of the function. In this note a method is described by which this class of totally symmetric function can be identified without any further decomposition. DEFINITIONS AND THEOREMS Definition 1: The terms disjunctive canonical form, minterm, totally symmetric function (tsf), x-number, and variables of symmetry have their usual definitions [1]. Definition 2: If the minterms whose sum constitutes a Boolean function are written in their binary forms (with 0 representing a complemented variable, and 1 an uncomple- mented or true variable) as rows of a table, each of whose columns represents a variable, then the table is known as the combination table, or the tabular form representation of the given Boolean function. Definition 3: The weight of a row in the tabular form is defined as the number of l's in the row. Definition 4: If the combination table is so partitioned that all rows of a particular weight are in a single partition, then the tabular form is known as the partitioned tabular form. Definition 5: If in each partition of the partitioned tabular form the minterms appear from top to bottom of the parti- tion in ascending order of their decimal designation, then the tabular form is called the ordered partitioned tabular form. Definition 6: A totally symmetric function with only one a-number is called an elementary symmetric function (esf). It is known that if the esf is written as a combination table, Manuscript received January 28, 1969; revised August 6, 1969. The author is with the Department of Electrical Engineering, St. Louis University, St. Louis, Mo. TABLE I N(0) a -NUMBERS N(0) WHEN n IS N(l) 2 ~~ ~ ~~3 4 56 1'1 (S) 03 04 05 06 2:2 3:3 12 2 4:4 5:5 0134 14 6:6 0145 15 7:7 123 0156 8:8 0:10 23 3 256 11:11 0235 036 014 15:15 1234 24 16:16 4) 17:17 01356 2356 21:21 1245 0135 22:22 012456 25:25 234 26:26 02346 31:31 12345 then the weight of each row is constant and is equal to a [4]. It can be calculated that the number of O's, N(O), and the number of l's, N(1), of each column of an esf are given by the binomial coefficients N(O) = (n - 1) N(l) = (n- (1) (2) where n =number of variables and a = a- number. Since a totally symmetric function of cx-numbers al, a2 * , c,. is the Boolean sum of the esf 's of a-numbers, 5 OC2~ . . . S XrS the number of O's and l's of each column of a totally symmetric function can be calculated from the knowledge of N(O)'s and N(1)'s of its constituent esf 's. Thus it is possible to prepare a chart from which the a-numbers of a totally symmetric function can be read when the ratio of the number of O's to that of I's of the columns are known, Table I gives the relevant portion of such charts for n = 2 through n = 6, where the ratio N(O): N(l) is unity.
Transcript
645SHORT NOTES
On Identification of Totally SymmetricBoolean Functions
NRIPENDRA N. BISWAS, SENIOR MEMBER, IEEE
Abstract-This paper shows how a "unity-ratio" totally sym-metric function can be identified without any further decomposition.The identification is carried out by writing the given function in itsordered partitioned tabular form. The method is general, straight-forward, and programmable on a digital computer.
Index Term-Symmetric function.
INTRODUCTION
The identification of a totally symmetric function is com-pleted when 1) its a-numbers, and 2) its variables of symme-try are determined. It is well known that when the disjunc-tive canonical form of a totally symmetric Boolean functionis written in its tabular form, the number of O's and l's ofone column is the same as that of another [1]. By findingout this ratio of number of O's to the number of l's, the a-numbers and the variables of symmetry of the function canbe determined [1]- [3]. However, when this ratio turns outto be unity, the variables of symmetry are found, accordingto the existing methods, by further decomposition of thefunction. In this note a method is described by which thisclass of totally symmetric function can be identified withoutany further decomposition.
DEFINITIONS AND THEOREMS
Definition 1: The terms disjunctive canonical form,minterm, totally symmetric function (tsf), x-number, andvariables ofsymmetry have their usual definitions [1].
Definition 2: If the minterms whose sum constitutes aBoolean function are written in their binary forms (with 0representing a complemented variable, and 1 an uncomple-mented or true variable) as rows of a table, each of whosecolumns represents a variable, then the table is known as thecombination table, or the tabularform representation of thegiven Boolean function.
Definition 3: The weight of a row in the tabular form isdefined as the number of l's in the row.
Definition 4: If the combination table is so partitionedthat all rows of a particular weight are in a single partition,then the tabular form is known as the partitioned tabularform.
Definition 5: Ifin each partition ofthe partitioned tabularform the minterms appear from top to bottom of the parti-tion in ascending order of their decimal designation, thenthe tabular form is called the ordered partitioned tabularform.
Definition 6: A totally symmetric function with only onea-number is called an elementary symmetric function (esf).
It is known that ifthe esf is written as a combination table,
Manuscript received January 28, 1969; revised August 6, 1969.The author is with the Department of Electrical Engineering, St.
Louis University, St. Louis, Mo.
TABLE I
N(0) a -NUMBERSN(0) WHEN n ISN(l) 2 ~~ ~ ~~34 56
1'1 (S) 03 04 05 06
2:2
3:3 12 2
4:4
5:5 0134 14
6:6 0145 15
7:7 123 0156
8:8
0:10 23 3
25611:11 0235 036
014
15:15 1234 24
16:16 4)
17:17 01356
235621:21 1245
0135
22:22 012456
25:25 234
26:26 02346
31:31 12345
then the weight of each row is constant and is equal toa [4].
It can be calculated that the number of O's, N(O), and thenumber of l's, N(1), of each column of an esf are given bythe binomial coefficients
N(O) = (n - 1)
N(l) = (n-
(1)
(2)
where n =number of variables and a= a- number.Since a totally symmetric function of cx-numbers
al, a2 * , c,. is the Boolean sum of the esf 's of a-numbers,5 OC2~. . . S XrS the number of O's and l's of each column of
a totally symmetric function can be calculated from theknowledge of N(O)'s and N(1)'s of its constituent esf 's. Thusit is possible to prepare a chart from which the a-numbersof a totally symmetric function can be read when the ratioofthe number ofO's to that of I's ofthe columns are known,Table I gives the relevant portion of such charts for n= 2through n= 6, where the ratio N(O): N(l) is unity.
IEEE TRANSACTIONS ON COMPUTERS, JULY 1970646
Definition 7: When each column of the tabular form of atsf has equal number of 0's and 1's, then the tsf is known as
a unity-ratio totally symmetric function.If a given n-variable Boolean function, when written in
the tabular form, exhibits any of the ratios as given inTable I, but its row weights do not match with the requiredoc-numbers, then the matching can be achieved (if the func-tion is at all totally symmetric) by double complementationof one or more columns of the function.
Definition 8: Double complementation of a column meansthat 1) the variable heading of the column is complemented,and 2) the l's of the columns are changed to 0's, and the 0'sto l's.
It can be easily verified that this process does not alterthe value of the particular given Boolean function [2], [3].
Theorem 1: The maximum number of columns that needbe doubly complemented to yield the correct set of row
weights is given by
k _ {n/2max l(n- i)/2if n is even
if n is odd.
Proof: Let k be the number of columns to be doublycomplemented in a given function to yield the correct set
of row weights. Then the same result would have beenachieved by doubly complementing (n- k) columns becauseofthe following property ofa totally symmetric function [1]:
SIX ,42, ,z(X1X2, X.)(
=S(n-al),(n a2) ... (n- a,) (5X529 .. X5n)-It should be seen from Table I that for every unity-ratio
tsf if {aIl, C2.* * OCr} is a set of correct row weights, then{(n - c1), (n - C2) (n - ocr)} is either the same set or anotherset of correct row weights, Hence, when n is even, kmax= n/2,and when n is odd, kmax=(n - 1)/2. Q.E.D.
Theorem 2: As a result of k number of double comple-mentation, a row weight rw is changed to a new row weight
rw as given by
r'w = rw ± k
or
r' frw±(0,2,4,** p)p=k-2 ifkiseven
)y+±(1,3,5,.-.q)q=k-2 ifkisodd.
Also
0 < r' < n.
Proof: Double complementation of a column changeseither a 0 of a row into a 1 or a 1 into a 0. If the k number ofdouble complementation changes all 0's of a row into l's,then r' = rw+ k. On the other hand, if it changes all l's into0's, then r' = rw- k. This proves the first part. If, however,the double complementations affect a combination of 0'sand l's, then when k is even, there may be only one pair of0 and 1 and the rest are all 0's or l's. Then r' = rw± (k-2).
If there are two pairs of 0's and l's and the rest are all 0'sand l's, then r' = rw± (k -4). Repeating this argument, thesecond part for the case when k is even is proved. In a similarmanner the other part when k is odd is also proved. Thelast part is obvious as r' can neither be negative norgreater than n. Q.E.D.
THE METHODAt this stage let us expound the method for functions of
up to seven variables only so that kmax is limited to 3, andonly the following three cases may arise.
Case 1-k =1: Here the given function which is requiredto be identified must have row weights which are + 1 fromthe set of correct row weights (Theorem 2). Thus, ifwe con-
sider only those rows of the given function which has a row
weight, say u, then by double complementation of only one
column, the weights of all these rows must change to u-Ior u+ 1. Now, if u-I is the correct row weight, then theremust exist one and only one column having an all I entry.On the other hand, if u+ 1 is the correct row weight, thenthere must exist one and only one column having an all 0entry. This will be true for another set of rows with weight,say w. Thus we can conclude that if in the partitioned tabu-lar form ofthe given function there exists a column for whicheach partition has either an all 0 or an all I entry, then thefunction is totally symmetric when this column is doublycomplemented.
Case2-k = 2: Here the row weights ofthe given functionmust differ from the correct row weights by 0 or + 2. Thoserows which differ by 2 or -2 must have only two columnswith all 0 or all 1 entries, respectively. From such rows therequired columns can be determined. But if the number ofsuch rows is less than two, then the required columns can-not be determined. In this case, consider the rows whoseweight has not changed and let the unchanged weight be u.Then u is also an a-number, and the original tsfhad all rowsof weight u. Now those rows which had a 0 in one of therequired column and a 1 in the other required column, musthave remained unchanged in weight. Hence no two columnscan have all 0 or all 1 entry. But there must be a subpartitionin the partition of rows of weight u where one of the re-quired columns has an all 0 and the other an all 1 entry. Inorder that the subpartition is available, it is imperative thatthe tabular form must be in its ordered partitioned form.Under this condition, one of the two subpartitions has rowswith 0's in the x1 column, whereas the other has rows with1 in the same column. Two typical cases are illustrated inExamples 1 and 2.Example 1:f(x1x2x3x4x5)= v(2, 5, 9, 12, 15, 16, 19, 22,
26, 29).The function has been written in its ordered partitioned
tabular form in Table II. Here the partitions of rows ofweights 2 and-3 indicate columns x1 and X4 as the requiredcolumns to be doubly complemented, whereas the partitionsof rows of weights 1 and 4 indicate that columns X2, X3, andx5 should be doubly complemented. It will be evident from
SHORT NOTES
TABLE II
ROW X X2 X3 X4 X5 rwNO.O IO O
3 o I^ I26 0 0° 37 0 1 0 38 I 0 0 39 0 44
10 45 5 5 5 55 5~ 5~ 5 5
TABLE III
ROW 5lIX2 X3 X4 x5 rwNO __ __ __ __ _
10 0002 0 0 0 103 1 0 44 II 1 0 45 i 0 46 000017 001008 0 1 0 009 1 0 1 40 0 I I 4
TABLE VI
RNOW X X2 X3 X4 X5 X6 rw
0 00 0 00 023456789101 1121314151617181920 I I I I I
(3) that both these indications are valid and there is no con-tradiction. Double complementing columns x1 and X4, thefunction appears as in Table III. The new row weights are 1and 4 which match the x-numbers of Table I (n= 5 andN(0): N(l) =5:5). Hence the function is a tsf, and can bewritten, by (3), as
f = S1,4(5X1X2X3X4XS)=-S--xx-1,4(XlX25X3X4 Jh5)
Example 2: f(XlX2X3X4)= v(0, 5, 6, 9, 10, 15).The function has been written in its ordered partitioned
tabular form in Table IV. Here the subpartitions consistingof second and third rows as well as those consisting offourth and fifth rows point out that x1 and x2 are the re-quired columns. Doubly complementing these (Table V),the new row weights become 2, which is the required o-number. Hence, by (3),
f=S2=x152X3X4)= S2(XlX25x3-4)-
Case 3-k= 3: Here the row weights of the given functionmust differ from correct row weights by ± 1 or by + 3. By
similar reasonings as was done in case k =2, if there is apartition with row weights differing by + 3, then this par-tition will show all 0 or all 1 entries in the required columns.On the other hand, if no such partition exists, then a sub-partition of a partition with row weights differing by ± 1will have three columns having all 0 in two of the columnsand all 1 in the third, or all 1 in two and all 0 in the third.The following example illustrates this case.Example 3: f(xlx2x3x4x5x6)=v(0, 3, 6, 9, 12, 15, 18, 24,
27, 30, 33, 36, 39, 45, 48, 51, 54, 57, 60, 63).The function has been written in the ordered partitioned
tabular form in Table VI. It can be seen that there are twosubpartitions indicating x1, X3, and x5 columns as those tobe doubly complemented. On so doing we get Table VII,where the new row weights turn out to be only 3. Hence thefunction which is identified as a tsf can be written, by (3), asfollows:
f = S3(xlx253X45XO= S3(Xl5C2X3x4X5X6)-
It will be appropriate to mention here that in cases where2< k <kmax when a partition or a subpartition has k columnswith all 0 or 1 entries, another partition or subpartition may
have the other (n - k) columns with all 0 or 1 entries. Thisis not true for k= 1, since no two rows which have all 0 or all1 entry in (n - 1) columns can belong to the same partition.The method can now be written in steps as follows. It is
assumed that the tabular form of the given function alreadyidentifies it as a unity-ratio function.
Step 1. Write the given function in the ordered parti-tioned tabular form.
Step 2: Detect column/columns having all 0 and/or all 1entries in one or more partitions.
Step 3: If the result of Step 2 is positive and does not haveany contradiction in case of more partitions indicating thecolumn/columns, then doubly complement the detectedcolumn/columns to obtain the new tabular form. If the newrow weights match the x-numbers as given in Table I, thenthe given function is a tsf, whose designation is then written.In case Table I is not available (as may be cases for n > 6),check if the number of rows with a new row weight w equals(n,). If this relation holds good for all new row weights, thefunction is a tsf whose designation is then written.
Step 4: If the result of Step 2 is negative, then detectcolumn/columns having all 0 and/or all 1 entries in one ormore subpartitions.
Step 5. If the result of Step 4 is positive and noncon-tradictory, then doubly complement the column/columnsto determine the xc-numbers and variables of symmetry ofthe given function as in Step 3.An optional step to save labor may be suggested. In case
the complementary function of the given function f has asmaller number of rows, then instead of identifying f, fmay be identified following the above five steps. Once f isknown, f can be identified from the following property ofthe tsfs [1]:
S{1a(X1X2 ... Xn) = S{fj}(X1X2 ... Xn) (4)where {ail and {a;} are two subsets of the entire set ofa-numbers {0, 1, *- , n}, so that
oci :A ot
and
fatil u {oCj} = {O, 1, ,n}.
CONCLUSIONIt is clear by now that although the method has been ex-
plained for values of k up to 3, Steps 1 through 5 as well asthe optional step are quite general. Even Table I is not ab-solutely essential. For a given function of n variable, the kcolumns to be doubly complemented will be detected assoon as the function is written in the ordered partitionedtabular form. It must be mentioned here that both this andthe decomposition method fail for functions with unityratio 1: 1. However, these cases are trivial so far as identifi-cation is concerned, as the two terms of the function arealready known to be minterms mo and m2n-1. Another sub-class of the unity-ratio tsf's, which has been circled inTable I, remains totally symmetric for all combinations ofvariables of symmetry [5 ]. These can be recognized withoutgoing through either this or the decomposition method.The method as outlined in this paper virtually presents thecolumns to be detected in a pictorial form. In fact, if a com-puter program is written to produce the ordered partitionedtabular form of the given function, the columns to bedoubly complemented will be immediately visible.
ACKNOWLEDGMENTThe author wishes to thank the anonymous referees
whose comments have been very helpful in improving thispaper.
REFERENCES[1] R. E. Miller, Switching Theory, vol. 1. New York: Wiley, 1965.[2] E. J. McCluskey. "Detection of group invariance of total symmetry of
a Boolean function," Bell Sys. Tech. J., vol. 35, pp. 1445-1453, Novem-ber 1956.
[3] M. P. Marcus, "The detection and identification of symmetricswitching functions with the use of tables of combinations," IRETrans. Electronic Computers (Correspondence), vol. EC-5, pp. 237-239, December 1956.
[4] M. A. Harrison, Introduction to Switching and Automata Theory.New York: McGraw-Hill, 1965.
[5] N. N. Biswas, "Special class of symmetric Boolean functions,"Electron. Letters, IEE (London), vol. 5, p. 72, February 1969.
