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UNIVERSITY OF ILLINOIS 
 GRADUATE COLLEGE 
 DIGITAL COMPUTER LABORATORY 
 
 INTERNAL REPORT NO. 57 
 
 SOME THEOREMS ON THE METRIC PROPERTIES OF 
 BOOLEAN ALGEBRA 
 
 by 
 
 SABURO MUROGA 
 August 6, I95I+ 
 
 This work has been supported in part by 
 the Office of Naval Research under 
 Contract NR OU4 001 
 
Some Theorems on the Metric Properties 
 of Boolean Algebra 
 
 Several theorems on the metric properties of Boolean Algebra 
 have been obtained. In this report they are described with the expectation 
 of reader's knowledge of the following report which gives the foundation of 
 this work: 
 
 Internal Report No. k6, Metric Properties of Boolean Algebra 
 and their Application to Switching Circuits, by D. E. Muller. 
 
 The theorems were suggested by computational results, using 
 ingenious coding tricks which were given in the above report. 
 
 THEOREM It If there is the following distance relation between two Boolean 
 
 12 m 
 members g and g in p variables, which do not include any of x , x , ..., x 
 
 and their complements: 
 
 I^g-L* S 2 ) = d, where p > m ; 
 then the relation 
 
 L[g 1 f(x , x , ..., x ), g 2 f(x , x , ..., x m )] = d/2 m 
 
 holds, where 
 
 12 m 
 
 f = X X , . . ., X 
 
 or some of x x , ..., and x can be complemented. 
 
 "T l 
 
 PROOF: g. = g. x + g. x 
 
 Since g. does not contain x , from the symmetrical structure of the variables, 
 we have 
 
 LCg.x 1 , 0) = LCg.x 1 , 0) , 
 
 ~T "T 
 L(h 1 x , g 2 x ) = d/2 , 
 
 L(g-[X , g 2 * ) = d/2 , 
 
 g.x and g.x are disjoint. 
 
 IT O m 
 
 Now g.x and g.x do not include any of x , ..., x . If we repeat the above 
 
 and then 
 
 and 
 
 since 
 
-2- 
 
 1 1 
 process on g.x or g.x , we can get the theorem for m = 2. Thus the re- 
 petition of the above process will produce the statement of the above theorem. 
 
 p-r 
 Assume that t is a member of the net of order d = 2 , where 
 
 p is number of independent variables . Write t in the following form: 
 
 1 2 3 23 13 12 123 
 
 t = u + v n x + v^x + v^x + W.X X + W,_X X + W^X X + SX X X 
 1 c. 5 ± 2 3 
 
 where 
 
 u = u + u : polynomial of x , 
 
 h 
 u : polynomial of x , 
 
 k 
 v.'s : polynomials of x , 
 
 1* 
 
 w.'s : polynomials of x _, 
 
 s : polynomials of x , 
 then t can be rewritten as follows: 
 
 , x^ of the r-th degree 
 , x of the (r-l)th degree 
 
 P 
 
 ., x of the (r-l)th degree 
 ., x of the (r-2 )th degree 
 . , x of the (r-3)th degree, 
 
 (1) 
 
 n \ 1 2 3 / v 1 2 3 
 
 t = L (n + v + v + v + w + w + w + s j x x x + (u + v + v + w ) x x x 
 
 IT 3 
 
 12 3- 
 
 + (u + v., + v_ + w^ ) x x x + (u + V n + V^ + W., ) X X X 
 13 2 12 3 
 
 rr n123 / ^l23 / n123 1 2 3, . . 
 
 + L(u + v.Jxxx + (u + v J x x x + (u + v ; x x x + u x x x J (2 ) 
 
 Consider the following h: 
 
 12 3 n~T 2 3 
 
 12 3 
 
 12 3 
 
 h=<Xxxx+(3xxx+7xxx +5xxx 
 
 (3) 
 
 li n 
 
 where oc a (3, 'i and 5 are the polynomials of x , ..., x of the (r-l)th degree. 
 But we make the assumption that any two of c< , P, , 1/ and S are not the same 
 and that the sum of any two of them is not equal to any one out of the rest. 
 Therefore: 
 
 12 3 
 
 (t+h) = [ (u + v 1 + v + v_ + w n + w + w^ + ex + s ) x x x 
 J. 2 3 1 2 3 
 
 + (u + V + V + W + (? ) XX) 
 
 X - 2 x 5 
 
 IT 
 
 + (u + V- + V + w + "/ ) X X X 
 
 123 
 + (u + v + v + w + 5 ) x x x ] 
 
 r / \123/ \ 1 2 3 , x T? 3 1 2 3, ,, s 
 
 + L(u + vjxxx +(u + v)xxx +(u + vjxxx + uxxxj(4j 
 
-3- 
 
 According to whether each term in the second bracket [ ] is equal to zero or 
 not, we can divide all the possible cases into several ones as in Table 1. 
 Take the case 1 where each term in the second bracket [ ] is 
 
 equal to zero, that is, u + v -u + v=u + v=u = 0. Then we get 
 
 12 5 
 u = v = v = v t so the first term in (k) reduces to (w + w + w + ex. + s )x x x , 
 
 k p 
 
 From the assumption, w.'s are the polynomials of x , ..., x of the (r-2)th 
 
 k p , . 
 
 degree and ex is the polynomial of x , ..., x of the (r-ljth degree. In short, 
 
 c* is of higher degree than w.'s. So though the terms of degree lower 
 
 then (r-l) in ot may vanish for some t, the terms of the (r-l)th degree do 
 
 12 3 
 not vanish. So from Theorem 1, the first term (w + w +w+oc+ s)xxx 
 
 "D- (r-l )-3 
 in (h) has the number of ones not less than 2 K . Similarly the second, 
 
 d— Cr-1 )-3 
 third and fourth terms in (4) have the number of ones not less than 2 . 
 
 Totally the number of ones in (t + h) in this case Is not less than 2 
 
 Next, take case 2 where u + v ^ 0, and u + v = u + v=u = 0. 
 
 Then we have u = v_ = v^ =0. Now the factors of the first, third and fourth 
 
 2 3 
 
 terms 
 
 in (h) reduce to v., + w., + w^ + w^ + <x + s, v n + w^ + Y and v n + w^ + 5, 
 
 1123 12 13' 
 
 respectively. Assume v + w + 7 = 0. Then from v. = w + 7 , we have 
 
 v n + w ? +w^+w + <X + s =w n +\r^+<x + y+s and v, + w 7 + 5 = w^ + w, + 5 + O' 
 1123 13 13 23 
 
 Each of them is a polynomial in x , •«•, XT of the (r-l)th degree and the 
 terms of the (r-l)th degree can not vanish from any member t by the assumption 
 that oc £ r and o £ y . So from Theorem 1, we can have at least two terms 
 out of the first, third and fourth termq in (^), each of which has a number of 
 ones not less than 2 ^ ~ '~ . But from v f (from u+v £ and u = 0), 
 the fifth term, the polynomial of the (r-l)th degree has a number of ones 
 
■K- 
 
 p-(r-l)-3 p-r 
 
 not less than 2 .So total number of ones is not less than 2 
 
 The other cases can he treated in similar ways as follows. 
 
 In the case 3, either 1 or k and then either 2 or 3 is not less than 
 2 .In the case h, one out of the terms 2, 3, and h is not less than 
 2 P-(r- j-3^ because ^ if v + v + w + /3 =0 and v + v + w + 7 = 0, 
 
 we have v., + v^ + w^ + 5 = w n + w^ + w^ + fi + 'V + 5 for the fourth term, which 
 12 3 12 3 
 
 is not zero from P + 7^5. 
 
 In the following three cases, we assume u n = and u / 0. In the case 
 
 1 2 
 
 5, either of 3 and k is not less than 2 . And if the third term is 
 
 zero, we have v = w + Y . Then the first term is v + w + w+<x + 7+s 
 and the second, v + w + [3 . So either of these two is not zero from the 
 assumption that o< + 7 £ R . In the case 6, one out of 1, 2, and 3 is not 
 
 have 
 
 less than 2 P ~^ r " 1 ' ) "? If v~ + w_ + /3 =0 and v, + w + J = 0, we 
 
 2 1/ 1 2 
 
 v..+v^+w.,+w r +W- ? +cx' + s=w_, + £<T+ R + 7* + s for the first term, 
 12 12 3 3 1 
 
 which is not zero from the assumption. The case 7 is ohvious . In the case 8, 
 two out of 2, 3 and h are not less than 2 ^ . 
 
 If u £ 0, each term in equation (h) is not less than 2 . So the 
 
 p-r 
 total number of ones is not less than 2 
 
 Those exhaust all possible cases. 
 
 Therefore in any case, we have L(t + h, ) > 2 . h itself contains a 
 
 p-r 
 number of ones not less than 2 . h can be rewritten as follows: 
 
 h = cxxVx 5 + (3(1 + x 1 )x 2 x 5 + 7x 1 (l + x 2 )x 3 + Sac 1 * 2 (1 + x 5 ) = 
 
 ( <X + p+ 7 + 5) x^x 3 + (3x 2 x 5 + 7x X x 5 + bxV. 
 
 As a special case, if <X = fi+ 7+5, we have h=#xx +'/ xx + 5 x x . 
 
 If ex! £ fi + '"y + 5, the first term in (5) can be of higher degree by one than 
 
 the last three terms. If the difference between o< and /B + 7+ & is particularly 
 

 
 
 
 
 
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-6- 
 
 a constant, h has a form: 
 
 123 R 2 3 ry 1 3 R 12 
 
 h = X x x + p x x + 7 x x +5xx 
 
 As a conclusion, we can get the following theorem. 
 
 THEOREM 2: The Boolean member of the following form: 
 
 h=( o< + B + <i + 5) x X x 2 x 5 + /?x 2 x 3 + 7x 1 x + 5x X x 2 (6) 
 
 where o< _, R , y and 5 are the polynomials of the (r-l)th degree [not less 
 
 than (r-l)th degree], has the minimum distance d = 2 from any member of the 
 
 p-r 
 net of order d = 2 , if any two out of oc } R y y and 5 are not the same and 
 
 then sum of any two out of them is not equal to any one of the rest. 
 
 Using the expansion (2), we can get the following corollary. 
 
 COROLLARY 1: If the relation 
 
 L(t, h' ) > d = 2 P " r (7) • 
 
 p-r 
 holds for any member t of order d = 2 , where h' is the polynomial of 
 
 12 3 
 degree not less than r + 1 and does not include any variable of x , x and x , 
 
 then we have 
 
 L(t + h, h' ) > d 
 
 for any member t, where h is given by (6) in Theorem 2. 
 
 1 2 
 PROOF: From (k) and the fact that h' does not include any of x , x 
 
 3 
 and x , we get 
 
 12 3 
 t + h + h' = [(h' + u + v, + v^ + v^ + w n + w + w^ + oc + s )x x x + (h' + u 
 
 12 3 12 3 
 
 v_ + v + w_ + P ) x x x + (h' + u + v. + v^ + w^ + 6 ) x x x 1 + [ Ch' + u + Y, ) 
 
 c. 3 l 12 3 ' 1 
 
 1 2 3 /^, . 12 3/^, ^12 3/^, x 1 2 3 n / rt x 
 
 xxx + (h' + u ' v x x x + (h* + u + v )x x x + (h T + u)x x x J . (9) 
 
 Take the factor (h* + u + v., + v + v_, + w_ + w^ + w_, + oc ) in the first term 
 
 123123 
 
 in (9)- The whole terms from the second up to the last o£ can be expressed as 
 
 p-r 
 a member of the net of order d = 2 , since all of them are polynomials of 
 
 degree not more than r. 
 
-7- 
 
 So from Theorem 1: 
 
 L(h* + u + v + v + v + w + w + w + (X + s )x x x , 0) > d/2 . 
 Similarly, any term in (9) has a number of ones not less than d/2" y . So totally 
 the relation (8) is satisfied. 
 
 ACKNOWLEDGEMENT 
 
 The author is greatly indebted to Professor David E. Muller for his 
 valuable suggestions during this work.