All solutions of finite equations

DISCRETE MATHEMATICS Discrete Mathematics 137 (1995) 1-6

ELSEVIER

All solutions of finite equations Dragi6 Bankovi6 Facuhy ~?f"Science, Radoja Domanovi~a 12, P.O. Box 60, 34000 Kragujevac, Yugoslavia

Received 2 June 1993

Abstract

The problem of solving (arbitrary) equations over arbitrary (finite) sets has been studied in the literature [l 3, 5, 7, 9, 10, 15 17, 19]. After determining one general solution in [-15], Pregi6 determined in [,17] all general reproductive solutions of a finite equation, supposing that particular solutions are known. In this paper we determine all general solutions (including all general reproductive solutions) of a finite equation without the above supposition. We also describe all general solutions of Boolean equations in Boolean algebra ~z. Keywords: Finite equation; Boolean equation; General solution

We firstly state the definition of the general solution. Definition 1. Let E be a given n o n - e m p t y set and ~ be a given unary relation of E. A formula x = ~o(t), where ~p" E ~ E is a given function, represents a general solution of the x-equation J/(x) if and only if

(v t)~(q,(t))

^

(v x ) ( ~ ( x ) =~ (3 t) x = ~o(t ) ).

A formula x = f f ( t ) , where ~" E--*E is a given function, represents a general reproductive solution of the x-equation ~ ( x ) if and only if (V t)~(~J(t)) A (V t ) ( ~ ( t ) ~ t = ~ (t)). Let Q = { q o , q l . . . . . q,,} be a given set of m + 1 elements and S = {0, 1}. Define the operations + , o and x r in the following way: +

0

1

o

0

1

0

0

1

0

0

0

1

1

1

1

0

1

X y ~-

10 if x = y , otherwise (x, y ~ Q ~ S ) .

0012-365X/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 001 2 - 3 6 5 X ( 9 3 ) E 0 1 59-2

D. Bankovi~'/ Discrete Mathematics 137 (1995) 1-6

Assuming x ^ 1°x=x)

that (Vx~SuQ)(x+O=x^O+x=x^x.O=O^O.x=O^x.l= Pre~i6 [15] considered the following x-equation:

SO ° Xq°"}- S1 ° xql-lt- "" "Jvsm° x q ' = o

(1)

(siES , x~Q).

In the sequel o will be omitted. Note that SOqj q° d- S l q j q' + " " + smqj q" = 0 ~

Lemma

1. For

every

vector

(2)

Sj = O.

(So,Sx . . . . . sm)~{O, 1} "+1

and

every

permutation

(Jo,J, . . . . . jm) of{O, 1. . . . . m}, the vector (to, t, . . . . . t,,)E{O, l} m+l defined by th=SjoSj'"S:h S °

( h = O , 1. . . . . m - - l ) ,

t,.=S~oSj, ...st._ ' has m components 0 and one component 1.

Proof. Let (Sjo,Sj . . . . . . sj._,)=(1, 1. . . . . 1). Then, obviously, th=S/oSj . . . s A _ s ° = O

(h=O, 1. . . . . m - l ) ,

tm= SjoSj, . . . sj,_ = l.

L e t s j 0 = s j = . . . . sA = 1 a n d s A = O ( O < k ~ < m - 1 ) . (a) If p < k then tp=SjoSj, ... sjp_,s~=O because s~=O. (b) If k < p < ~ m then tp=O because tp contain s),, which is O. I f p = k then tp=SjoSj,...sj, , s ° = l . [] Theorem 1. L e t So . . . . . s,, be given. E v e r y function A : Q ~ Q A(t)=

k=0

[si,,oq . . . .+. .s. .i . . .s. o. q.,

can be written in the f o r m

+ s .,. . .s.. ~. . .s°. . . q ~k.2

+ si,.osi,,,'"s° . . . . qo . . . . . + si,.osi~.l""si ..... qg,..] tqk'

(3)

where f o r every k e {0, 1. . . . . m}, (ik,o, ik, 1. . . . . ik, m) and (g,. o, gk, 1 . . . . . gk,.,) are permutations o f {0, 1. . . . . m}.

(4)

Proof. Let a function A : Q--*Q be written as A=

. FLqjoqo qxqj, "-] ... qJ.

If A:qk--~qj, (k = 0 , 1. . . . . m) and sih.oSi,' 1"'" si..... s~., = 1. (there is only one element in the sequence s°Ik, O' s k,O s~°k, 1 , Sik.oS~k . I "~ . . . , s °| k , m - 1 ,s.Ik.O s k,l ° ' ' S j k , m - | being 1, by L e m m a 1),it is sufficient to take g,,h =Jk

(k = 0, 1. . . . . m)

and we get the function A in the form (3).

[]

D. Bankovi{/ Discrete Mathematics 137 (1995) 1-6 Theorem

3

2. l f E q . (1),

So Xq° + $1 Xqt + -'" + SraXq" = O,

is consistent (i.e. SoS x ... s,,=O), then a function A defines the general (reproductive) solution of Eq. (1) if and only if it is of the form o k=O 0 + Si,. oSi,,1 "'" S i~,._1 q' . . . . + S,.. oSik. l...Sik ' ' - ,q'k,. ] tq" .

(5)

under the following conditions: Ok, o, ik, 1..... ik, ~) are permutations of {0, 1.... , m}, (io, o, il,o .....

ira,o) is a permutation of {0, 1,..., m},

(6) (7)

(under condition (6) and (io,o, ik o..... im,o) = (0, i ..... m)).

(8)

Proof. Let A(t) be of the form (5), and (6) and (7) be fulfilled. For arbitrary t~Q there is k~({0,1 . . . . . m}) such that t = q , . Let sik., be the first element of the sequence si.... si,,,, .... si,., being 0 (there is such an element because of SoSl...Sr,=O), i.e. si,.o =si~., . . . . . S~.... = l and s~.,=0. Bearing in mind Lemma 1, the formula x = A ( t ) gives x=qi,. ,. Besides, qi~., satisfies Eq. (1) because of (2) and si~.,=O. If q, satisfies (1) then s, = 0. In accordance with (7) there is k E {0, 1. . . . . m} such that ik, o = r. Taking t = qk, formula (5) gives x = q, because of Lemma 1. Let the formula x = A ( t ) represent a general solution of (1) and the function A be written as A=[qo q, qJo qJl

"'" "'"

qm]. qi.

Let R = {it s,=O ^ i~{0,1 . . . . . m} }, i,e, {q~li~R} is the solution set of (1). Let us determine i~,j in the following way: • if A: qo---,qj ° then io,o=Jo and Mo = {0, 1. . . . . m}\{jo}, • if A :qk~qjk (k~{l . . . . . m}) then (a) i f j k ~ M k - i then ik, o=jk and M k = m k - ~ \ { j k } (b) ifjk(EMk-1 then ik.l=jk and ik, o=Z for some Z ~ M k - l \ R

and M k = M k _ l \ { z } .

4

D. Bankovi{/Discrete Mathematics 137 (1995) 1 6

One can prove that there exists Z E M k - I \ R in (b). Namely, the a s s u m p t i o n 7(3z)(z6Mk I \ R ) gives Mk-1 o R . Further, jkddMk_l implies jk6{io, o . . . . . ik 1.0} which gives Jk E { Jo . . . . . Jk- 1 } because of (a) and Jk ~ R. Therefore, card({ jo . . . . . J k - l } W { j k . . . . . j,,}) = card({ jo . . . . . j k - x } W { j k + l . . . . . j,,}) < c a r d ( { jo . . . . . j k - 1 } ) + c a r d ( m k

1) ~
because c a r d ( { j k + l . . . . . j m } ) < c a r d ( M k 1) and {Jo . . . . . J k - t } , M k - 1 o R . Thus, A is not the general solution, which contradicts our supposition. Then

i

[SiO o qik, ° 4- S k, oSO, 1qi~,

o 2qi~,2 4-"" 4- S¢.oS¢.l Sik,

k=0

+ si~,oS¢.l"" .so .... qi . . . . + s¢,osi~,,...si ..... qi~.~] t q~= A(t).

Let A(t) be of the form (5), and (6) and (8) be fulfilled. If t = qk and qk ~ R then x = A (t) gives x = qk because q k e R ¢~ Sk = 0 and (8). Thus, (5) defines the reproductive solution. Let x = A(t) represent a general reproductive solution of (l). If we determine the indices ik. j in the following way: ik, o = k

and

ifqkCR

A:qk--*qw

thenik, l = w ,

we have qk e R implies S° oqik.o = s~ q k = qk

and qkCR implies s o~,oqi,.o +si ,.o sO~,1qik.t =swqw=qw" o

Obviously, condition (8) is fulfilled. Bearing in mind L e m m a 1 A ( t ) can be written in the form (5) and (6). [] Let {Ao, A1 . . . . . Ap} = {0, 1}", where n is a natural n u m b e r and p = 2 " - 1 . Theorem 3. Let f: {0, 1}"~{0, 1} be a Boolean function. I f the equation f (x~, ... , x , ) = 0 is consistent then the n-tuple G ( T ) = ( g l ( t l . . . . . t.) . . . . . g.(tl . . . . . t.))

( T = ( t l . . . . . t.))

determined by p

G ( T ) = (J [f'(Ai~,o)A¢,o~f(Ai~.o)f'(A¢.,)Ai,.~ k-0

u f(Ai~.o)f(A¢.,)f'(A¢,:)Ai,.~ f(Ai,.o)f(A¢.~)f(Ai~, ~ ) ' " f ' ( A ¢ . , _ 1)Ai,.,_l u f (Ai~.o) f (Ai,.~) f (Ai~,~) ... f (Ai,.,_l)Ai~.~] T A~,

(9)

D. Bankovi~'/ Discrete Mathematics 137 (1995) 1-6

5

where for every k~ {0, 1. . . . . p} (Ai~.o, Ai~.,, .... Ai~,. ) is a permutation of{O, 1}", defines the general (reproductive) solution of f ( x a . . . . . x . ) = 0 if and only if it fulfils (Ai .... Ai ........ Aio.p) is a permutation o f {0,1}" (it fulfils (Ai .... Aio,,, .... Aio.,)= (Ao, A 1 . . . . . At,).

Proof. Since al = a and a0=~, (a~{0, 1}), i.e. p

~1 0

if ~ = f l if ~ # f l

the operations u , c~ and :~a in Boolean algebra ~z have the same properties as the operations + , • and x y, respectively, in Theorem 2. Therefore, it is easy to see that Theorem 3 holds. Z Example. Let us solve the equation a x ' y ' u b x ' y u c x y ' u d x y = O algebra ~ . Note that f ( O , O ) = a , f ( O , 1 ) = b , f ( 1 , O ) = c , f ( 1 , 1)=d. The general solution, for instance, is

in

Boolean

(x, y) = (a'(O, O) u ab'(O, 1) u abc'(1, O) u abc(1, 1))tl t; u(c'(1,O)ucd'(1, 1)wcda'(O,O)wcda(O, 1))t~ t 2 u(b'(O, 1)~bc'(1,O)t3bcd'(1, 1)ubcd(O,O))tlt'2

u(d'(1, 1)wda'(O,O)udab'(O, 1)•dab(1,O))tlt2 or, in scalar form, x = (abc' ~ abc)t'l t'2 u (c' u cda')t~ t 2 w (bc' w bcd')tl t'2 u (d' u dab)ta t2 y = (ab' u abc)t'l t'2 w (cd' w cda)t'l t2 u (b' u bcd')tl t'2 ~ (d' u dab')t1 t2.

The general reproductive solution, for instance, is (x, y) = (a'(O, O) u ab'(O, 1) ~ abc'(1, O) ~ abc(1, 1))t~ t~ u (b'(O, 1) w bd'(1, l) ~ bdc'(1, O) u bdc(O, 0))t'l t2 u (c'(1, O) w ca'(O, O) u cad'(l, l) u cad(O, 1))t 1t~ u (d'(1, l) u db'(O, 1)udba'(O,O)udba(1,O))tlt2.

References [1] D. Bankovi6, On general and reproductive solutions of arbitrary equations, Publ. Inst. Math. (Beograd) 26(40) (1979) 31-33. [2] D. Bankovi6, Equations on finite sets, Bull. Soc. Math. Belg. S6r. B 41 (1989) 47 53. [3] D. Bankovi6, All general solutions of finite equation, Publ. Inst. Math. (Beograd) 47 (61) (1990) 5-12. [4] D. Bankovi6, A generalisation of L6wenheim's theorem, Bull. Soc. Math. Belg. S6r A 44 (1992) 59-65.

6

D. Bankovib/Discrete Mathematics 137 (1995) 1-6

[5] M. Bo2i6, A note on reproductive solutions, Publ. Inst. Math. (Beograd) 19 (33) (1975) 33-55. [6] M. Carvallo, Espaces vectoriels bool6ens et postiens, Formules d'interpolation, C.R. Acad. Sci. Paris S/~r. A 272 (1971) 1366-1368. [7] J. Chvalina, Characterizations of certain general and reproductive solutions of arbitrary equations, Mat. Vesnik 39 0987) 5-12. [8] M. Davio, J.P. Deschamps and A. Thayse, Discrete and Switching Functions (Georgi Publ. Co and McGraw Hill, St. Saphorin and New York, 1978). [9] C. Ghilezan, M6thode a r6soudre des relations dont les r6solutions appartiennent ~i un ensemble fini, Publ. Inst. Math. (Beograd) 10(24) (1970) 21 23. [10] C. Ghilezan, Une g6n6ralisation du th6or+me de L6wenheim sur le 6quations de Boole, Publ. Inst. Math. (Beograd) 1I(25) (1971) 57 59. I l l ] C. Ghilezan and S. Rudeanu, Interpolation formulas over finite sets, Publ. Inst. Math. (Beograd) 25(39) (1978) 45-49. [12] Gr.C. Moisil, Th+orie structurelle des Automates Finis (Gauthier-Villars, Paris, 1967). [13] E.L. Post, Introduction to a general theory of elementary propositions, Amer. J. Math. 43 (1921) 163-185. [14] S. Pre~i6, Une classe d'6quations matricielles et l'6quation fonctionnelle f 2 = f Publ. Inst. Math. (Beograd) 8 (22) (1968) 143-148. [15] S. Pre~i6, Une m6thode de r6solution des 6quations dont toutes les solutions appartiennent ~t un ensemble fini donn6. C.R. Acad. Sci. Paris Ser. A 272 (1971) 654-657. [16] S. Pre~i6, Ein Satz fiber reproductive L6sungen, Publ. Inst. Math. (Beograd) 14(28) (1972) 133-136. [17] S. Pre~i/:, All reproductive solutions of finite equations, Publ. Inst. Math. (Beograd) 44 (58) (1988) 3-7. [18] S. Rudeanu, Boolean Functions and Equations (North-Holland, Amsterdam, 1974). [19] S. Rudeanu, On reproductive solutions of arbitrary equations, Publ. Inst. Math. (Beograd) 24(38) 0978) 143-145. [20] R. Wille, Algemeine Algebra zwischen Grundlagenforschung und Anwendbarkeit, Technische Hochschule Darmstadt, Preprint No. 107, 1973.