AN ANALYTICAL METHOD OF SOLUTION FOR SYSTEMS OF BOOLEAN EQUATIONS

Transcription

1 CHAPTER 5 AN ANALYTICAL METHOD OF SOLUTION FOR SYSTEMS OF BOOLEAN EQUATIONS 51 APPLICATIONS OF DE MORGAN S LAWS A we have een in Secion 44 of Chaer 4, any Boolean Equaion of ye (1), (2) or (3) could be reduced o heir correonding logically equivalen Claical Boolean Equaion Thu ye, (1) f(x j n,, H 2, H 1) gh j(,, H H, ) ^ 2 1 (2) f(x,, H, H ) gh (,, H H, ) j n 2 1 j ^ 2 1 (3) f(x,, H, H ) = gh (,, H H, ) correond o, reecively, j n 2 1 j ^ 2 1 (4) [f (X,,X,X)] v g (X,,X ) = j n 2 1 ' j n 2 (5) f(x j n,,x 2,X) 1 v [g j (X n,,x 2,X)] 1 ' = (6) [f j (X n,,x)] 1 ' v g j (X n,,x) 1 & f j (X n,,x) 1 v [g j (X n,,x)] 1 ' = 123

2 In hee condiion, a formal Boolean Equaion Syem i a yem of a finie number of imulaneou logical inerrelaion of Boolean Funcion of any given variable In a given Syem of "k" Simulaneou Boolean Equaion wih "n" variable (where "k n; n k") i i oible o calculae all oluion of "" variable (1 n), called "main variable (or main incognio)" Thee variable are a funcion of he remaining "(n-)", called "non-main variable (or non-main incognio)" If we have a Syem of "k" Simulaneou Boolean Equaion of "n" variable, i mean ha we begin wih "k" of any of hee relaion of (1), (2) or (3) ye Each of hee equaion can be relaced by he correonding logically equivalen Boolean Equaion (4), (5) or (6) ye Thu, we mu make a logical addiion of hee "k" equaion of he yem o find he final "Solving Equaion of Syem of Boolean Equaion" (SESBE) of he given Syem, ha correond o he following exreion: (7) f(x n,, X n-+1, X n-, X n--1,, X 2, X 1 ) = (SESBE) Therefore, we aly o hi exreion he Law of De Morgan, alo called "Exanion Theorem", relaed o he fir "" main incognio, a we have een in Secion 32 of CHAPTER 3, Proerie (24) and (24) : 1 Cae: Through Proery (24) of Secion 32 Chaer 3, exreion (7) become: (8) X n v v X n vx n - + 1vf ( 1,, 11,, X n -,, X2, X1) & & X n v v X n - + vx ' 2 n vf( 1,, 1,, X n-,, X2, X1 ) & & & & X ' ' ' n v v X n vx n - + 1vf(,,,, X n -,, X2, X1 ) = where we have he following claical Produc Term "P " in exreion (9) of he fir "" main variable and "F ", wih =,1,2,, ( 2-1 ), are given in exreion (1): ( 9) Xn Xn + 1 = P Xn X' n + 1 = P 2 2 X' n X' n + 1= P ( 1) f 1,, 11,, Xn,, X2, X1 = F f 1,, 1,, Xn,, X2, X1 = F 2 2 f,,,, Xn,, X2, X1 = F 124

3 Subiuing he fir member of exreion (9) and (1) for heir reecive econd member in exreion (8), we have: (11) P v F & P v F & & P v F = ( SESBE) or, ( 12) P v F = (SESBE) =o where we have: a) "P ", wih =,1,2,,(2 1) rereen all he Claical Produc Term - ee exreion (9) - relaed o he fir "" main variable = Xn, Xn 1,, Xn + 1B; he cardinal number of "" variable will deermine he quaniy of facor of hee Claical Produc Term ` b) " F ", wih =,1,2,,(2 1), rereen all Boolean Funcion (2`) which deend on he oher remaining non-main variable (n-), { X n, X n--1,,x1} Thee Boolean Funcion " F " can be obained if i i aumed ha he main incognio = Xn, Xn 1,, Xn + 1B correond o he "" binary value Thi "" value mu have a correonding binary exreion in " " I i convenien o oberve, ha he "Claical Produc Term" can be analyically exreed by he following exreion: ( 13) P = (X ), where: =,1,2,, (), 1 ha i: #( ) 1 = #( ) 2 or = 2 wih: = -1 2, where: if =, i will be (X) and ' = X if = 1, i will be (X ) = X 1 ha i: # () 1 = # (-121) 2 or = 2 125

4 exreion: The Boolean Funcion " F ", alo can be analyically exreed by he following (14) F = f(, -1,, 2, 1, Xn, Xn 1,, X1), where =,1,2,,(), ha i: 1 # () 1 = # (-121) 2 or = 2 Taking he value of P in exreion (13) and " F " in exreion (14) and ubiuing hem in exreion (12), we have: 2 1 (15) f(, -1,, 2,, 1, X, X,, X ) v (X ) n n 1 1 =, where = = 2 1, or #( ) 1 = #( -1 21) 2 where, if =, i will be (X ) ' = Xi and if = 1, i will be (X ) = Xi However, we wan o find he "General Soluion of he given Syem", which correond o he "General Deduced Theorem of he given Syem" In hee condiion, o obain hi "General Deduced Theorem of he given Syem", one ha only o exre he "" main incognio Xn, Xn 1,, Xn = B a a general funcion of heir correonding "(n-)" non-main incognio { X n, X n--1,,x1}, a we can ee in he following exreion: 1 (16) X = C, v F, wih = 1,2,, = 2 where, " C, " i arbirarily conan o "/1", whoe value will be deermined by aking he value of " X " of exreion (16) and " P " of exreion (13) and ubiuing hem in exreion (12) we have he following: (17) F v C, v F = # = =! " # $ #

5 Develoing he fir member of exreion (17), in relaion o heir "n-" no-main incognio { X n, X n--1,,x1}, we have ha heir coefficien, deending on he arbirary conan " C, ", hall all be necearily null Thi fac, allow he deerminaion of all value "/1" for hee conan, ha hall deermine he general oluion of he yem, and herefore, he general deduced heorem from he given Syem 2 nd Cae: Through he Proery (24) of Secion 32 Chaer 3: Alo, we aly o exreion (7) he Law of De Morgan, relaed o he fir "" main incognio, a we have een in Secion 32 of CHAPTER 3, Proery (24), hi exreion become: ( 18) ' ' ' [ Xn& & Xn + 2 & Xn + 1 & f( 1,, 11,, X n -,, X2, X1 )] ' ' [ Xn& & Xn + 2 & Xn + 1& f( 1,, 1,, X n -,, X2, X1 )] [ Xn& & Xn + 2 & Xn + 1 & f(,,,, X n -,, X2, X1 )] = where we have he following claical Sum Term " S " of he fir "" main variable and F, wih =,1,2,, ( 2-1), are given in he following exreion, (19) and (2): ( 19) X' n&& X' n + 1 = S X' n&& Xn + 1 = S 2 2 Xn&& Xn + 1 = S 126 f31,, 11,, Xn,, X2, X18= F f31,, 1,, Xn,, X2, X18= F 2 2 f3,,,, Xn,, X2, X18= F Subiuing he fir member of exreion (19) and (2) for heir reecive econd member in exreion (18), we have: ( 21) S & F S & F S & F = (SESBE) o or, ( 22) S & F = (SESBE) = where we have: 127

6 a) "S ", wih =,1,2,,(2 1) rereen all he Claical Sum Term - ee exreion (19) - relaed o he fir "" main variable = Xn, Xn 1,, Xn + 1B; he cardinal number of "" variable will deermine he quaniy of arcel of hee Claical Sum Term ` b) " F ", wih =,1,2,,(2 1), rereen all Boolean Funcion (2`) which deend on he oher remaining non-main variable (n-), { X n, X n--1,,x1} Thee Boolean Funcion " F " can be obained if i i aumed ha he main incognio = Xn, Xn 1,, Xn + 1B correond o he "" binary value Thi "" value mu have a correonding binary exreion in " " I i convenien o oberve, ha he "Claical Sum Term" can be analyically exreed by he following exreion: ( 23) S = 1X 6, where: =,1,2,, (), ha i: 1 #( ) 1 = #( ) 2 or = 2 wih: = -1 2, where: if =, i will be (X) and = X if = 1, i will be (X ) = X ' Taking he value of " S " in exreion (23) and " F " in exreion (14) and ubiuing hem in exreion (22), we have: (24) f(, -1,, 2,, 1, Xn, Xn,, X ) & (X ) 1 1 =! where, = 2 1, or #( ) 1 = #( -1 21) 2 where, 128 " $ # =

7 if =, i will be (X ) = X and if = 1, i will be (X ) = X However, by anoher form, we wan o find he "General Soluion of he given Syem", which correond o he "General Deduced Theorem of he given Syem" In hee condiion, o obain hi "General Deduced Theorem of he given = B a a Syem", one ha only o exre he "" main incognio Xn, Xn 1,, Xn + 1 general funcion of heir correonding "(n-)" non-main incognio { X n, X n--1,,x1}, a we can ee in he following exreion: 2 1 (25) X = C, & F, wih = 1, 2, 3,, = where, " C, " i arbirarily conan o "/1", whoe value will be deermined by aking he value of " X " of exreion (25) and " P " of exreion (23) and ubiuing hem in exreion (22) we have he following: (26) = &K 'K F& C, & F = 1 = ( )K *K = Develoing he fir member of exreion (26), in relaion o heir "n-" no-main incognio { X n, X n--1,,x1}, we have ha heir coefficien, deending on he arbirary conan " C, ", hall all be necearily null Thi fac, allow he deerminaion of all value "/1" for hee conan, ha hall deermine he general oluion of he yem, and herefore, he general deduced heorem from he given Syem Thu, in he 1 or 2 nd Cae above, we obain he fir "" main variable = Xn, Xn 1,, Xn + 1B (or deenden variable), in funcion of he oher indeenden variable, { X n, X n--1,,x1}, in heir differen aec of noaion The oluion obained by exreion (7) will coniue one SOLUTION MEMBER n n(n 1)(n - +1) (or THEOREM), wihin he oible combinaory oluion, " = ", of ( 1) 21 a general " h " GROUP 129

8 To obain all he oluion of he fir "" main variable " {X n, X n 1,X n + 1}", where hee variable are funcion of he oher remaining "n-" non-main variable, " {X n, X n 1,, ( 27) or, ( 28) X 2, X 1 }" Xn = fn ( Xn,, X2, X1), Xn + 2 = fn + 2( Xn,, X2, X1) Xn + 1 = fn + 1( Xn,, X2, X1) Xn ( Xn,, X2, X1) Xn + 2 ( Xn,, X2, X1) Xn + 1 ( Xn,, X2, X1), we need o imoe he following incognio yem: or, in anoher abbreviaed manner: (29) { Xn,, Xn + 2, Xn + 1 (Xn,, X2,X1) 13