Louis H. Kauffman- De Morgan Algebras - Completeness and Recursion

8/3/2019 Louis H. Kauffman- De Morgan Algebras - Completeness and Recursion

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D e M o r g a n A l g e b r a s - C o m p l e t e n e s s a n d R e c u r s i o n

L o u i s H. K a u f f m a n

U n i v e r s i t y o f Il l i n o i s a t C h i c a g o C i r c l e

A n e l e m e n t a r y p r o o f i s g i v e n o f a c om -p l e t e n e s s t h e o r e m f o r D e M o r g a n A l g e b r a s .T h e p r o o f i n v o l v e s a c o n s t r u c t i o n t h a ta s s o c i a t e s t o a D e M o r g a n a l g e b r a B , an e w D e M o r g a n a l g e b r a ~ . T h e c o n s t r u c -t i o n o f ~ b e a r s a c l o s e a n a l o g y t o t h ec o n s t r u c t i o n o f t h e c o m p l e x n u m b e r s f r o mt h e r e a l n u m b e r s . S i m i l a r l y , De M o r g a na l g e b r a s m a y b e c o n s t r u c t e d f r o m B o o l e a na l g e b r a s . R e l a t i o n s h i p s w i t h r e c u r s i o na n d p e r i o d i c s e q u e n c e s a r e d i s c u s s e d .

I . I n t r o d u c t i o n

A D_e M o r s a n a l g e b r a i s a n a l g e b r a t h a ts a t i s f i e s m o s t o f t h e p r o p e r t i e s o f a B o o -l e a n a l g e b r a e x c e p t f o r t h e l a w o f t h e e x -clud ed middle , e xpre sse d as x + x' = 1o r x x ' = 0 i n t h e u s u a l B o o l e a n n o t a -t i o n. T h e s e a l g e b ra s h a v e b e e n s t u d i e d b yv a r i o u s a u t h o r s (s e e [ 1 ] , [ 2 ] , [ 5 ] ) . T h ep u r p o s e o f t h i s p a p e r i s t o g i v e a n e l e -m e n t a r y p r o o f o f a c o m p l e t e n e s s t h e o r e mf o r D e M o r g a n a l g e b r a s , a n d t o i n d i c a t es o m e i n t e r e s t i n g e x a m p l e s o f t h e s e a l g e -b r a s .

T h e c o m p l e t e n e s s t h e o r e m t h a t w e p r o vem a y b e d e d u c e d a t o n c e f r o m d e e p e r r e -s u l t s a b o u t t h e s t r u c t u r e o f D e M o r g a n a l -g e b r a s ( s ee R e m a r k 2 . 1 0 ) . N e v e r t h e l e s s , Ib e l i e v e t h a t t he p r o o f g i v e n h e r e i s o f

i n t e r e s t b e c a u s e i t i s q u i t e e l e m e n t a r y ,a n d i t i s a g e n e r a l i z a t i o n o f a c o r r e s -p o n d i n g a r g u m e n t f o r B o o l e a n a l g e b r a s . D eM o r g a n a l g e b r a s g r o w o u t o f B o o l e a n a l g e -b r a s .

I n s e c t i o n 2 we d e f i n e D e M o r g a n A l -g e b r a s a n d g i v e a c o n s t r u c t i o n t h a t a s s o -c i a t e s t o a D e M o r g a n a l g e b r a B a n e wD e M o r g a n a l g e b r a B . T h e c o n s t r u c t i o no f ~ b e a r s a c l o s e a n a l o g y t o t h e c o n -s t r u c t i o n o f t h e c o m p l e x n u m b e r s f r o m t h er e a l n u m b e r s . W i t h t h e a i d o f t h i s c o n -s t r u c t i o n , t h e c o m p l e t e n e s s r e s u l t ( T h e o -r e m 2 . 5 ) i s p r o v e d . I n s e c t i o n 3 w e s h o wh o w ~ i s r e l a t e d t o r e c u r s i o n i n B .S e c t i o n 4 o u t l i n e s t h e c o n s t r u c t i o n o f a na l g e b r a o f p e r i o di c s e q u e n c e s , a n d d e l i n e -a t e s d i r e c t i o n s f o r f u r t h e r i n v e s t i g a t i o n .

2 . T h e C o m p l e t e n e s s T h e o r e mI t i s p o s s i b l e t o c h o o s e a v e r y c o n -

c i s e s e t o f a x i o m s f o r t h e a l g e b r a s w es h a l l s t u d y . I s h a l l g i v e a l o n g e r l l st ,a n d t h e r e b y a v o id e x t r e m e l y d e t a i l e d d e -m o n s t r a t i o n s .D e f i n i t i o n 2 . 1. A D e M o r g a n a l g e b r a Bi s a s e t B t o g e t h e r w i t h a u n a r y o p e r a -t i o n a ~ a ' ( i n v e r s i o n ) , a n d t w o b i n a r yoper atio ns a,b ~ a+b, a,b ~ ab thats a t i s f y t h e f o l l o w i n g a x i o m s :

( i) T h e b i n a r y o p e r a t i o n s a r e e a c hc o m m u t a t i v e a n d a s s o c i a t i v e .(ii) (a')' = a, aa = a, a+a = a fora l l a E B.

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(ill) (a+b)' = a'b', (ab)' = a'+b'for all a,b E B.(iv) There exist elements O,1 E Bsuc h that aO = O, a+O = a, al = a,a+l = 1 for all a E B.(v) a(b+c) = (ab)+(a c), a+(bc ) =(a+b)(a+c) for all a,b,c E B.

A B o o l e a n a l g e b r a i s a D e M o r g a n a l g e b r athat satisfies the axioms abov e plus

(vi) a+a' = 1 and aa' = 0 forall a E B.

D e f i n i t i o n 2 . 2 . L e t B b e a D e M o r g a nalge bra and let ~ = BxB. Thus

= [ ( a ,b ) i a , b E B j . D e f i n e o p e r a t i o n sin B as follows:

(a,b)(c,d) = (ac,bd), (a,b)+(c, d) :(a+c,b+d)( a , b ) ' : ( b ' , a ' ) , O = ( 0 , 0 ) , i = ( l , 1 ) ,@= ( o , l ) f o r a , b , c , d E B .

With these definition s, it is easy to seet h a t B b e c o m e s a D e M o r g a n al g e b r a .Note that Z I = Z a nd ~ ' = ~ . Hence, ifB is a non- triv ial (0 ~ I) Boo lea n al-gebra, then ~ is a De Mor gan alge brat h a t is not Boolean.

If V = [O,I} is the small est non-triv ial Bool ean algebra, then V = [o,~,t.~]is a small De Mor gan algebra. Note that~ , ~ = ~ a nd ~ = O , ~ + ~ . ~ . .

If, for a,b,c E B, we adopt the con-ven tio n a(b,c) = (ab,ac), then we may

Aident ify a E B with (a,a) E B. Ino t h e r w o r d s , t h e d i a g o n a l m a p A : B ~ ~ ,defin ed by A(a) = (a,a), is a De Mor gana l g e b r a h o m o m o r p h i s m t h a t e x h i b i t s B a s^a s u b a l g e b r a o f B . E v e r y e l e m e n t o fis of the form a{ + b~ for a,b E B.Def ini tio n 2~3. Let S be any set. Thefree De Mo rga n alg ebr a o__ S, deno tedB(S), is obtaine d as follows: The primi -tive expres sions in B(S) are O,1 andelements s E S. Let E(S) denot e thes e t o f e x p r e s s i o n s d e f i n e d b y t h e f o l l o w -ins rules:

( i) P r i m i t i v e e x p r e s s i o n s ar e i nE(S). That is, O,1 E E(S) ands c E ( S ) .(ii) If x,y E E(S), the n x', x+y, andx y b e l o n g t o E ( S ) .

L e t = d e n o t e t h e e q u i v a l e n c e r e l a t i o n o nE(S) that is gener ated by the axioms(i) - (v) of Def ini tio n 2.1. LetB(S) = E(S) /= . It is easy to ver ify thatB ( S ) i s a D e M o r g a n a l g e b r a w i t h o p e ra -t i o n s i n h e r i t e d f r o m t h e f o r m a l o p e r a t i o n sof rule (ii).D e f i n i t i o n 2 . 4 . A h o m o m o r p h i s m : B - Cof De Mor gan algebr as B and C is a set-m a p p i n g s u c h t h a t ~ ( x y ) = ( x ) ( y ) ,(x+y) = (x) + (y) and (x') = (x)'for all x,y E B.

A s u s u a l , a h o m o m o r p h i s m f r o m a f r e ealgeb ra B(S), :B(S) - C, is dete rmine du n i q u e l y b y i t s v a l u e s ( s ) f o r s E S .

We are now prepa red to state the com-p l e t e n e s s t h e o r e m .The ore m 2. . Let V = {o,~, {,~] be thef o u r e l e m e n t D e M o r g a n a l g e b r a d e s c r i b e da f t e r D e f i n i t i o n 2 . 2 . L e t B ( S ) b e af r e e D e M o r g a n a l g e b r a o n a s e t S . T h e nfor a,B E E(S), ~ = B if and only if ( s ) = ( B ) f o r e v e r y h o m o m o r p h i s m

: B ( S )In other words, an equal ity ~ = B is

a c o n s e q u e n c e o f t h e a x i o m s f o r a D e M o r g a nalge bra if and only if it is true about them o d e l V .

I n o r d e r t o p r o v e T h e o r e m 2 . 5 w e s h a l ln e e d s o m e p r e l i m i n a r y l e m m a s.Lemma 2.6. Let f(x) E E(S) be an expres-s i o n i n v o l v i n g x E S . T h e n f ( x) =Ax + Bx' + Cxx t + D where A,B ,C,D aree x p r e s s i o n s t h a t d o n o t c o n t a i n t h e e l e me n tX .

T h e p r o o f o f t hi s l e m m a p r o c e e d s J u s ta s i n o r d i n a r y B o o l e a n a l g e b r a , a n d w i l lt h e r e f o r e b e o m i t t e d .

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L e m m a 2 . 7 . T h e f o l l o w i n g e q u i v a l e n c e h o l d si n E ( S ) :(Ax + Bx' + Cxx')' = A'x + B'x' + xx' +A'B'C' .Proo f: (Ax + Bx' + Cxx') a= ( A x ) ' ( B x ' ( C x x ' ) '= ( A ' + x ' ) ( B ' + x ' ' ) ( C ' + ( x x ' ) ' )= ( A ' + X ' ) ( B ' + x ) ( C ' + X ' + x )= (A'B' + A' X + B'X' + X'x) (C' + x' + x)= A'B'C ' + (A'C' + A'B' + A')X

+ (B'C' + A'B' + B')X'+ (A' + B' + C' + 1 + 1) xx'

= A'B'C ' + A'(C' + B' + 1)x+ B'(C' + A' + 1)x' + xx'

= A'B'C ' + A'X + B'X' + XX'T h i s c o m p l e t e s t h e p r o o f o f t h e l e mm a .

F o r t h e n e x t l e m m a, w e s h a l l p r o c e e di n f o r m a l l y , r e g a r d i n g f ( x) a s a f u n c t i o nof x, and eva luat ing f(O), f(~), f(2),f ( ~ ) . T h i s c a n a l l b e m a d e m o r e p r e c i s ei n t e r m s o f h o m o m o r p h i s m s , b u t o n l y atsome loss in clarit y.Lem ma 2.8. Let f(x) = Ax + Bx' + Cxx' + Db e a n e l e m e n t o f E ( S ) a s d e s c r i b e d i nL e m m a 2 . 6 . T h e n

f(O) = B + Df(~) = A + D

+ i = Df(~ ) + f(~ ) = A + B + C + D.

AH e r e e q u a l i t y m e an s a s f u n c t i o n s o n V .Proof: Cert ainl y f(0) = B + D andf ( 1 ) = A + O . N o wf(Z ) = (A + B + C)Z + D (s in ce ~t =Z ), andf(~) : (A + B + C) ~+ D. Hen cef(Z) +~ = (A + B + C + I )~ + D = i + D,andf(~) + ~, = ~.~ + D. Th er efo re( f( ~) + Z ) ( f ( ~ ) + & ) = ( Z + D ) ( ~ + D )

= ~ + 2 D + ~ D + D= 0 + ( ~ + ~ ) D + D= D + D = D .

Fi nall y, f(i) + f(~) = (A + B + C)(# +~D + D= A + B + C + D .

T h i s c o m p l e t e s t h e p r o o f o f t h e l e m ma .

Proof of The ore m 2. 5 . Let m,6 E E(S).B y L e m m a 2 .6 w e m a y a s s u m e t h a t ~ = A x+ Bx' + Cxx' + D and that ~ = Ax ++ ~x' + ~xx' + ~ where A,B ,C,D andA , B , ~ , D a r e e x p r e s s i o n s t h a t d o n o t c o n -t a i n x . T h e p r o o f w i l l p r o c e e d b y i n -d u c t i o n on t h e t o t a l n u m b e r N o f v a r i -ables (elements of S) that occur in tht w o e x p r e s s i o n s . I f N = O , t h e n a = Dand ~ = ~ whe re D and D are equ alto 0 or I. Sin ce 0 ~ 1 in a freeD e M o r g a n a l g e b r a , t h e t h e o r e m i s t r i v i a lfor the case N=0. If N > O, then eithe

or ~ conta ins a var iab le x and wem a y u s e t h e e q u i v a l e n c e i n d i c a t e d a b o v e .

T h u s w e m a y a s s u m e b y i n d u c t i o n ( u s i nL e m m a 2 .8 ) t h a t t he f o l l o w i n g e q u i v a l e n c eh o l d i n E ( S ) :

(,) + D_ A += D

A + B + C + D = A + ~ + ~ + DWe now use (*) to show that ~ = ~ inE ( S ) .

= Ax + Bx' + Cxx ' + D= (A' x + B'x' + xx' + A'B' C')' + D,

b y (Z.7): ( A ' x ) ' ( B ' x ' ) ' ( x x ' ) ' ( A ' B ' C ' ) ' + D= ((A'X)' + D)((B' x')' + D)((XX' )' + D

((A'B'C')' + D)= ((A + D) + X')( (B + D) + X)((XX') '

+ O ) ( A + B + C + D )N o w m a k e th e s u b s t i t u t i o n s i n d i c a t e d b y

(* ), rever se steps, and conc lude that= 6 . T h i s c o m p l e t e s t h e i n d u c t i o n s t e

and the proof of the theorem.N o t e t h a t T h e o r e m 2 . 5 h a s t h e f o l l o w -

i n g c o r o l l a r y :Coro llar y 2.9 . Let B(S) be a free DeM o r g a n a l g e b r a . T h e n B ( S ) i s i s o m o r p h it o a s u b - a l g e b r a o f a p r o d u c t o f c o p i e s t h e f o u r e l e m e n t a l g e b r a V = [ o ~ , { , ~ JProo f: Let ~ = (:B(S) - ~I is ah o m o m o r p h i s m ~ . L e t V b e a c o p y o f V

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indexed by an element of ~, and letA= U V. Defin e F : B(S) - W by

F(e) = U (a) where : B(S) - V. The nby 2.5 F(e) = F(0) if and only if c~-6.Hence F injects B(S) as a subal gebraof ~.

R e m a r k 2 . ! 0. I n f a ct , C o r o l l a r y 2 .9 ist r u e f o r a r b i t r a r y D e M o r g a n a l g e b r a s .This dee per resu lt may be found in [2] or[5].~ . R e c u r s i o n a n d F i x e d P o i n t s

T h e A c o n s t r u c t i o n l e a d i n g f r o m aB o o l e a n a l g e b r a B t o i t s c o r r e s p o n d i n g

AD e M o r g a n a l g e b r a B i s c l o s e l y r e l at e dt o t he s t r u c t u r e o f r e c u r s i o n i n t h e g i v e nB o o l e a n a l g e b r a .

Let T : B - B be a map pin g of thefor m T(x) = ax + bx' whe re a,b E B. Thas period two; in fact

T 2 ( x ) = T ( T ( x ) ) = ( a + b ) x + a b x 'T 3 ( x ) = a x + b x ' = T ( x )o ,.

T n + 2 ( x ) = Tn(x).Hence there may be no element X E B suchthat T(X) = X. However, B does cont ainsuch fixed points. For example, ifT(x) = x', then x = x' has no solution si n t h e B o o l e a n a l g e b r a B , b u t i s s a t i s -fied by { and & inp r o p o s i t i o n 3 . I. L e t B b e a B o o l e a n a l -gebra, and let T : B - B be the mappi ng

Adescri bed above. Let ~ : B - B be thec o r r e s p o n d i n g m a p p i n g o n B . T h e n t h e r eexist elements X of B such that~(X) = X. In partic ular, we may takeX = (T(x), T2(x) ) or X = (T2(x), T(x))for any x E B.P r o o f : T h e f o l l o w i n g i d e n t i t i e s i n Ba r e e a s i l y v e r i f i e d :

T(x) : aT(x) + bT2(x),, T2(x) : aT2(x)+ b T ( x ) ' .Let X = (T(x), T2(x )). The n~ ( X ) = aX + bX'

= a ( T ( x ) , T 2 ( x ) ) + b ( T 2 ( x ) ' , T ( x ) ' )= (aT(x) + bT2(x)' , aT2(x) + bT(x)')= ( T ( x ) , T 2 ( x ) )---- X

Thus the algebraic structure of B re-flects the proper ties of period two sequen-c e s r e c u r s l v e l y g e n e r a t e d f r o m B . T h en e x t s e c t i o n d e s c r i b e s a m o r e g e n e r a l D eM o r g a n a l g e b r a o f s e q u e n c e s .

~ . S e q u e n c e M o d e l sC o r r e s p o n d i n g t o a D e M o r g a n a l g e b r a B

let 8(B) denote the set of all sequence sof elements of B with an assigned evenperiod (possibly of period 0). That is8(B) consists of sequences b = [bnJ suchthat n ranges over the integers andbn+ p = b n for all n, whe re p = p(b) isa n e v e n n o n - n e g a t i v e i n t e g e r a s s o c i a t e dw i t h t h e s e q u e n c e .

8 ( B ) h a s th e s t r u c t u r e o f a n a l g e b r aas follows:

and

H e r e t h e s y m b o lc o m m o n m u l t i p l e .

(ii) (a')n = (an_k)' whe re k = p(a)/ 2,p ( a ' ) = p ( a ) .Note that in the sequence algebra, inver-s i o n i s o b t a i n e d b y o r d i n a r y i n v e r s i o np l u s a h a l f - p e r i o d s h i f t . T h e s u b - a l g e b r aof period two sequences in 8(B) is iso-morphi c to B. As it stands, the seque ncea l g e b r a i s n o t a D e M o r g a n a l g e b r a .

(1) (ab) n = (an bn) , Ca + b) n = a n + b nPCab) = P(a + b) = Icm(p(a), p(b))

if p(a) ~ oand p(b) + O,0 otherwise,

l c m d e n o t e s l e a st

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Axio ms (1 ), (ll), (lii), and (v) are satis -fied but there is no available choice for0 or 1. If 8 (B)p denot es the sub-algebra of sequences of period p thenwe may take 0p and lp to be the cons-tant sequences of zeroes and ones respect-ively, with assigned period p. This gives8(B) p the structure of a De Morgan alge-bra for each p. We could force 8(B) in-to the mold by taking a quotient constr uc-tion, but I belie ve it is more interest-ing to leave it as it stands . Our rulesfor combining sequences of differentperiods provide a simple model of inter-ference phenomena that bears investigat ionon its own grounds.

We may regard 8 (B) as a set of peri -odic oscilla tions. In this regard it isintere sting to compare our ideas with thesuggestions of G. Spencer Bro wn in hisbook Laws of Form ([4]). Spencer Brownsuggests that an element X satisf yingX' = X might be seen as an oscil lati on(if its O, the n its l, th en its O, the nits l, ...). Tak ing this liter ally, wemight form a diagram:-n vq F -LY-L_In each case, the spatial s ense in whic h

i and ~t = ~ involves ordinary in-vers ion plus a left-right shift. With Z"and ~ synchr onized (and note that herethe synchron ization has become the spatialrelati onshi p of the two sequences) as

~= O . This is the moti-bove, w e havevation for our constructio n of 8 (B), andalso for the cons truc tion B ~ of sec-tion 2.

There are many connections be tween ourdiscussi on and Brown's work. These willbe explored in another paper. I wouldlike to remark here that it is strikingthat once one lifts the law of the exclu-ded middle f rom Bool ean algebra, there isopened up the possibility of infinitemodels involvin g simple analogs of wave-

forms and interfe rence phenomena . Thistemporal, mus ical aspect is prec isel ywhat is prohibited by the stark all ornothing of two-valued logic. When we dropthese restrictions, the result is notfuzziness and ambiguity, but the preciseemergence of patterned forms, spatial andtemporal.

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B i b l i o g r a p h yBialynicki-Birula, A. and Raslowa,H., On the repres entat ion of quasi-Boolea n algebras. Bull. Acad. Polon.Sci. Ser. Sci. Math. Astronom. Phys.(1957), 259-261.Balbes, R. and Dwinger, P., Distr i-butive Lattices, Univ. of MissouriPress (1974).Berman, J. and Dwinger, P., De Mor-gan algebras - free products andfree algebras, (unpublished).Brown, G. Spencer, Laws of Form, TheJuli an Press, Inc., New Yor k (197 2).Kalman, J. A., Lattices w ith involu-tion, Trans. Amer. Math. Soc. 8 7( 1 9 5 8 ) , 4 8 5 - # 9 1 .Kauffman, L. H., Net work synthes isand Varela's calculus, (to appear inInt. J. Gen. Syst.).Varel a, F. J., A calculus forself-ref erence, Int. J. Gen. Syst. 2(1975), 5-2#.Vare la, F. J., The arithme ticof clos ure, Pro6 ress i__n Cyb ern eti csand Systems Research, V ol. 3, edit.by R. Treppl, G. Klir and L. Riccl-ardi, Hemisp here Publicatio ns (Wiley)N.Y. (1977).

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Louis H. Kauffman- De Morgan Algebras - Completeness and Recursion

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Transcript of Louis H. Kauffman- De Morgan Algebras - Completeness and Recursion