Zeitschr. f. neath. Logik ztiid RrundEagen d. M d h . H d . 32 , S. 393 - 330 ( 1 9 8 6 )

A RELXTIOSSHIP BETWEEN ORDISARY FUNCTIONAL COMPLETENESS ASD -4 GENERALISATION O F THE CONCEPT

by ALAS ROSE in Xottingham (England)')

U'e shall consider the relationship between t'lie classical concept of functional com- pleteness and a modified form of the concept of D,-completeness. This was recently introdnccd by the author [5J and is very similar to concepts investigated by ROSEN- BERG [9] and several othxs. The topic was first introduced by KUDRJAVCEV [I] and an extensive list of references is given in [g]. A set' of gates corresponding to functors G', . . . . ~ G, of n l , . . . ~ n,, arguments respectively and dela.ys of d i , , . . . , din, units of time between the timcs of signals to the ni inputs of the ith gate and the tim: of it,s output ( i = 1. . . . , h ) detcrmines a definition of a D,-formula as follows, a D,-formula k i n g an ordered pair ( P . 6) where P is a formula of the convtmtional propositional cal- cnlus n-lth G I , , . . , G, as t'he primitivr functors and 6 is a non negative integer.

(I) If P is a pivpositional variable: then ( P , 0) is a D,-formula. (11) If ( P , , E ~ , ) , . . . ~ (Pni, tin,) are D,-formulae and Sij + e i j = lei ( j = 1,

then (GiP, . . . Pni ~ i t) is a D,-formula ( i = 1, . . . , h ) . (HI) If (P , d) is a D,-formula and t,he truth-value of P is independenL of the truth-

d u e s of its proposit'ional variables, thon ( P , E ) is a D,-formula (& = 6 + 1, d + 2 , . . .). ( 1 1 7 ) An expression cannot be a D,-formula except in virtue of (I), (11) and (111). A set of gates is said to be D,-conzplete if, given an arbitrary positive integer n and

an arbitrary truth-t.able Jn relating to the combination of ?z formulac. there exists, for some uon-negative integer 6, a I ) , -formula (P , 6) such that, the formiila P corresponds to J,, .

, G, which det.:>rmiw n func- tionally complete systcni, do there exist positive integers a,,. ( j = 1, . . ., ni: i = 1, . . . . 6) such that the corrosponding set of gates is D,-complete. We first show that the integers never exist if 6 = 1.

In this case therc cannot exist D,-formulae (P. 6) for which 6 > 0 unless %he in- tegers 6 , , , . . . , a,,, are all equal. Thus if we make t'hc definition

We shall consider the question, given functors G I ,

W-P = d f GI€'. . . P

the only D, -formulae containing no propositional variables other than t'hat, denoted by the syntactical variable P are the formulae (W'P, nd,,) (n = 0, 1, . . .), unless one of the formulae W"P is of constant, truth-value. In that' case, if this constant truth- valuc is denoted by c . the formula W P takes t'he truth-value c when P takes this truth- value and G , P , P,, takes the truth-value c when P, . . . . P,, t'ake the tmt'li- valip c . (Let 2 (1; . . . , m} --t {I. . . . . m> bc the associated function and let

= ran(w), Q i + , = ran(w I Q i ) (i = 1, 2, ). An easy induction shows that (7, 5 ,(?,-, [ i = 2, 3, I I ,), If Gn is a singleton [ c ) but NC(Sj,,-,) > 1, then Q , c Q i - , ( i = 3, 3, . . . . n ) and ~ ( x ) = G (x E In particular W ( G ) = a , ) Thne t'hc functor Q,

I ) Dr. ALAS ROSE died suddenly in April 1986. Ol*

324 A. ROSE

is not' a Sheffer funct'ion (even if functors of more than two arguments are allowed), contrary to our hypothesis. If none of the D,-formulae (W"P, nd,,) correspond to formulae P of constant truth-value, then, of course, I),-completencss fails.

The main object of this note is to establish, subject to a slight modification of the concept of a D,-formula, the opposite result for the Z-valued case whenever b > I . (The change does not invalidate the previous proof.) We shall then revcrt to thc original definition of a Ill-formula and consider th? effect. on the results whcn b > 1, first in t'he 2-valued case and subsequent'ly in many-valued casm. Wc introduce a new clause (denoted by (IIIA)) and make the obvious modification to clause (IV).

( I I IA) If ( P k : F : ~ ) are D,-formulae, P, always takcs the trut,h-value ck ( k = 1, . . . , ec - 1, u + 1, . . . , ni), G,&, . . . Qni takes the trut,h-value d whenever Q1, . . . Qufl, . . ., Qni take the truth-values c I , . ., c u p , , c , + ~ , . . . , cni re- spectively and cik + dik = A, ( k = 1, . . . , u - 1: u + 1, . . . ~ ni), then (GiP, . . . Pni , &) is a D,-formula irrespective of the values of the syntactical variable P, and the dalay di, (i = 1, . . ., b) .

ilTe first show that it will be sufficient to establish the exist'ence of positive integers 6 , ,, . . . , Bbn, in t'he following 5 cases, where gi(xl, . . . , xni) is t'he truth-value of the formula G,P, . . . Pni when P I , . . ., Pni t,ake .the truth-values xl, . . ., xni rcspectively and Qi denotes t'he ordered pair (gi(T, . . ., T), g i (F , . . . , F ) ) (i = 1, . . ., 6). We shall denote the set (1, . . . , 6 ) by J .

(I) For some i E J , Ei = ( T , F ) . (11) For some i, j E J , Ei = (T, T), E j = ( F , F ) and t8here cxist truth-values

(111) For soinc i, j E J ; g i ( x l . . . . , xni) = T in all the 2 " ~ cases, O j = ( F , T) and, n j , GjP, . . . Pnj does not always t.ake t . 1 ~ same trut,h-value as some

(IV) For some i , j E J ; ( X i = ( T , T ) , Ej = ( F , T) and the formula GiP, . . . Pni

(V) For somc i , j E J ; Q i , O j = (a, T) and i $. j .

If c F i E { (T . T ) , ( F , F ) } for all values of i in J , then functional completmess cannot hold unlecs, for a t least one value of i in J , the formula G,P, . . . P,,: is not of con- stant truth-value. Thus if the conditions of (I) and (11) both fail and those of .thg dual of (11) (relating to the equation gj (x , ~ . . . ~ xn j ) = T ) fail also, then thcre must be a member i of J for which Qj = ( F , T). If j is unique, t'hen, by the above result for the cast when b = 1, there must be a member i of J for which Q i E ((T, T ) , ( F , F ) } . If GiP, . . . Pni takes, for each such value of i, a-trut'li-value wi which is the same in all the 2"i cases and, for some 6 E (1, . . ., n j } , Q j P , . . . P,,] takes a truth-value deter- mined uniquely by that of P , then functionayl completeness must fail. Thus if j is unique, (111) (or its dual) or (IV) (or its dual) must hold. If j is not unique, then (V) holds. We do not, of course, need t o consider duals of cases alr,eady discussed.

(I) We note that if HP = T P , then HP =:T FiP . . . P . Thus, if we* assign the value 1 t o d,, ( j = 1, . . ., ni ) , we may construct the D,-formula ( H P , 1). Any signa.1 may be delayed for k units of time by constructing (HkP, k ) ( k = 1, 2, . . .) and D , - completeness follows a t once.

xl, . . ., x,,~ such that gi(xl , . . ., xni) = F .

formula built out of functors and Pa exclusiv.ely.

takes the truth-value F in a t least one of the 2"i cases.

ORDISARY FUNCTION.4L COMPLETEXESS A S D -4 GENERSLISATION OF THE CONCEPT 325

(11) lye first make the definition

G'PQ =df G i Z , . . . Zni

where 2, = P or 2, = Q according as x, = T or xu = F (v = 1, . . ., ui). If we assign t'he value 2 or 1 to Si, according as x, = T or xu = F we may simulate a decision ele- ment for the functor G' with delays of 2 , 1 of the first and second inputs resp-ctively. If. further: we assign t,he value 1 to a j , (v = 1, . . ., nj) and maks the definition

FP =df F j P . . . P

n-c may coiistruet the U,-formula (FP, 1) and, since FP is of constant truth-value, t,lie D,-formula (FP, k ) (k = 2 , 3 , ). We may then make the definitions

N P = d f G'PFP. HP = d f NNP, V P =df GFPFP,

H I P = d f G'VPHP, HiP = d f G'VPH,_,P ( i = 2 , 3, . . .)

and construct the D,-formulae

( J P > 2 ) , (HP, 4), (VP , a) (a = 3 ,4 , . . .), ( H i p , i + 4) (i = 1 ,2 , . . .).

Since H,P = P ( i = 1, 2 , . . .) we have only to delay all primary inputs by 4 units of time and all further delays required for the synchronisation of inputs are available corresponding to the D,-formulae (Hip> i + 4).

(111) We may assum?! wit,hout loss of generality, that gj(xl, . . . , xnj) depends on all of x1 ~ . . . . x n j . (Otherwise n-e may set the delays corresponding to the irrelevant variables and those corresponding t o the functor Gi (as we do in bot'h cases below) at 1 and simulatc a gat'e with a smaller number of inputs by filling the irrelevant argunicnts with the formula VP.) The non-existence of the integer 6 ensures that

\Ye consider first the case where GjP, . . . Pnj takes the truth-value T whcsnever exactly one of P , , . . . . Pnj takes t'lie truth-value F. We assign the value 1 to d i l . . . . . dini> dj1 , b j 3 , . . . , ajnj and the value 2 to dj2. We make the definitions

) I j >= 2 .

V P = d f P I P . . . P:

N,P = d f GjH,PVP. . . VP,

HkP = d f GjNk-lPVP. . . V P , NkP = d f GjHk- ,PVP. . . V P

N 2 P = d f GjVPPVP. . . VP, H4P =df N2N2P,

N6P = d f N2H4P, H6.P = d f H2H4P, ( k = 7 , S, . . .)

so that the D,-formulae (VP , a) (a = 1 , 2 ; . . .), (HkP, k ) , (NkP, k ) ( k = 6, 7 , . . .) are all constructable and D, -completeness follon-s at, once.

If the condition of tlic previous paragraph fails we ma,y assume, without. loss of generality. that g j (F . T. . . . , T) = P and make the definitions

V P =df P i p . . . P , FP =df GjPvP . . . VP.

We may therefore assign t,he value 1 to 6,, , . . . , B i n , , a j 2 , . . . , ajnj and (except where otherwise stated) the value 4 to B j , (cf. clause (IIIA) of the definition of a D,-formula) and construct t'he D,-formulae (VP , a) (01 = 1,2, . . .), (FP, B) (B = 2 ,3 , . . .). Sincc the truth-value of GjP, . . . Pnj depends on that of P, , there must exist Q 2 , . . . , Qnj

in the set, { VP: FP) such that GjPQ2 . . . Qnj = R for some R E {P: N P ) . Thus we may construct one of the D,-formulae

0) ( N P , 4), (ii) (H4P, 4).

326 A . ROSE

Similarly we may construct one of the D,-formulae

(iii) (N,P, k ) , (iv) (H,P, k )

whenever (H,-,P, k - 1) is constructablc ( k = 6, 7 . . . .). If (iv) holds. we may construct (H,P. k ) ( k = 8, 9. . . .) and D,-completeness fol-

1on.b a t once. If (i) and (iii) hold, we may construct (H,P, k ) from (N,-,P, k - 1) SO

uc may construct (N,P, 9), (H,,P, 10). (N,,P. 11) and (from N , and H,) ( N , , P , 12) so the result fo l lo~s easily. If (ii) and (iii) hold, me mag construct (H,P, 3) (provided that we give d,, the value 3) and then (N,P, '7). (H,P, 8 ) ( H J , 9 ) so the r-sult fol- lows easily.

(IV) We let all delays corresponding to G", and G, be equal to 1 and make the fol- loiiing definitions. where 5 , . . . ., xni are truth-values surh that g,(x, . . . ., r,,) = F.

N P = d f G,P. . . P. HP = d f NNP, V P GlP Y, H5P = d f G , X , . . . X n i j

where X , = V P or X , = HHP according as J * ~ = T or x, = F ( a = 1, . . . . n,) We may tbereforr- constiixt the D, -formulae

( N P , l ) , ( H P , 2 ) , (HHP, 4). ( V P . a ) (z = 1 . 2 . . . .) (H5P. 5).

Thus if we makc the definition

HkP = d f G',Yk, . . . Ya,,

wheie Y,, = VP or Ykv = H k _ ,P according as x , = 1' or x, = F (v = 1 . . . . n,) then me mag construct the D,-formula (H,P, k ) ( k = 6, 7 . . . .). Since H,P = T P ( k = 5, 6, . . .), D,-completcness follows at once.

(V) We let all d*?lays cow-sponding to G, . G, lo- equal to 2. 1 respectively and make thc definitions

N I P = d f F,P . . . P ,

H,P = d f N , N 2 P ,

N,P = d f F,P . . P , H2P = d f N,hT,P. HkP = d f N , N , H , - , P ( k = 4 , 5 . . .).

D,-completeness then follon-s by arguments similar t o those used previously. Thus the theorem is proved.

We considrr finally the question mheth-r t h y extension of the conccpt of a D l - formula is necessary. We shall show that the theor-m remains true when clause (IIIA) is deleted except in th? case xihere the truth-table of thy functor G, of case (HI) is deflncd by the tqiiation G,P, . . P,, = T NA"J- 'P, . . . P,, in mliich case it does not remain true.

In all othcr case5 wliich involved the iisc of clause (IIIA) (th.+-, b-ing sub-cases of case (111)) w e may assume, without loss of generality. that for somp positiw integer u, g , ( T . . . .. T , F . . . ., F ) = T , the number of T's on th- left hand side, of the equa- tion heing equal to u. We let d , , , make (using an ohvious notation) the dpfinitions

1 h,, = l ; 6,,+, , . . . , d,,, = 2 and

V P = d f G , P . . . P. N , P = d f G,I'I' . . T'Pf'. . P. H 4 P = d f N 2 N 2 P , FP =df N2VP.

Thu. n e may construct the D, -formulae

(VP.31) (a = 1.2.. . .). (N,P, 21. (H,P. 4). ( F P , p ) (p = 3. 4.. . .).

ORDINARY FUNCTIONAL COMPLETENESS AND A GENERALISATION OF THE CONCEPT 327

By hj-pothesis there exist truth-values x 2 : . . ., xnj such that gj(T, x 2 , . . . , xn,) + + g j (F . x 2 , . . . ~ xn j ) . Thus if X , = VP or X , = FP according as xu = T or x, = F (,zi = 2 . . . . , n j ) ? the formula GjPX, . , . X,, will take the truth-value of P or of hTP according as gj(T, x 2 , . . ., xnj) is equal to T or to P.

In the case wherc it is equal to T me may make the definitions

HkP = d f GjfIk_ ,PX2 . . . Xnj ( k = 5 , 6, . . .) and const.mct, the D,-formulae (HkP, k ) (k = 5 , 6, . . .). The result, then follows a t once. In the other case we may make the definitions

L V ~ P = d f GjH4PX2 . . . Xnj.

HkP = d f GjNk_,PX2 . . . X n j .iikP = d f GjHk_,PX, . . . xnj

NGP = d f N2H4P: ( k = 6, 7 , . . .), ( k = 7 , 8, . . ,).

Thus we may construct the D,-formulae (HkP, k ) ( k = 6, 7 , . . .) and the result fol- Ion-s at once.

== NAnj-lP1 . . . Pnj and we can construct a D,-formula (Q, 6) s not of constant truth-value and (ii) 6 > 0, then Q must be of the

form G j R , . . . Hnj whew (B,, 6 - a j U ) is a D,-formula (v = 1 , . . ., nj). If 6 is the least integer satisfying (i) and (ii), t'hen, for w E (1, . . . , nj} , either R, is of constant truth-value or 6 = a j , . In t'lie lat'ter case R, is a propositional variable. The former case niust, arise a t least once unless the delays for G j are equal. If tho constant truth- values are all equal to T, then Q takes the constant truth-value F, (assuming inequal- ity of delays) cont'rary to hypothesis. If at, least one constant truth-value is equal to F a then, for the corresponding value of v .

6 - d j , 2 d j , , . . . ) ajnj and none of R, , . . . ~ Rni is a propositional variable. Thus Q is of constant truth-value and KP should have a contradiction.

If the delays are equal and (P, 6) is a D,-formula such that P contains no proposi- tional variablrs other than those denoted by Q, R, then P must take the same truth- value as one of the formulae t . /, Q, R, NQ, NR, AQR: NAQR, KQR, NAQR. Thus D,-conipletencss would fail, contrary to hypothesis.

The ewe where D,-completeness fails and b = 2 is, of course, sliglitly artificial as the primitives are not, independent. Furthermore the functor F , is ZL close approxima- tion to a logical constant, which could reasonably be regarded as available in machine logic. However, in many-valued proposit.iona1 calculi, we can give examples of sets, of all finite cardinalit,ies, of functors which are functionally complete and independent, but to nhich no positive integers 6 , I , . . . , abnb giving D,-completeness correspond. These examples relate to m-valued propositional calculi for particular values of m, but n-e shall also consider the case where b > 1 and 3 5 m < R ~ . Let us consider first the ni-valued propositional calculus where m = a, . . . ak + 1 ; a , = 2 ; a,2, and a 2 , . . . , a , are distinct primes. Let b = k + 1 ; n,, . . ., nk = 1 ; nk (using an obvious notation)

f i ( l ) = f i ( 2 ) = 1 + (m - l ) / a i , Fkf lP lP2 = T NCNP,P, .

f i (x) = 1 (x = 3,

Theorem. The functors P,, , F,,, form a complete set of independent connectives for the wb-valued propositional calculus and there do not exist positive integers 6, ,, . . , , d,, , d,,, ,, , d,,,,, such that the corresponding set of gates i s D,-complete.

Since m = a , . . . a, + I , m - 1 is not divisible by 3 and the propositional calculus with F,+, as the only primitive funct'or is, by duality [6] and a result of [ 7 ] , a system in which the functors C, N of LUKASIEWICZ 121 are definable. Thus the formula F,CPP always takes the truth-value 1 I- (m - l) /q or, renaming the truth-values I , 2 . . . . , m as I , (m - 2) / (m - I ) , . . ., 0 respectively, the truth-value 1 - a;, ( i = 1. . . . , k ) . Since the least common denominator of t'hese k rational numbers is m - 1. n.o can, by a theorem of MCNAUGHTON [4], const'ruct a formula @(F,CPP, . . . , F,CPP) which always takes the trut'h-value (m - 2)/(m - 1) (or 2 ) and functional complet'eiiess fol- lows from a generalisat,ion, by ROSSER and TURQUETTE [lo], of a theorem of SEUPECKI [ 1 21.

If Ei is the set of those (rational) trut,li-values of the %-valued propositional calculus which are of the form aa i / (m - I ) , where a is art integer, then Qi is closed with respect to t,he functions f l ( .), . . . , .), f i + l ( .), . . . , f k + , ( .) but 1. E @,, f i ( l ) = 1 - a;' and not f i ( l ) E Qi (i = I , . . ., k ) . Thus the functors F , , . . ., F , are independent and the independence of F,+, follows a t once since t.he ot,her k functors are unary.

The proof that positive integers d,, . . . , 6,+,,, do not' exist is somewhat similar to the proof for the functors F , , F, of the 2-valued logic referred to in the first para- graph. We shall suppose first that &+,,, = S,,,,, and denote the common value by 6. We shall suppose further that the formula P is built' out of the symbols p , F , ~ . . . . F,+, exclusively and that P takes the trut'h-value f ( x ) when p takes the truth-value z. We shall establish D,-incompleteness by showing that if E > 0 and (P , E ) is a D,- formula, then f (1 ) = f (2) . If f(1) + f ( 2 ) , let E be the least positive integer such that, for some P, (P , E ) is a D,-formula and f(1) + f (2 ) . Since E > * O : P is of one of the forms Pip' ( i E (1, . . ., k } ) , Fkf lP 'P ' ' . In the first case (using an obvious notation) / ( I ) = f i ( f ' ( l ) ) = / , ( / ' (a ) ) = f(2),unlessP'isp, inwhicli case f(l)= f ( 2 ) = 1 + (m - 1) a;'. I n the second case f(1) = f,+,(f'(l), f"(1)) = f k t l ( f ' ( 2 ) , f " (2) ) = f ( 2 ) , unless E = 6, in which case f(1) = fk.tl(l, 1) = m = fk+, (2, 2 ) = f ( 2 ) .

If 6,+,,, + 6k+1,2; we consider, as in the previous paragraph, the D,-formula ( P , E )

for the least positive integer E such that f (1) =+ f ( 2 ) . If P is of the form F,P' (i E ( I , . . . , k } ) , the proof is as above. If P is of the form F,+,P'P" and d,,,,, > > S,+,,,, thcii Pr' is not p and f"(1) = f"(2). If P' is not p , then

f ( ' ) = f k + l ( f ' ( l ) , f " ( l ) l = f k + l ( f ' ( 2 ) , f " (3) ) =

If P' is p , t,hen P" must be built out of p , F, , . . . , F , exclusively. (Otherwise the corresponding D,-formula (P", 2) would be such that L >= 6,+,,, and (P , E ) could not be formed from ( p , 0) and (P", A).) Hence f"(2) + m and

f ( ' ) = f k + l ( l , f " ( l ) ) = f k + l ( 2 , f " (2 ) ) = f ( 2 ) .

The proof for the case where 6,+,,, < is similar. Thus the theorem is proved.

We have shown that the number of independent primitives is unbounded by giving a set of cardinality k + 1 for the case where m = a , . . . a, + 1 . This does not, of course, establish any result for the cases where b > 1 and m > 2 , unless m is of this

ORDINARY FUNCTIONAL COMPLETENESS AND A GENERALISATION O F THE CONCEPT 329

form. However a set of cardinality 2 can be constructed in all the remaining cases. Tl’e let f l ( l ) = f l ( 2 ) = 2 ; f l ( z ) = 1 (z = 3 , . . ., m), F,PQ = T NCPNQ, unless m 3 1 (mod 3), in which case the truth-table of the functor F, is as abovs except that (du- alling a result of [S]) f2 (m - (m - 1)/3, m - (m - 1)/3), f2(m - 2(m - 1)/3, m - (m - 1)/3) = 1. The proofs of functional completeness, independence, and D,- incompleteness are very similar to those given above since (a) f l ( z ) 5 2 (z = 1, . . . , m)

TI~L? changes to the function i2(., .) when m 3 1 (mod 3) are not strictly necessary. Although (as, using rational truth-values, f,(1/3, 1/3) = 1/3) the functor F , does not generate the Lukasiewiez propositional calculus, the functors F , , F , do, together, genzrate it and functional completeness follows as in the proof of the theorem. This is liecause we may makc the definitions

and (B, f 2 ( Y . z ) = m (Y, z E { 1 , 2 ) ) .

V2P = d f F,F,P, V,,,P = d f F,V,PV,P, N p = d f F,V,,,PP,

CPQ = df NF,NPQ,

the formula V,P always taking the truth-value i (i = 2 , . . . , m). Similarly the in- equalities a,, . . . , ak > 3 in the definition of the integer m of the theorem arc un- necessary, the only essential requirements being that a , , . . . , a, 2 2 and af , . . . , a, are distinct primes. As immediately above, F,F,PF,P is a suitable definition of V,,,P and the rest of the proof is unchanged, as is that of D,-incompleteness.

If the sets of functors generated by the functor F , and by the functors C, N are denoted by 3, h j respectively, we have so far considered only cases where 3 = $ or 3 c 8. Corresponding results may, however, be established for ccrtain cases whcre 3 2 @ and the sets 3, @ (with respect to inclusion) are incommensurable. For example let us consider the cases where m = 5 , 9 and the truth-table bf the unary functor W corresponds to the transposition (34). Let F, be as in the previous 2 paragraphs and the truth-table of the functor F , be defined by the equation F,PQ = NCNP WQ.

We first shom that

(i) The original proof of D,-incompleteness remains valid. (ii) If m = 5 , then 3, ,@ are incommensurable. (iii) If m = 9, then 3 2 @.

Result (i) is immediate since (00, (/I) are unaffected by changes to the function f 2 ( . , . ) and (ii) follows from the equations f2(5, 3) = 2 (or, using rational truth-values, f2(0, 1/2) = 3/4), f2(4, 4) = 4. In order to establish (iii) it will be sufficient to define C and N since f2(9. 3) = 6 (or, using rational truth-values, f2(0, 3/4) = 3/8). We make the definitions

V,P =df F,PP, V7,P =df F,DDDPDDDDP, N P = d f F,PV,P, W P = d f NF2V9PP, CPQ = d l NF,NPWQ.

In the 9-valued case functional completeness follows a t once from (iii). In the 5-valued case we make the definitions

v,r -d&Flr , v,r =dfP2vlrvlr, ivr =df P ~ P J J J I , WP = df N F , V p Y . CPQ = df NE”,NPW&

and functional completeness follows at once.

1 3 3 3 P 2 3 3 1

3 3 1 2

R,eftwnces

, 5'. B., Completeness theorem for a class of automata without feedback conplings (Russian). Dokl. Akad. Xauk SSSR 182 (1960), 272-274. English translation: Soviet Math.

[a ] L~-IL~SIEWICZ, J . , and A. TARSKI, Untersuchungen iiber den Aussagenkalkiil. Comptes rendus (!Tarsaw), Classe 111, 23 (1930), 30-50. (For an English translation see [13].)

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(Eingegangen am 13. April 1984. nherarbeitete Fassunp vom 31. Oktoher 1984)