Zt'ilschr. 1. math. h g i k und Urundlagen d . Math. Bd. SO. S. 177-182 ( 1 9 8 4 )

GENERALISED FUNCTIONAL COMPLETENESS OF SETS OF rn-VALUED SHEFFER FUNCTIONS

by ALAN ROSE in Nottingham (England)

I t has been shown [l] that if G , H are') non-conjugate Sheffer functions of the 3-valued propositional calculus then the propositional calculus whose only primitive symbols are propositional variables and binary variable functors with values in the set {G, E I } has generalised functional completeness [3]. It has also been shown [4] that this result may be used as the basis of an induction, thus proving that if G , . . . , G, are non-conjugate 3-valued Sheffer functions, the generalised functional completeness property holds for the set { G , , . . . , an} ( n = 2, 3, . . .). The argument of the induction step of [4] remains valid if the integer m is such that 4 5 m < et0

so, in order to establish the generalised functional completeness of the propositional calculus corresponding to the set { G , , . . . , G,} of non-conjugate Sheffer functions (n = 2 ,3 , . . .) in the m-valued case (4 5 m < N ~ ) , it will be sufficient to consider the m-valued propositional calculus corresponding to the set { G, H } of non-conjugate Sheffer functions. As a first step we shall prove that, if 3 5 m < u0, we may construct a formula GXy from the symbols p , S exclusively which takes the truth-value z ( y ) whenever S takes the value G ( H ) (x , y E (1, . . . , m } ) This has been proves already for nL = 3,4, [l], but the present proof is simpler. We shall then prove generalised functional completeness whenever 3 5 m < eta, the proof being simpler than that given previously for the case where m = 3 [l]. We shall, however, make use of a thr- orem related to the m-valued case proved in [I]. (See also [Z].)

Theorem 1. I n the m-valued propositional calculus whose only primitive symbols are propositional variables and binary variable functors with values in the set ( G , H } of non- conjugate Sheffer functions we may construct a formula ex,, containing no symbols other than S, p , which takes the truth-value z ( y ) whenever S takes the value G ( H ) (z, y E { 1 , . . . , m } ; nl = 3, 4, . . .).

Lemma 1. There exists a functor W in the set { G , H } such that, if X is the other mem- ber of this set, then, for some truth-values x , y , z , ( y =t= z ) we m y wnstruet formulae QXg, QXz from the symbols p , 6 exclusively which take the truth-value x whenever 6 takes the d u e W and the truth-values y , z respectively whenever S takes the value X .

Since G is a Sheffer function we may construct, for some truth-value k , a formula !P,,(S, p ) which takes the truth-value 1 whenever 6 takes the value G and which takes the truth-value k when 6 takes the value H , for some truth-value w of p . Since H is a Sheffer function we may construct a formula AJS, p ) which takes the truth- value w whenever S takes the value H . Thus we may take Glk to be the formula !?',,(S, AJd, p ) ) Since G is a Sheffer function we may construct a formula @,(S, p ) which takes the truth-value u whenever 6 takes the value G (u = 1, . . . , m ) . Thus the formula @,(S, @,&) takes the truth-value u whenever 6 takw the value Q arid some truth-value v (= v(u)), independent of the truth-value of p , when 6 takes the value H .

') We shall often abbreviate G( , ), H ( , ), 6( , ) by G, H, 6 respectively. 1" Xtschr. f. math. Logik

178 A R O S E

Thus we may construct m formulae cDuU (u = 1 , . . . , m). If the values of v correspond- ing to them values of u are not all distinct we may construct formulae CD",~ , CDu2v(uL + u,) and then take W = H , x = v , y = ul, z = u,. If v( l ) , . . . , v(m) are all distinct then, since G , H are not conjugate, they are, in particular, not conjugate with respect to the permutation corresponding to the function v( ). Thus, for some truth-values a, #? v(g(ar, #?)) + h ( v ( a ) , w(p)). Hence the formula 6@uv(u)CDpu(B) takes the truth-value g(a, #?) whenever 6 takes the value G and the truth-value h(w(a) , w@)) whenever 6 takes the value H . Thus we have constructed the formulae @ g ( u , p ) , , ( ~ ( u ) , u ( f l ) ) ,

and the lemma is proved. Lemma 2 . There exists a truth-value w such that the formulae a,, , . . . , Qwm are all

constructable. Since X is a Sheffer function we may construct a formula E(6, p , , . . . , p k ) which

takes the respective truth-values al, . . . , ar2t under the 2k assignments of truth-values from the set {y, z> to p , , . . . , p k when 6 takes the value X . Thus, if k 2 log, m, we may construct a formula E(6, p , , . . . , pk) which takes the truth-value i when 6 takes the value X and p , , . . . , pk take the truth-values B,,, . . . , pkl respectively (i = 1, . . ., m), where (#?,,, . . is the i th ordered n-tuple (with respect to a given ordering) of truth-values in the set (y, z}. Hence E(6, Qxpl i , . . . , Q x B k r )

takes the truth-value i when 6 takes the value X and the truth-value E ( W , x, . . . , x) when 6 takes the value W ( i = 1, . . . , m). (Here E(W, x, . . . , 2) denotes the truth- value of z(6, p I , . . . , p k ) when 6 takes the value w and p , , . . . , pk all take the truth- value z. Similar notations will be used elsewhere without comment.) Thus we may take w = E(W, 2,. . ., z) and the lemma is proved.

Proof of t h e Main Theorem. We may suppose, without loss of generality, that W = G . Since G is a Sheffer function we may construct a formula t , (S , p ) which takes the truth-value x whenever 6 takes the value G (z = 1: . . . , m). Let us denote by @,(6, p l , . . . , p , ) the formula obtained from [,(S, p ) by replacing the i th occurrence of p (reading from left to right) by p i (i = 1, . . . , T), where T is the number of occur- rences of p . Since H is a Sheffer function, there exists an assignment alY , . . . , arY of truth-values to p,, . . ., p, respectively under which the formula @,(H, p l , . . . , p,) takes the truth-value y (cf. [41). Thus @,(6. @ w u l y , . . . , @,,,,) is a suitable choice for QXy (z, y E {I, . . . , m}) .

Theorem 2. If G , H are non-conjugate Sheffer functions, the propositional calculus of Theorem 1 has generalised functional completeness (m = 3 , 4, . . .).

Let V,( ) denote the functor such that V , ( p ) always takes the truth-value x, Wmm-,+,( ) denote t,he functor V,( ) (x = 1, . . . , m) and W,( ), . . ., Wmm-,,,( ) denote the remaining unary functors. Furthermore let W , ( p ) = p .

Lemma 1 . We may construct, far some integer i i ? z the set (1, . . . , mm), the mm for- niulae i l i j ( S , p ) such that Ai,(G, p ) == W , ( p ) , - ; l i j ( H , p ) == W , ( p ) (j = 1 : . . . , d").

Since H is a Sheffer function we may construct a formula E(6, p , y, , . . . , q",) such that, if al,, . . . , m,, is the jth assignment (with respect to some given ordering of the assignments) of the mm assignments of truth-values to q, , . . . , ym,

Thus the formula E(B, p , @ , u , , , . . . , @ , u m j ) takes the truth-value of W , ( p ) whenever B takes the value H and the truth-value of ;(a, p , 1, . . . , 1) whenever 6 takes the

@ g ( u , p ) , v ( g ( u , f l ) ) and we may take W = G , x = g(a , #?,I y = h(w(a) , .(#?)), = v(s(., 8,)

( ( H , x , a , , , . . .,a,,,) = w,(x) (x = 1 , . . . , m ; j = 1 , . . . , m m ) .

GENERALISED FUNCTIONAL COMPLETENESS OF SETS OF m-VALUED SHEFFER FUNCTIONS 179

value G. Thus we may choose i in such a way that Z(G, p , 1, . . . , 1) = T W,(p) and 3 6 , p , G I a r J ' . . ., GlmmJ) is a suitable choice for A,,(S, p ) .

Lemma 2. For some integer i in the set { I , . . . , mm - m} we may construct the mm formulae A , , ( S , p ) which take the truth-values of W, (p ) , W j [ p ) when S takes the values G , H respectively ( j = 1, . . ., mm).

If the integer i of Lemma 1 is at most mm - m there is nothing to prove. We now assume that i > mm - m and prove first that the formulae &(d, p ) ( k = m" - m + 1, . . ., mm; j = 1 , . . ., mm) are all constructable, noting that this part of the proof does not make use of the inequality (cf. the definition of &,(a, p ) below). Since G is a Sheffer function we may construct a formula !Pk(S, p ) which takes the truth-value k - mm f m whenever S takes the value G. Let p occur 1 times in this formula and let Ok(S, p , , . . . , p , ) be the formula obtained from it by replacing the nth occurrence of p by p,, (n = 1, . . . , 1) . Since H is a Sheffer function we may ([4]) choose integers PI,, . . . , P I , from the set {1, . . . , mm} in such a way that

@ k ( H , w j , J ( P ) ~ . . ' 9 w/ l , , (P ) ) = T w , ( P ) . Hence the formula @,[d, Aijl,[d, p ) , . . ., AIjlJ(d, p ) ) is a suitable choice for &(a, p ) ( k = mm - m + 1,. . . , m . ; j = 1, . . . , m m ) .

Let the total numbers of occurrences of p and of q l , . . . , qn, in the formula ~ ( d , p , q l , . . . , q,,,) of Lemma 1 be h , m' respectively and let the formula obtained from it in the usual way be denoted by S'(S, p l , . . ., p,, , r l , . . ., r,,,,), the variables replacing q,, having lower suffixes than those replacing q b if a < b . Since G is a Sheffer function there exist truth-values xl,. . . , x,,+,,,,, y, . . ., yh+,,,, such that

Hence there exist truth-values z, , . . . , zh+,,,, , 01 such that, for some integer s in the set { I , . . ., h + m'}, ('(G, z , , . . . : Zh+m') =k t ' (G, zl, . . . , z,-,, 01, z,+,, . . ., z , , + ~ , ) . Thus

(A) the formula Z ( G , z,, . . ., z,-,, p , z,+,, . . . , zh+,,,,) takes a t least 2 distinct truth-values

and, if the multiplicities of al,, . . .,am, are the numbers of occurrences of q l , . . ., qm in E[d, p , q, , . . . , qm) respectively,

(B) Let

-

s 1 5 . . . ? xh+nr') * t'(G? y1 7 . . - 7 yh+m').

S ' ( H , p , . . - 3 p7 aljJ . . ., ~ 1 1 , . . . )anr j , . . .) an,;) W , ( P ) .

CB(d, P ) = ~ ~ m n ~ - m + z g , l ( d ~ P )

C,(% PI = p (B € ( 5 ) n (1 , . . . I h ) ) I

(B E { I , . . . , h } - {a}),

c h + , ( d , p ) = @zhrpapr, (B c h + / l l S , p ) = A k " ' n r + ~ p ~ , , l ~ d ~ p ) (B

{ I 3 ' . ' J '"'} - (' - h } ) :

{' - h} A {l, . ' .> m ' } ) l

where &,(d, p ) is constructed in the same way as &,(d, p ) except that the riles of G, H are reversed (a , b E { 1, . . . , m } ) and, in (B), occurs in argument number 1 + h + #I of E(, . . . ~ ). The formula

%'(d, 5 1 (6, P ) , . . ., Sh+m'(Sj P I ) then takes the truth-values of E'[a, zI , . . . , z S - , , p , a,,, , . . . , zh+,,,,), Z ( H , p , . . . , p , n , ; , . . .. nI.,, . . . , a,,,,, . . .!a,,,) when 6 takea the values Gs H respectively and the lemma follows at once from (A) and (B).

12.

I MI A. HOSE

We define a proper subset of a set r5 to be a set 9 which strictly includes the empty set and which is strictly included in &.

Lemma 3. There exists a partition of the set {I, . . , , m} into proper subset8 { n , . . . ., nk), ink+,,.. .,A,) such that we may construct mm+2 formulae YX,,(d, p ) (.c. , ~ j E 11, . . ., m } ; j = 1, . . ., mm) which satisfy the equatiom y+,,,(G, na) = x (n = 1, . . . , k ) , yxy,(G, n,) = y (a = k + I , . . ., n ~ ) , yxYr(H, OC) = wr(a ) (OC = 1, . . ., m).

Let the formula W,(p) of Lemma 2 take the truth-value /? when p takes the truth- values n, , . . .! nk but not otherwise. (Since i 5 m’” - m we can choose k , n, , . . . , nk , /? for which 1 k <= m - 1.) Since G is a Sheffer function we can construct a formula @(b. p ) such that @(G, p ) takes the truth-value x when p takes the truth-value /? and otherwise takes the truth-value y . Let @(d, p ) contain exactly h occurrences of p and let a,(s, p , , . . ., p h ) be the formula obtained from it in the usual way. Let

Then a suitable choice for Px,,,(d, p ) is Wulj( 1, . . . , Wuh;( ) be functors ([41) such that Q,W, W a I J ( p ) , . . . , Wuh,(p)) = T W,(p).

Q,(d, L1imlJ(d, P I , * . . / 1 , u h , ( d , P I ) . Lemma 4. If n E (2, . . . , m - l} and the set { I , . . . , m} ,is partitioned into ?L proper

subsets 8, , . . . , &,, then there exist (not necessarily distinct) truth-values a , , . . . , a, such thut, if the formula W ( p ) takes the truth-value ak whenever p takes a truth-value in &, ( k = 1 , . . ., n) , then, for some unary functors W‘( ), W”( ), the formula W”(G(W(p) , W ‘ ( p ) ) ) takes any R + 1 (not iiecessarily distinct) truth-values corresponding to some sets S,, . . . , and there exist truth-values a’, , . . . , a:, such that, in a corresponding way. for some unary functors W”’( ), W””( ), the formula W””(G(W”’(p), W ( p ) ) ) takes a q 21. + 1 (not necessarily distinct) truth-values corresponding to some sets 3, , . . . , 3,,+, , the I I + 1 sets being independen.t of the choice of truth-values.

\Ve shall prove only the existence of the truth-values a,,. . .,a,,, functors W’( ), W”( ) and sets S,, . . . , S,+, as the proof for a ; , . . . ,a:, W’”( ), W””( ), Y l , . . . , 3,,+, is similar. Since Q is a Sheffer function some row of its truth-table must contain a t least 2 distinct entries (i.e. g(x, y ) + g(z, z ) for some truth-values x , y , 2). Hence there exist truth-values y , , . . . , y,, such that the corresponding rows contain a t least n + 1 distinct entries. We may suppose, without loss of generality, that these entries are of truth-values P I , . . .,/?,+, and that . .,Pa, occur in row

ye ( k = 1, . . . , n’), where a, = 0 , a,. = n + 1, a,. - a,,,-l 2 2 and that 8, 2 2 (a, < a, for 0 5 i < j

- -

n’). Thus we may take

mak_ ,+ 1 3 . . . 9 mak = Y k ( k = 9 * . . I ?t’ - I ma,#_l+l 9 * * 9 = y n

and the numbers of occurrences of y , , . . . , y,,, in the truth-table of W ( ) will be at least a l , a2 - a , , . . .,a,,. - a,.-, respectively. (As a , , a, - a,, . . ., a,,-l - a,.-, of tho classes b , , . . . , 8, correspond to y , , . . ., Y, , , -~ respectively, then, a fortiori, the numbers of rows of the W( ) truth-table will bc sufficient. Since an, = n + 1 the number, n - a,,-, , of classes corresponding to y,,, will be equal to a,,,. - u , , - ~ - 1, of which, since a,, - a,,,-, 2 2, the last is &,,. As 2, 5 2 thcre will be at least a,. - a,,-, rows corresponding to y,,..) Hence we can choose the truth-table of W’( ) so that G ( W ( p ) , W , ( p ) ) takes the truth-values P I , . . . , /?,+, (and no others) for some truth-values of p . We can then choose W “ ( ) so that /?,, . . . , b,,+, may be replaced

-

GENERALISED FUNCTIONAL COMPLETENESS O F SETS O F 771-VALUED SIIEFFER FITNCTIONS 181

by any n + 1 truth-values. Thus the sets induced by our choice of al, . . . , 6,. I+"( ) are independent of the n + 1 truth-values.

Lemma 5. If .F1 , S2 is a partition of the set {l , . . . , m} into proper subsets then there mist truth-values c , x , y such that either h(c, x) E F1, h(c, y ) E 9, or h ( z , c ) E -F, ,

We shall suppose that the lemma is false and that h(1, 1 ) E F i . It follows at once that h(1, a ) E 9, (a = 1 , . . . , m) and therefore that h(p, a) E Sti (a, b E {1, . . . . N L ] ) .

Thus H ( p , q) cannot take a truth-value in S3-l, so H is not a Sheffer function and we should have a contradiction.

Lemma 6. If W , ( ) is the functor of Lemma 2 with the roles of G, H reversed and .rs y are distinct truth-values then there exist functors W:y( ), W,( ) ( k E {m" - m + 1, . . . . m"}) such that the formula W,(H(W:y(p), JVJp))) takes distinct truth-vaZues when p takes the truth-values x, y or the formula W, (H(Wk(p) , W:y(p))) takes distinct truth-values in these circumstances (5, y E { 1, . . . , m}).

We shall suppose that the second alternative of Lemma 5 holds and prove the first alternative of the present lemma. The proof for the other case is omitted as it is similar. We choose k = mm - m + c where c i s the integer of Lemma 5. Let S, be the set of truth-values y such that w,(y) = w , ( l ) and S, = { 1 , . . . , m} - 9,. (6ince i 5 mm - m, Sl, 9, are proper subsets.) Let a , p be truth-values such that h(a, c ) E 9, , h(8, c) €9r2 and W:y( ) be a functor such that W : ~ ( X ) = a, W ; ~ ( Y ) = 8. Since we have chosen a , p so that wi(h(a, c ) ) + w,(h(p, c ) ) , the lemma is proved.

Lemma7 . I f n ~ { 2 , . . . , m } t h e r e e x i s t a p a r t i t i o n o f t h e s e t { l , . . . ,m} in tonpropar subsets I,, . . ., 8, and a formuh 2, ,,(& p ) which takes the truth-value xk u~hen 8 takes the value G and p takes a truth-value in rFk ( k = 1 , . . . . n) and which takes the truth-vaEue of W,(p) when 8 takes the va2ue H (xl, . . . , x,, E ( I , . . . , m } ; j = 1, . . . , NP).

We shall prove the lemma by induction on n. If n = 2 it reduces to Lemma 3. We now assume the lemma for n and prove if for n + 1 , assuming that the first alternative of Lemma 6 holds, as the proof for the second alternative is similar. We shall denote by Aab(d, p ) a formula which takes the truth-values of W a ( p ) , Wb(p) whenever 6 trikes the values G, H respectively (a , b E (1 , . . . , m}). (We have not, of course, yet pruvcd the existence of / lab(& p ) in all cases.)

We shall show first that, for suitable choices of truth-values a, b, c, d , e , a1 , . . . , a, and a suitable partition of the set {1 , . . . , m} into proper subsets S1, . . . , S,,+l the formula

h(y, C ) E $ 2 .

z4ab(6~ b(Z~, . . . L Z ~ C ( ~ , Po)! Ade(8 , P I ) ) has the required properties when S takes the value G and, when 6 takes the value H , takes distinct truth-values when p takes the distinct truth-values x , y of Lemma 6 (F1 , . . . , being independent of x1 , . . . , x.,~, x, y ) . We choose a to correspond to the functor W"( ) of Lemma 4 (cf. infra) and b to correspond to the functor Wi( ) of Lemma 2 (with the rilles of G, H reversed). Thus, by Lemma 2, we may construct the formula Aab(d, p ) . We then choose a l , . . .,a,, as in Lemma 4, the sets 8 , , . . .. 6, of Lemma 4 corresponding to our induction hypothesis, constructing W'( ) to corre- spond') to the truth-values xl,. . ., x , ,+~, the sets P1,. . ., S,,,, being chosen as

l ) We note that Lemma 4 allows a completely free choice here.

182 A.ROSE

those proved to exist in Lemma4 and we choose c to correspond to the functor W:y( ) of Lemma 6. We may, by the induction hypothesis, construct Za,.,.ac(B, p ) . Finally we choose d to correspond to the functor W'( ) of Lemma 4 and e as the integer k of Lemma 6. Thus (ide(8, p ) exists by Lemma 3.

We denote the formula thus constructed by X x 1 ...Xn+lXY. Let Y,, denote the &m(m - 1) formulae taking the truth-values of X X l , , . X n + l X Y when 6 takes the value H . (Here Y,, is a formula built out of the symbols H , p . We note that it is independent of xl , . . . , xn.+, .) In the propositional calculus with H as the only primitive functor we may, for all choices of w l , . . . , w,," (LX = fm(m - l)) , construct a formula @ ( p , , . . . , p,) taking the truth-values w, , . . . , w," under the ma assignments of truth-values to p 1 . . . , p , (with respect to a given ordering of the assignments). Since no two distinct truth-values of p will give rise to the same assignment of truth-values to the LX for- mulae Y,,, the formula @ ( A , , . . ., A,) ( A , , . . ., A , being these LX formulae in some given ordering) can be constructed to correspond to any of the mm unary truth-tables. Let ! P ( p , , . . ., p h ) be the formula obtained from @ ( A , , . . . , A , ) by replacing A , , . . . , A , by p , , . . . , p , , where h is the sum of the numbers of occurrences of A , , . . . , A , . (As is replaced by variables with lower suffixes than those replacing A , if . . . , z , , + ~ , , , . . . , z l h r . . ., z . + , , ~ such that

< y.) Since H is a Sheffer function, there exist (cf. [4]) truth-values zl

y(zk1 I . . . , Z k h ) = z k (k = 1, . . . , 72 + 1).

Thus the formula

~ ~ x z l I . . . z ~ + l , l ~ l b l ~ . . - 1 Xzlr...zm+l,hahbh)~

in which a,, b , ; . . .; a,, b, are the values of x, y respectively for A , (z being the num- ber of occurrences of A , in @(Al , . . . , A, ) ) and a,,, , b,+l ; . . .; a,, b, correspond similarly to the numbers of occurrences of A , , . . ., A , respectively, is a suitable choice for ZxI. . .xn+lj(S, p ) , provided that @ ( A , , . . . , A,) is constructed to correspond to the truth-table of W,( ).

Proof of t h e Main Theorem. If, in Lemma 7, we let n = m it follows a t once that the mzm formulae ( i a b ( B , p ) (a, b E (1, . . . , m}) are all constructable. Hence, by a theorem of B. J. LOWESMITH ([l], [Z]), our theorem follows a t once.

References [l] LOWESMITH, B. J., Sheffer functions and related concepts. Ph. D. Thesis, University of Not-

[2] LOWESMITE, B. J., and A. ROSE, A generalisation of Stupacki's criterion for functional comple-

[3] ROSE, A., A generalisa,tion of the concept of functional completeness and applications to modus

[4] ROSE, A., Completeness of sets of three-valued Sheffer functions. This Zeitschr. 29 (1983),

tingham, 1981.

teness. This Zeitschr. 80 (1984), 173-175.

ponens. This Zeitschr. 28 (1982), 317-322.

481 - 483.

Alan Rose University of Nottingham Department of Mathematics University Park Nottingham NG 7 2RD Great Britain

(Eingegangen am 24. MLrz 1982)