The use of universal decision elements as flip-flops

Zeitsehr. 1. math. Logik und Grundlagen d . Math. Bd. 4, S. 169-174 (1958)

THE USE OF UNIVERSAL DECISION ELEMENTS AS FLIP-FLOPS By ALAN ROSE in Nottingham, England

It has been shownl) that there are several functors @ ( P I , P 2 , . . ., P,) (n 2 4) such that all functors of one and two arguments are definable in terms of @ and the logical constants 0 , l in such a way that @ occurs in the definiens only once. Thus a logical decision element for the functor will act as a “universal decision element”, i. e. an element which can be used as a decision e!ement for all of the functors of one and two arguments.

We first give settings for a class of 16 universal decision elements. These are simpler than those given previously.2) We consider those functors @ whose truth- tables are those of the formulae of the form

[ F ( P , Q ) , R,A( fJ , 331 where each of the functors Y , A is one of 3 , 3 , == , + . We first note that the functors of two variables may be defined in terms of conditioned disjunction, negation, 1 and 0 as follows: .

I p V Q = d f . [ I , &, PI7

P i t Q =df. [ P , Q ) 01,

P Q =df. [ P , Q ) N PIf

p + Q = d r . [ N P , Q , P l ,

i ( l )

P I Q = d f . [ 1 , Q , -PI9 p/Q =df. [- p , &, 11 7

P I ; Q = d f . [ O , Q , P J , P J Q = d f . [ O , & , -PI . For each functor W (or A ) we can find propositions R,, R , , . . ., R, such that

R,, R2, * -, R, E { P , 1 , o>, yV(R1, R 2 ) = T p , Y ( R 3 , R 4 ) = T N p , W ( R 5 ? & ) = T 1 ) y(R, , R s ) = T o *

We have : 1 3 P = T P D o = T 1 P = T O f P = T P ,

P 2 0 = ~ D P = T O - P = T ~ ~ P = , N P , (2)

T

0 3 0 ~ T 1 D O = T O - O = T O f 1 = T 1 j

1 ~ o = T o ~ o ~ T o ~ ~ = T O + o = T o .

l) B. SOBOCI~SKI, On a universal decision element. J. Computing Systems I , 71-80 (1953). A. ROSE, Sur les BlBments universels de d6cision. Comptes rendus (Paris) 244, 2343-2345

(1957). A. ROSE and J. E. PARTON, British Patent Application, No. 28685/56. This application and

the application in the USA were made by the National Research Development Corporation (Great Britain).

J. M. PUGMIRE and A. ROSE, Formulae corresponding to universal decision elements. This Zeitschr. 4, 1-9 (1958).

2) A. ROSE, Op. cit.

12 Ztschr. f . math. Logik

170 ALAN ROSE

Thus, by appropriate choice of the R’s, we may use the definitions ( 1 ) t o define the functors of two1) arguments in terms of @, 1 , 0 in such a way that @ occurs in the definiens only once. Thus, for example, if

@ ( P , Q, R, X, T) = T [ P 9 Q , R , X TI we note that we may make the definition

and that

Hence we may make the definition:

PlQ = df. “ P ) Q , 11

l D P = , - P , O - o = T 1 .

P l Q = d f . @ ( l , P,Q,0 ,0 ) - We now define a flip-flop as a device with two inputs and one output. Initially

both inputs and the output correspond to the truth-value P. If the first input is changed to T the output changes to T and remains at T even if the input reverts to F. If now, however, the second input changes to T the output changes to P and remains a t F even if the input reverts to P. This type of mechanism may, in certain circumstances, be constructed from a decision element with three inputs, by connecting the third input to the output.

Let us denote the functor represented by this element by F ( P , Q R ) , the inputs P , Q, R representing respectively the input connected to the output and the first and second inputs. Thus R is the “re-set’) input. Hence, from the above definition of a flip-flop, we obtain successively:

The last condition (output remains at P even if second input reverts to F ) merely gives us again the condition

Since six of the eight entries in the truth-table of F ( P , Q, R) are determined by the equations (3) there are exactly 22 or 4 functors P which satisfy the conditions. F (P, Q, R ) must have the same truth-table as one of the following four formulae :

P(0 , 0 , 0) ‘ T 0 .

P v Q 3 R , (P 3 R ) v Q , [ - R , P , &I, L 2 ( P , Q, - R ) . L, ( P , Q , R ) takes the truth-value T if and only if at least two of P, Q, R take the truth-value T. Let us denote these functors by Fl (P, Q, R), F 2 ( P , Q, R), F,(P, Q , R ) , F,(P , Q, R) respectively. Then, for us to be able to use a universal decision element as a flip-flop, it is sufficient for one of the functors P, , F, , E”, , F4 to be definable in terms of O 7 0 , 1 in such a way that @ appears in the definiens only once.

l) For the treatment of functors of one argument see A. ROSE, Op. cit. or J. M. PUGMIRE and A.RosE, Op. cit.

THE USE OF UNIVERSAL DECISION ELEMENTS AS FLIP-FLOPS 171

172 ALAN ROSE

has the same truth-table as one of the formulae:

@ ( P , Q , R , PI, @ ( P , Q , R , Q ) , @ ( P , Q , R , R ) , @ ( P , P , Q , R ) , @ ( P , Q , P , R ) , @ ( P , Q , Q , R ) , @ ( P , Q t R , 01, @ ( P , Q , R , 11, @ ( P , Q , 0, R ) , @ ( P , Q 3 1 , R ) , @ ( P , O , Q , R ) , @ ( P , 1 , Q , R ) , @ ( O , P , Q , R ) , @ ( I , P , Q , R ) ,

i. e. as one of the formulae:

(B) 1 - R , ( P + Q ) z R, - R , ( P = Q ) v ( Q - A ) , ( P f Q ) & ( Q + R ) ,

Clearly none of the formulae ( A ) has the same truth-table as N R. Before considering the remaining twelve formulae we shall make the following

Def in i t ion : If, when P I , P,, . . ., P, take the truth-values x,, x,,. . ., xn respectively the formula i2 (Pl , P, , . . . , P,) takes the truth-value w (x, , x2, , . . , xn) (xl, x2, . . . , x,, w = T, P) and, for each of the 2n-1 (n - 1)-tuples of truth- values (xl,. . ., xi-l, xi+l, . . ., 2,) a t least one of the equations

L,(--P, - Q , 4, L ~ ( N P , -Q , N R ) , L,(P, N Q , N R ) ,

( Q I P ) f R , ( P I Q ) f R , P & Q = R , P v Q + R , ( P I Q ) + R , ( Q I P ) + R .

~ ( 2 1 , . . a j ~ i - ~ ~ ~ , x i + l , . . ., u ( z ~ , . . ., ~ i - 1 , T, ~ i + l , . . ., xn) = T

holds, then the truth-table of i2 ( P , , P, , . . . , P,) is monotonic non-decreasing with respect to Pi.

The definition of “monotonic non-increasing with respect to Pi” is similar. Thus, for example, the truth-table of P I Q is monotonic non-increasing with respect to P and monotonic non-decreasing with respect t o Q.

Thus the truth-tables of the first three formulae (B) are monotonic non-increasing with respect to a t least two of P , Q , R and the remaining nine truth-tables are not monotonic with respect to any of P, Q , R . On the other hand, F d ( P , Q , R ) is monotonic non-increasing with respect to R only (i = 1 , 2 , 3,4). Hence we cannot find a formula ( A ) which has the same truth-table as one of the formulae ( B ) and the decision element for the functor @ cannot be used as a flip-flop.

Difficulties may arise in relay versions of the universal decision element i f T, F correspond to connection to positive potential and to earth respectively. If the effect of leaving the input Q (or R ) of F i ( P , Q , R) on open circuit is different from the effect of connecting it to positive potential and different from the effect of connecting it to earth then the intermediate stage occurring during the alteration of a truth-value may interfere with the working of the flip-flop. However this diffi- culty cannot arise in the case of a relay with only one winding. In the case of the circuit shown below, which uses a relay with two windings, and which corresponds t o the formula [ P , (Q D R) v ( R 3 X), T ] we cannot use the definition

F,(P, Q , R ) =df. @ ( I , 6, P , R , 0)

TnE USE OF UNIVERSAL DECISION ELEMENTS AS BLIP-FLOPS 173

OUT

0 O P

Q R S

since the above difficulties arise for the third argument place of @, We may, however, overcome the difficulty by making use of the definition

F , ( P , Q, R) z d f . @ ( I , 0 , P7 3, Q ) . The circuit then becomes:

’ First input

v 0

O +

Second Input [Re-set)

174 ALAN ROSE

A method has been developed’) for testing formulae to determine whether they correspond to universal decision elements. The method may be extended to determine those decision elements which are universal and may, in addition, be used as flip-flops. By methods similar to those used2) for the construction of the formulae A , , . . ., A, we may construct formulae A,, A, , , Al, , A , , which correspond to the statements “Fi is definable in the required manner” (i = 1 , 2 , 3 , 4). Thus we have to take as an additional input t o the last conjunction decision element, an output corresponding to the formula Ct.“9 A<.

1) J. M. PUCMIRE and A. ROSE, Op. cit. 2) See the previous footnote.

(Eiagcgangen am 27. Jamiar 1958)