Computability complexity and Languages- Fundamentals of Theoretical Computer Science

5. Quantifiers 7
Thus, let P(t, x 1 , ••• , xn) be an (n + 1)-ary predicate. Consider the predicate
Q(y, x 1 , ••• , xn) defined by
Q(y,x 1 , ••• ,xn) -P(O,x 1 , ••• ,xn) V P(l,x 1 , ••• ,xn)
V ··· V P(y,x 1 , ••• ,xn).
Thus the predicate Q(y, x 1 , ••• , xn) is true just in case there is a value of
t ~ y such that P(t, x 1 , ••• , xn) is true. We write this predicate Q as
The expression "(3 t), y" is called a bounded existential quantifier. Similarly,
we write (Vt), YP{t, x 1 , ••• , xn) for the predicate
P(O, XI' ••• ' xn) & P(l, XI' ••• ' xn) & ... & P(y, XI' .•• ' xn).
This predicate is true just in case P(t, x 1 , ••• , xn) is true for all t ~ y.
The expression "(Vt), y" is called a bounded universal quantifier. We also
write (3t) < YP(t, x 1 , ••• , xn) for the predicate that is true just in
case P(t, x 1 , ••• , xn) is true for at least one value of t < y and
(V t) < Y P(t, x 1 , ••• , x n) for the predicate that is true just in case
P(t, x 1 , ••• , xn) is true for all values oft < y.
We write
for the predicate which is true if there exists some t E N for which
P(t, XI' ••• ' xn) is true. Similarly, (Vt)P(t, XI' ••• ' xn) is true if
P(t, XI' ••• ' xn) is true for all t EN.
The following generalized De Morgan identities are sometimes useful:
-(3t),YP(t,x 1 , ••• ,xn)- (Vt),Y -P(t,Xp···•xn),
-(3t)P(t,x 1 , ••• ,xn)- (Vt) -P(t,x 1 , ••• ,xn).
The reader may easily verify the following examples:
(3y)(x + y = 4) - x ~ 4,
(3y)(x + y = 4)- (3y), 4 (x + y = 4),
(Vy){xy = 0) -X= 0,
(3y),z(X + y = 4)- (x + z ~ 4& X~ 4).