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9. Upper and Lower Bounds
9. Upper and Lower Bounds We have s e e n t h a t e v e r y f i n i t e set o f d e g r e e s h a s a l u b .
We s h a l l now see t h a t t h i s i s f a l s e f o r g l b ' s , and t h a t very simple i n f i n i t e sets o f d e g r e e s may f a i l t o have a l u b . An a s c e n d i n g sequence of d e g r e e s i s a n i n f i n i t e sequence {anj such t h a t an
<
an+l f o r a l l n.
F o r example, 0, O f ,
011,
...
i s a n ascending sequence.
Theorem (Kleene-Post-Spector) sequence of d e g r e e s .
.
L e t {an] be a n ascending
Then t h e r e a r e upper bounds b and c
for \an] such t h a t no upper bound o f f a n ) i s a lower bound o f pLcj
Corollary Corollary
1.No ascending sequence o f d e g r e e s h a s a l u b . 2. There a r e d e g r e e s b and c such t h a t i b , c \ h a s
no g l b . P r o o f . Take any ascending sequence Ian], and l e t b and Then a g l b of [ b , c j would be an upper
c be as i n t h e theorem. bound of
[.,I.
Q.E.D.
Now we t u r n t o t h e p m o f of t h e theorem. t i o n from N t o N o f d e g r e e an.
Let Hn b e a f u n c -
We c o n s t r u c t f u n c t i o n s F and G
from N X N t o N and t a k e b = dg F, c = dg G .
To i n s u r e t h a t
no upper bound of {an) i s a lower bound o f ]b,c), we must s a t i s f y t h e conditions:
F
(lI,J) If [I]
=
G [J]
= L, t h e n L
SR Hn f o r some n .
Let (Rs3 be a l i s t i n g of t h e space of c o n d i t i o n s . w e w i l l i n s u r e Rs a t s t e p s .
Again
A t s t e p s , w e w i l l first d e f i n e
f i n i t e l y many v a l u e s of F and G; we w i l l t h e n s e t
43
UPPER AND LOWER BOUNDS
44 (1)
F ( s , k ) = H ( k ) i f F ( s , k ) i s undefined,
(2)
G ( s , k ) = Hs(k) i f G ( s , k ) i s undefined.
S
T h i s w i l l f i r s t of a l l i n s u r e t h a t F and G a r e completely d e Moreover, F ( s , k ) can be d e f i n e d
f i n e d by t h e c o n s t r u c t i o n .
f o r o n l y f i n i t e l y many v a l u e s of k b e f o r e t h e l a s t p a r t o f s t e p T h i s shows
s; s o F ( s , k ) = H s ( k ) for a l l b u t f i n i t e l y many k . t h a t Hs
SRF; s o
b i s a n upper bound of
c i s an upper bound
ian]
.
Similarly,
of [an{. and l e t FS Let R, be (~I,J),
W e now d e s c r i b e s t e p s .
and G s be t h e p a r t s of F and G a l r e a d y d e f i n e d . Case 1: There a r e u and and [I]'
7~
such t h a t Cornp(u,Fs), Comp(r, G ' ) ,
T
and [J] a r e i n c o m p a t i b l e .
Ve t h e n choose such a
0
and
f o r a l l ( n , k ) i n t h e domain of
0
T.
W e set F(n,k)
=
u(n,k)
such t h a t F s ( n , k ) i s undefined;
and we s e t G ( n , k ) = rr(n,k) for a l l ( n , k ) i n t h e domain of such t h a t G s ( n , k ) i s u n d e f i n e d .
7~
We t h e n proceed t o (1) and
F We have i n s u r e d t h a t 0 C F and TT C G . Hence [ I ] ' c [ I ] T F G and [ J l C [JIG; s o [I] and [ J ] a r e i n c o m p a t i b l e . This (2).
implies t h a t (1-
I,J
) holds.
Case 2 . Otherwise. We t h e n proceed immediately t o (1) and ( 2 ) .
We must show
t h a t ( 1 1 , ~ )h o l d s . We f i r s t show t h a t t h e set of u such t h a t Comp(u, Fs) i s r e c u r s i v e i n H,.
We make a l i s t o f t h e f i n i t e number of v a l u e s
of FS n o t a s s i g n e d by ( 1 ) and ( 2 ) .
T o d e t e r m i n e whether or n o t
UPPER AND LOIGR BOUNDS
45
Comp(u, F s ) , i t w i l l s u f f i c e t o d e t e r m i n e f o r e a c h ( n , k ) i n the domain of u w h e t h e r or n o t ( n , k ) i s i n t h e domain o f FS, and, i f i t is, d e t e r m i n e F s ( n , k ) .
If Fs(n,k) i s i n out l i s t ,
we can do t h i s w i t h o u t a n o r a c l e .
Otherwise, ( n , k ) i s i n t h e
domain of FS i f f n Now f o r n
<
s, an
< s; and i n t h i s c a s e , F s ( n , k ) = H n ( k ) . < a s and hence Hn LR Hs. Hence a n o r a c l e
for Hs w i l l s u f f i c e . Now assume t h a t
[IIF
= [J]
G
= L.
We f i r s t show t h a t
L ( i ) = j c-i,3o(Comp(oJ F') & [11'(i) = j). F If L ( i ) = j , t h e n [ I ] (i) = j ; so by ( 2 ) of $4, t h e r e i s a
(3)
S i n c e u and FS have t h e common
u C F such t h a t [ I ] O ( i ) = j . e x t e n s i o n F, Comp(o, Fs). [I]'(i) T
=
j f o r some B.
C G such t h a t [ J ] " ( i )
Now suppose t h a t Comp(u, Fs) and G
S i n c e [ J ] ( i )= L ( i ) , t h e r e i s a =
L ( i ) ; and Cornp(a,Gs).
hypothesis, C0mp([1]~,[J]"); so L ( i ) Now Comp(u,FS), as a r e l a t i o n n f
Hs and hence RE i n Hs; w h i l e [ I ] ' ( i ) i , j , i s RE and hence RE i n H,.
z a t i o n s of t h e r u l e s of L
SR Hs
=
By t h e c a s e
j. 0, i , j ,
= j,
i s recursive i n
as a r e l a t i o n of
0,
Hence by ( 3 ) and t h e r e l a t i v i -
f 7 , t h e g r a p h of L i s RE i n H,.
by t h e r e l a t i v i z e d Graph Theorem; s o ( 1 1 , ~h)o l d s .
Thus
Q.E.D.
We s h a l l now see t h a t t h i s i s f a l s e f o r g l b ' s , and t h a t very simple i n f i n i t e sets o f d e g r e e s may f a i l t o have a l u b . An a s c e n d i n g sequence of d e g r e e s i s a n i n f i n i t e sequence {anj such t h a t an
<
an+l f o r a l l n.
F o r example, 0, O f ,
011,
...
i s a n ascending sequence.
Theorem (Kleene-Post-Spector) sequence of d e g r e e s .
.
L e t {an] be a n ascending
Then t h e r e a r e upper bounds b and c
for \an] such t h a t no upper bound o f f a n ) i s a lower bound o f pLcj
Corollary Corollary
1.No ascending sequence o f d e g r e e s h a s a l u b . 2. There a r e d e g r e e s b and c such t h a t i b , c \ h a s
no g l b . P r o o f . Take any ascending sequence Ian], and l e t b and Then a g l b of [ b , c j would be an upper
c be as i n t h e theorem. bound of
[.,I.
Q.E.D.
Now we t u r n t o t h e p m o f of t h e theorem. t i o n from N t o N o f d e g r e e an.
Let Hn b e a f u n c -
We c o n s t r u c t f u n c t i o n s F and G
from N X N t o N and t a k e b = dg F, c = dg G .
To i n s u r e t h a t
no upper bound of {an) i s a lower bound o f ]b,c), we must s a t i s f y t h e conditions:
F
(lI,J) If [I]
=
G [J]
= L, t h e n L
SR Hn f o r some n .
Let (Rs3 be a l i s t i n g of t h e space of c o n d i t i o n s . w e w i l l i n s u r e Rs a t s t e p s .
Again
A t s t e p s , w e w i l l first d e f i n e
f i n i t e l y many v a l u e s of F and G; we w i l l t h e n s e t
43
UPPER AND LOWER BOUNDS
44 (1)
F ( s , k ) = H ( k ) i f F ( s , k ) i s undefined,
(2)
G ( s , k ) = Hs(k) i f G ( s , k ) i s undefined.
S
T h i s w i l l f i r s t of a l l i n s u r e t h a t F and G a r e completely d e Moreover, F ( s , k ) can be d e f i n e d
f i n e d by t h e c o n s t r u c t i o n .
f o r o n l y f i n i t e l y many v a l u e s of k b e f o r e t h e l a s t p a r t o f s t e p T h i s shows
s; s o F ( s , k ) = H s ( k ) for a l l b u t f i n i t e l y many k . t h a t Hs
SRF; s o
b i s a n upper bound of
c i s an upper bound
ian]
.
Similarly,
of [an{. and l e t FS Let R, be (~I,J),
W e now d e s c r i b e s t e p s .
and G s be t h e p a r t s of F and G a l r e a d y d e f i n e d . Case 1: There a r e u and and [I]'
7~
such t h a t Cornp(u,Fs), Comp(r, G ' ) ,
T
and [J] a r e i n c o m p a t i b l e .
Ve t h e n choose such a
0
and
f o r a l l ( n , k ) i n t h e domain of
0
T.
W e set F(n,k)
=
u(n,k)
such t h a t F s ( n , k ) i s undefined;
and we s e t G ( n , k ) = rr(n,k) for a l l ( n , k ) i n t h e domain of such t h a t G s ( n , k ) i s u n d e f i n e d .
7~
We t h e n proceed t o (1) and
F We have i n s u r e d t h a t 0 C F and TT C G . Hence [ I ] ' c [ I ] T F G and [ J l C [JIG; s o [I] and [ J ] a r e i n c o m p a t i b l e . This (2).
implies t h a t (1-
I,J
) holds.
Case 2 . Otherwise. We t h e n proceed immediately t o (1) and ( 2 ) .
We must show
t h a t ( 1 1 , ~ )h o l d s . We f i r s t show t h a t t h e set of u such t h a t Comp(u, Fs) i s r e c u r s i v e i n H,.
We make a l i s t o f t h e f i n i t e number of v a l u e s
of FS n o t a s s i g n e d by ( 1 ) and ( 2 ) .
T o d e t e r m i n e whether or n o t
UPPER AND LOIGR BOUNDS
45
Comp(u, F s ) , i t w i l l s u f f i c e t o d e t e r m i n e f o r e a c h ( n , k ) i n the domain of u w h e t h e r or n o t ( n , k ) i s i n t h e domain o f FS, and, i f i t is, d e t e r m i n e F s ( n , k ) .
If Fs(n,k) i s i n out l i s t ,
we can do t h i s w i t h o u t a n o r a c l e .
Otherwise, ( n , k ) i s i n t h e
domain of FS i f f n Now f o r n
<
s, an
< s; and i n t h i s c a s e , F s ( n , k ) = H n ( k ) . < a s and hence Hn LR Hs. Hence a n o r a c l e
for Hs w i l l s u f f i c e . Now assume t h a t
[IIF
= [J]
G
= L.
We f i r s t show t h a t
L ( i ) = j c-i,3o(Comp(oJ F') & [11'(i) = j). F If L ( i ) = j , t h e n [ I ] (i) = j ; so by ( 2 ) of $4, t h e r e i s a
(3)
S i n c e u and FS have t h e common
u C F such t h a t [ I ] O ( i ) = j . e x t e n s i o n F, Comp(o, Fs). [I]'(i) T
=
j f o r some B.
C G such t h a t [ J ] " ( i )
Now suppose t h a t Comp(u, Fs) and G
S i n c e [ J ] ( i )= L ( i ) , t h e r e i s a =
L ( i ) ; and Cornp(a,Gs).
hypothesis, C0mp([1]~,[J]"); so L ( i ) Now Comp(u,FS), as a r e l a t i o n n f
Hs and hence RE i n Hs; w h i l e [ I ] ' ( i ) i , j , i s RE and hence RE i n H,.
z a t i o n s of t h e r u l e s of L
SR Hs
=
By t h e c a s e
j. 0, i , j ,
= j,
i s recursive i n
as a r e l a t i o n of
0,
Hence by ( 3 ) and t h e r e l a t i v i -
f 7 , t h e g r a p h of L i s RE i n H,.
by t h e r e l a t i v i z e d Graph Theorem; s o ( 1 1 , ~h)o l d s .
Thus
Q.E.D.