HIERARCHIES
OF SETS AND DEGREES
BELOW 0'
by
Richard L. Epstein,
We examine
of changes
two hierarchies
a recursive
generalizations
by asking what
[4].
[8].
ordinal,
[6].
hierarchies
of sets below 0' based on the number
approximation
functions
error predicates
by Ershov
and Richard Lo Kramer*
to a set needs
of the notion of being r.e.
set out in Epstein
constructive
Richard Haas,
dominate
as previously
The second views
the changes
as first suggested by Addison
a translation between
to the degrees of u n s o l v a b i l i t y
as dominated by a
[i], and developed
them and relate these
i 0'.
All n o t a t i o n
< 0' which is based on the jump operator
to Cooper.
See Epstein
We indicate
by
comes
[4].
The reader should be aware that there is another hierarchy
degrees
sets
the ideas of Putnam on trial and
We first review some facts about sets iT 0'
from Epstein
Both are
The first classifies
the number of changes,
This extends
We provide
to make.
of
and is due primarily
[4], Chapter XI for that.
the end of a proof by
a , and the end of a subproof
D.
*We are grateful
to H. Hodes
an earlier version of this paper.
suggestions
concerning
diagrams.
for correcting
a number of errors
Roger Maddux also aided us w i t h
in
We notate
0' = {(x,y): ~ (y)+}.
Recall that
0' ~T K = {x : ~ x ( X ) + } .
First we note the Q u a n t i f i e r C h a r a c t e r i z a t i o n of Sets Below 0':
A ~T 0'
iff
there are two r e c u r s i v e predicates
that
Proof:
=_J
w E A
Vy S(x,y,w)
Therefore
Similarly,
is r.e.
in 0' so
A
is r e c u r s i v e in
=_J
some
Y ! s}
Let
f
Vx~y R(x,y,w)
iff
XxVy S(x,y,w)
is recursive
ment of an r.e. predicate.
~ x V y NR(x,y,w)
iff
such
in 0' since its the comple-
~ x V w S(x,y,w)
in 0'
R,S
Thus
is r.e.
A
and
X
in 0'
are r.e.
0'
enumerate 0' and let
be 0' e n u m e r a t e d to level
s.
0's = {x : f(y) = x _< s,
For some
e,
A = ~e(0')
the e th f u n c t i o n partial r e c u r s i v e in 0'.
Define
As(X)
w i t h calculations truncated
by
= ~e,s(0~)(x)
s).
(where
+e,s
is
e
Then
x E A
iff
~tVs > t
~e,t(0')(x)
= 1
iff
Vt~s > t
~e,t(0')(x)
= 1
m
As a c o r o l l a r y to the proof we have
The Limit L e m m a
sive function
(Shoenfield
g
[II]):
f iT 0'
iff
there is some recur-
such that
f(x) = lim s g(s,x)-
Here lim s g(s,x) = f(x) means
for all sufficiently large
g(s,x)
~tVs ~ t
= a = f(x).
That is,
W h e n we have
g(s,x)
g(s,x) = a.
as in the Limit Lemma we call g(s,x) = fs(X)
and call that a r e c u r s i v e a p p r o x i m a t i o n to
we'll v i e w it as a set and say
If
As(X)
s
(if
A
is r.e.
"changes
(if
x E A
x ~ A,
then
A
there is one)
The r.e.
g(s,x)
= 0
f.
If
such that
f
is 0-i v a l u e d
= As(X).
has an a p p r o x i m a t i o n
its m i n d at most once."
As(X)
s,
As
such that
That is, there is at most one
As(X)
= 0
and
As+l(X)
= 1
always).
sets have always been c o n s i d e r e d a d i s t i n g u i s h e d class of
34
sets b e l o w 0'.
This is partly because the special p r o p e r t y just
d e s c r i b e d is easy to utilize,
tion in logic between axioms
set),
as "proving"
and p a r t l y because it reflects the rela(a r e c u r s i v e set) and theorems
(an r.e.
is just a recursive enumeration.
We w i s h to generalize the notion of r.e.'ness
so that r.e.
sets
are seen as part of a continuum.
Let us classify
A
by m e a s u r i n g how often an a p p r o x i m a t i o n to
A
changes before it settles down.
Definition:
A
is n-r.e,
iff
there is a recursive a p p r o x i m a t i o n
to
A
A0(x)
such that for all
= 0
A
s
x,
and
I{s : As(X)
# As+l(X)} I ~ n.
Note that this d e f i n i t i o n can be extended in an obvious w a y to
apply to functions,
predicate.
too.
The only 0-r.e.
The n-r.e,
[7], but note well that
is quite different from ours.)
set is
~
and the l-r.e,
sets are the usual r.e.
sets are those that arise after
ting the Boolean algebra of r°e.
A
[8] calls an n-trial
Similar ideas are p r e s e n t e d in Gold
his d e f i n i t i o n of 2-r.e.
sets.
(This is what Putnam
sets.
Rogers
n
[9], p. 317 shows that
is in the Boolean algebra g e n e r a t e d by the r.e.
A <
0'
(!btt
-btt
Putnam [8].
If
G ~ 2~
means b o u n d e d - t r u t h - t a b l e
denote by
~T
the set
steps in genera-
sets
iff
reducible).
{deg(A)
: A
See also
E ~}.
we always m e a n T u r i n g degree unless o t h e r w i s e specified.
class of partial functions from
T h e o r e m I:
{A : A
~T
any r.e°
m o r e than any r.e.
freedom we need.
to
is n - r . e . } T ~ {A : A
To prove this we construct,
w h i c h is
~
set.
~
define
If
~
is a
~T = (~ N 2~) T.
is n+l-r.e.} T.
for e.g., n = I, a 2-r.e.
We use the fact that
set to execute
By degree
A
the diagonalization.
This proof is given in E p s t e i n
set
A
may change once
That's all the
[4], A p p e n d i x 2, and
35
is due to R. W. Robinson.
To the best of our knowledge,
the first to prove this,
Recalling
whether
X, X
that
in [2].
X, X
n+l-r.e,
r.e.
implies that
implies
we'll m o d i f y the question
A
that
a little.
is w e a k l y n-r.e,
X
X
is recursive we ask
is n-r.e.
To answer this
Say that
if
A = lim s As(X)
l{s : As(X)
That is
A0(x)
= 0
Cooper was
is no longer required.
and
# As+l(X)}l
A picture will help:
s
0
n-r.e.
0
i
2
3
4
x
Diagram
I
I
2
0
0
0
0
0
S
3
4
0
I
weakly
n-r.e.
0
0
~ n.
i
x
0
0
1
2
0 0
i 1
2 i
3 0
4 1
0
~n changes
allowed
allowed
Note that the weakly n-r.e,
4
6
i 1 i
Pn changes
all zero
3
R0,
a recursive
sets are closed under complementation:
set
just
reverse the 0' and l's.
The w e a k l y
T h e o r e m 2:
0-r.e.
A, A
Proof:
sets are the recursive
are n+l-r.e,
=]
Easy.
_~
Suppose
approximations.
iff
they are weakly n-r.e.
A = lim s As,
At(x)
all
Then we'll only allow
# At+l(X)
x.
Formally:
A = lim s B s
To obtain a w e a k l y n-r.e,
we will go to the first stage at which
C0(x) = As(X).
# Bt+l(X)
sets.
approximation
As(X)
Cs(X)
are n+l-r.e.
# Bs(X)
# Cs+l(X)
C
s
to
A,
and set
if we see
since we know that in the end
A(x)
# B(x)
Then there will be at most n changes.
let
s o = ~is (As(X) # Bs(X)) , and,
Sm+ I = ps
Then define
(s > sm
Cs(X)
and
= As0(X)
for
A s (x) # As(X)
m
all
s i so •
m < n+l,
and
As(X)
# Bs(X)).
36
As(X)
Cs(X)
if
sm ! s < Sm+ 1
=
We leave to you that
ASm+l(X)
if
Sm+ 1 ! s
lim s Cs(X)
= A(x).
Clearly its n-r.e.
An e s p e c i a l l y important fact about the r.e. degrees is that they
are dense
(see Sacks
T h e o r e m 3:
Given
recursive r.e.
Proof:
procedure.
such that
out.)
Let
Cooper
A
What can we say about the n-r.e,
n+l-r.e,
and not recursive,
If
A
there is some non-
n = I; the rest follows by an inductive
is r.e. w e ' r e done.
F-E = A.
f
degrees?
C iT A.
We show this for
(F
So there are r.e.
is the numbers put into
be a I-I e n u m e r a t i o n of
yourself that
Corollary:
[I0]).
C
F,
is the required set.
No n-r.e°
and
A,
E
sets
E,F
the ones taken
C = f-l(E).
Convince
U
degree is minimal.
(unpublished) was
the first to prove the Corollary,
but by
quite different means.
By r e l a t i v i z i n g the proof of T h e o r e m 3 we can get that if
0 < d < a
and
~
and b o t h are n+l-r.e,
is r.e.
in d.
w h e t h e r the n-r.e,
then there is some
d < c <
This does not solve for us the q u e s t i o n of
degrees are dense:
that's still open.
us ask w h e r e in our c l a s s i f i c a t i o n schema such a
We
c,
c
But it makes
will lie.
I
can extend our h i e r a r c h y by raising the bound on the n u m b e r of
changes allowed.
Definition:
Given any
A
is
f
f-r.e,
w h i c h is total
iff
there is some r e c u r s i v e a p p r o x i m a t i o n
A
s
to
A
I{ s : As(X)
such that
# As+l(X)}I
S i m i l a r l y we may define what it means for
f-r. e.
°.. c o n t i n u e d
! f(x).
hiT
0'
to be
$7
We say a degree
a
is f-r.e,
Note that this extends
What do we know?
Indeed
A
is f-r.e,
the last place
Let's
As
Certainly
for some
changes
T h e o r e m 4.
{A : A
f iT 0':
Proof:
{A : A
is f-r.e for some
we can spot recursively
class.
it)
in 0'
Abbreviate
identity-
degree.
a given r.e.
argument
for some
~=
D(< 0')
then
is undecidable.
a given r.e.
# Cs+l(y)
identity-r.e,
is the construction
set, by allowing
y ! x.
Bs(X)
Any permitting
Hence the construction
degree produces
an id-r.e,
of a set B
# Bs+l(X)
only if
argument produces
an
of a minimal degree below
minimal
degree
(see e.g.,
[3]).
All the degrees used in the proof in Epstein
T h ( ( D ( < 0'), ~ ~
is undecidable
Hence they are id-r.e.,
arithmetic
goes through as for
It is open w h e t h e r
(see Soare
are constructed
T h ( ~ D ( < 0');
Th(({A
: A
is n-r.e.} T
Th(${A
: A
is n+l-r.e.} T
<7)
: A
Theorem 5:
A
Let
arguof
<7)
is undecidable
We can also ask w h e t h e r
is the same as
<))
sets.
We say
A
is
~-r.e.
if
is f-r.e.
$
be a class of total recursive
we can recursively
in
by permitting
<7).
is r.e.} T
Let us look at classes of f-r.e,
f ~ ~,
[4] and [5] that
and the translation of fragments
[12] for a discussion).
Th(~{A
some
f.
is not dense.
is i d - r . e . } T =
A permitting
which is <T C,
ments.
definition.
its mind.
is id-r.e.} T
Th(~;
Epstein
A iT 0'
is f-r.e.
as id-r.e.
If
Cs(Y)
A ~ a
the w e a k l y n-r.e,
every
look at a simple f-r.e,
function-r.e,
if some
$.
Let
g
enumerate
functions
such that
an index for every function
be a function which dominates
every func-
38
tion in
not
5.
Then there is an
A
w h i c h is g-r.e,
but is
~-r.e.
The idea of the proof is just a m o d i f i c a t i o n of the p r o o f that
there is a 2-r.e.
an f-r.e,
set
degree w h i c h is not l-r.e.
B, the set
A
w h i c h we are c o n s t r u c t i n g is allowed,
for s u f f i c i e n t l y large
x,
The proof is in E p s t e i n
[4], A p p e n d i x 2.
Corollary:
Here we k n o w that given
to make one m o r e change than
B
does.
Any of the usual h i e r a r c h i e s of r e c u r s i v e functions
a h i e r a r c h y on the recursive-r.e,
But the recursive-r.e,
T h e o r e m 6:
There is an
induces
degrees.
degrees aren't the w h o l e story.
A iT 0'
for any r e c u r s i v e
w h i c h is not f~r.e.
f.
The essential step in the proof is to show that we can "enumerate"
all the recursive-r.e,
approximations.
[4], A p p e n d i x 2 and is due to Posner
The proof appears in Epstein
(private c o m m u n i c a t i o n 1973).
How can we extend this hierarchy?
Ershov
[6].
2
We take another tack as in
Instead of b o u n d i n g the n u m b e r of changes by functions
w h e n we pass from the n-r.e,
C o n s i d e r that
A
case, we'll bound them by ordinals.
being w e a k l y n-r.e,
given by a collection of
n
can be v i e w e d as
partial recursive
functions,
A
being
~0 . . . . ' ~n"
Here
A(x)
= ~k(X)
Of course
say that
functions
G
where
A
k
is the largest
need not be 0-i valued.
is w e a k l y n - a p p r o x i m a b l e
~0 . . . . ' ~n
k = max t(~t(x)+).
such that
such that
For any total
A n d surely
t
~t(x)+.
G
we could
if there are partial recursive
G(x)
= ~k(X)
How could we extend this to
there needn't be a largest
smallest one!
t
for w h i c h
~t(x)+.
where
w?
In that case
But there is a
39
{G
: G
sive
is w e a k l y
such
After
that
G(x)
all we m a y
Definition:
n-approximable}
= ~k(X),
collect
~ - - ÷ G,
k = ~t(~t(x))+}.
these
~
= {G : ~ 0 . . . . .
~i's
partial
as
~n
partial
Let's
clean
recurthis
up.
@((i,x)).
recursive
if for all
x,
(n)
G(x)
Now
sive;
it's
then
= ~(~k,x>)
easy
~ ~
where
to d e f i n e
G
if
k = ~t < n ( ~ ( f t , x ~ ) ~ ) .
an w-r.e,
G(x)
set.
= ~(~k,x))
Let
such
@
be p a r t i a l
that
recur-
k = ~t(~(~t,x))+).
(~)
From
(~tt
this
means
Let
us
definition
truth-table
it f o l l o w s
that
A
is w-r.e,
iff
A
<
-tt
0'
reducible).
define
V n = {A
: some p a r t i a l
recursive
~,
~
~ A}
(n)
V
= {A : some
partial
recursive
~,
~--+
A].
(~)
Theorem
7:
I.
A
E Vn+ 1
2.
A
E V
Proof:
x
we
our
guess
guess
2.
idea
iff
We o u t l i n e d
at
n-I
A(x)
A
where
is simple:
s i n c e we k n o w
I.
A
A
is w e a k l y
is f-r.e,
above.
at m o s t
n
Note
times,
n-r.e.
for
some
recursive
that
A ~
Vn
iff
which
is the
same
f.
for
every
as c h a n g i n g
times.
Suppose
A = lim s A s ( x )
The
iff
that w e
is f-r.e,
the
we
changes
reverse
can
for some
label
for
recursive
each
x
That
are b o u n d e d
the o r d e r
of the
our
guess
first
f.
labels
by
is
by
on the
f(x).
f(x).
changes
Formally,
define
undefined
¢(<r,x))
=
if
A0(x)
if
r = f(x)
Ay(X)
if
r < f(x)
was
chosen
to be
y = ~t > z
Now
let us s u p p o s e
that w e
r > f(x)
are g i v e n
Az(X)
(At(x)
~ ~
all
computations
truncated
at stage
s.
@(<r+l,x))
and
~ Az(X)).
A.
(~)
with
and
Define
Let
~s
D
be
40
0
if
s = 0
~(~r,x>)
A
s
if
0
Then
l i m s As(X)
r,
as for all
Corollary:
Proof:
Theorem
class of sets
There
for a.
0 , }T ~
is no r e a s o n
recursive.
By
As(X)
makes
for any
w e n o t e that
f
q,
must
0'} T "
the o b s e r v a t i o n
algebra generated
V ,
yield
that the f o r m e r
by the r.e.
to stop this h i e r a r c h y w i t h
(see R o g e r s
a-S
[9], C h a p t e r
w e have a r e c u r s i v e
z = (Y)S
II).
~.
and
ordering
and s u c c e s s o r
z
a
be any
by a-S,
systems
distinguish
we w i l l m e a n that
Let
Denote
related univalent
and the p r e d e c e s s o r
sets,
the c o r o l l a r y .
w h e r e w e can r e c u r s i v e l y
a n d limit o r d i n a l s ,
that
~s(q,x)+
r = f(x)
{A : A J t t
the list of r e q u r s i v e l y
Thus g i v e n
of c h a n g e s
Zt ~((t,x>)+.
is the B o o l e a n
set of n u m b e r s ,
system
that
5 and 7, plus
ordinal
and t h e r e is s u c h an
for the first time
Calling
class of sets is
constructive
a-U . . . .
x,
: A Jbtt
{A
s > 0
A n d the n u m b e r
where
r = ~t ~ s ( ~ s ( < t , x ) ) + ) .
the l a t t e r
r = ~t j s ~ s ( < t , x > ) +
otherwise
= A(x).
can't be m o r e t h a n
be total
where
(x) =
e-T,
of n o t a t i o n
on a r e c u r s i v e
0, s u c c e s s o r s
functions
are
denotes
T
in the
where
y
is the
S.
Definition:
Given
a-S,
define
iff
(s-S)
~(x)
= ~(((y)s,X>)
least o r d i n a l
< a
s u c h that
~( <(~)s,X >)+.
We r e a d
~ -
+ ~
as " ~ is the
a-limit of
in n o t a t i o n
S."
~
in n o t a t i o n
S"
(a-S)
or "~ a - a p p r o x i m a t e s
notation we will often
We n e e d not
~
delete
suppose
that
reference
~
When working with
to " n o t a t i o n
is total
a fixed
S."
in this d e f i n i t i o n .
41
Lastly define
Va_ S = {~ :
some partial recursive
As the next theorem will demonstrate,
be notation-dependent.
n ! 0
and
V
We
are not.
as previously
Corollary
(to Theorem 7):
See Theorem
More importantly,
Theorem 8:
some of these classes will
leave to the reader,
defined.
Th(fV( Ta-S); j >)
Vn_ S = Vn,
we really do have all the functions
f iT 0 ' t h e n for some notation
then
and
if
a j w.
a-S
below 0' now,
f iT 0'
given any
from
Vn,
4.
If
a-S,
to
B-U
f ~T 0'.
As
S, f ~ Vw2_S"
we may find
recursively
Proof: 3 Suppose that for some
informal proof that
that
is undecidable
f ~ Va_ S
that we ~an pass
however,
Immediately we can conclude
If
Moreover,
~ -~ ~}
(a-S)
That is, for any system S,
V _S = V
Proof:
4,
a-S,
S
B-U,
and
B J a
such
f ~ VB_ U.
f ~ Va_ S.
We will give an
is fixed we will suppress
refer-
ence to it throughout.
Diagram 2
0
i
....
w,
~+i
....
B
0
1
2 .........
3
4
x
..a . ~ . . . . . ~ ' ~ , . . . ~ . .
~
guess at
priority
f(x)
with
B;
priority of guesses
is well-ordered.
To calculate
if there is a
If not,
then
f(x)
look for some
C < B
such that
~((B,x>)
= f(x).
B
with
,(<~,x>)+,
~((
S,x>)+.
Then ask
a question recursive
If there is such a
~
in 0'.
go to it and ask
the same question.
Since
a
is w e l l - o r d e r e d we can be sent to a smaller
~
with
42
~((~,x>)+
then
in
only a finite number of times.
f(x).
This procedure
for finding
The least one we reach is
f(x)
is clearly recursive
0'.
Note that if
x,
f(x)~
f ~ V
iff
f
is partial the same proof works.
we simply never make a guess at
and is partial or total then
Now assume
that
f ~
f ~T 0'.
Thus if
O
We w i l l obtain a system of n o t a t i o n
S
such
V~2_ S
Since
f iT 0',
X = {<x,s>
: s = 0
well-ordering
(x = y
f iT 0'
f(x).
Then for each
and
R
f(x) = lim s f(s,x).
or
on
f(s,x) # f(s+l,x)}.
X
s ! t)o
Let
given by
Consider the recursive
fx,s>R~y,t)
iff
x ! Y
or
This has order type w.
Diagram 3
e.g.
012
.
32
,
s
1
•
6
)
54
8
13
7.
12 II
i09
X
As in the proof of T h e o r e m XX, Chapter Ii, Rogers
[9] we obtain a sys-
tem of n o t a t i o n
as follows:
xSy
S
for
iff
2
by p a d d i n g out
x = (v,n>
and if
and
v # u,
RIX
y = (u,m>
vRu;
and
<v,u> ~ X
otherwise if
v = u, n < m.
Now define
I f(s,x)
~(<<x,s>,x>)
if
=
otherwise
s = 0
or
f(s,x) # f(s-l,x)
43
Clearly ~ ~
f.
w -S
Lastly,
O
given any e-S, we can o b t a i n a n o t a t i o n
the same m a n n e r as d e s c r i b e d above,
for w h i c h
Note that if we w e r e to define classes
only that e-R
V~_ R
for
~+~2-U
a+~
•
2
f iT 0',
there is some
[]
w h e r e we require
R
such that
f ~ V _ R.
G i v e n a n o t a t i o n ~-S.
The classes
For ~ = B+I
vBT S
for
form a hierarchy.
B ~ ~,
That is,
this is a m o d i f i c a t i o n of the proof of T h e o r e m 1 that
there is a 2-r.e.
set w h i c h
allowed for each v a r i a b l e
~T
x
any l-r.e,
set.
A f t e r all, w e ' r e
one more change than before.
The changes
in the proof are really only to accomodate the ordinal notation.
consider
~
a limit ordinal.
to construct an
for each
< ~.
notation
in
is a recursive w e l l - o r d e r i n g on a recursive field,
then given any
T h e o r e m 9:
f E V
U
x,
A
Essentially
Now
in this proof w e ' r e allowed
w i t h an a p p r o x i m a t i o n w h o s e numbers of changes,
dominates the n u m b e r of changes to any given
Y ~ V~,
for
Thus the proof is, up to m o d i f i c a t i o n to accormmodate ordinal
(which is not simple),
the proof of T h e o r e m 5.
Note that for m a n y - o n e degrees the h i e r a r c h y is better behaved:
if
A ~ m B ~ Va_ S
then
A ~ V _ S. This fails for T u r i n g degrees: by the
C o r o l l a r y to T h e o r e m 3 any A iT 0' w h i c h has m i n i m a l degree provides a
4
c o u n t e r e x a m p l e for V I.
T h e o r e m I0:
Let
If
S
A
be a n o t a t i o n for
~ V .n_ S
Conversely,
W
A
is
A
x = y
~×n
then
is
Vn-r.e.
is the n o t a t i o n for
o r d e r i n g on
if
if
we have
w-n.
~-n
(namely
r < t).
Vn-r.e.
then
A ~ V .n_ W
where
given by the canonical
(x,r) ( (y,t)
iff
x < y
or
44
We will
denote
by
the
V .n
class
of
Vn-r.e.
functions.
Proof:
Let
given by
f
~ Vn
We'll
A
~
where
which
show
~ V .n_ S.
at m o s t
n
informal
proof
and
note
can recursively
(z) S = ~om+s,
with
f(x)
leave
that
determine
s
deleting
and
....
012
makes.
recursively
We present
an
that
< k
ordinals
j ~-n
(q)s = ~ - k + t
or
m = k.
we
and
We proceed
legibility.
at
We have
first
u-block we know
f(0,x)
= s.
we
in a n o t h e r
and
If w e
stay
We will
for e a c h
@
have
time
block.
goes
changes
our
changes
in
block
y = ~-m+s
is
+
(in a c o m p u t a t i o n
As
l o n g as w e
its
guess
~(~.x>)+
and
~
So
s
block
change
but now
to c h a n g e
~
t
this
later have
in t h i s
....
in this
m < n.
can
block,
012
0,.m+t
which
for s o m e
that
.......
most
~(<y,x>)
¥ = ~.m+s
~
at m o s t
=
in
at the
n copies
f
~
once
m
for
0J
~
times
to g u e s s
notate
such
or
subscripts
function
approximate
correct.
which
t
k < m
that
a total
4
012
are
construct
the i n f o r m a t i o n
to the r e a d e r .
q,z
w
Look
are
the d e t a i l s
any
via
that we need
until we
given
A
of changes
by showing
and whether
the p r o o f ,
Diagram
at
And we'll
the number
f E Vn
times
approximate
~ A.
(~.n-S)
dominates
that
First
~
We will
at m o s t
guess
to a n o t h e r
at
th
i n the m - -
s
times.
~ = w-k+t
can change
f(l,x)
stay
search).
and
k < m
its m i n d
at m o s t
at m o s t
n
= s+t.
f(x)
~-block.
The
final
times,
time
So
t
45
shifts
is, say
~(~,x)+
f(r,x) = f(r-l,x)
C = w.r+u,
+ u = f(x).
Now suppose that
our guess at
and
A(x)
we have
rl
A = lim s As(X)
is dominated by
where
f(0,x) . . . . , f(m,x)
be the
to
f E Vn-
A schematic p r e s e n t a t i o n will suffice
Let
the number of changes
m
to show that
A E Vw. n.
guesses we make at
f(x),
m < n-l.
~(x)
the first
kl-changes
of
As(X)
where
f(0,x)
= kr
~I (x)
k1
the next k2-changes
of
As(X)
where
f(l,x)
= k 2.
2 (x)
~k2
A(x) =
~(x)
the m th km-changes where
f(m-l,x)
= km
n
%k (x)
m
That is, we have for
t j kr,
~(<(r,t>,x))
= ~kr_
tm-r (x).
W
Similarly we may prove
Theorem
II:
If
A E Vw.e_ S
And if
notates
w×~
A
is
w.~
(namely
(r) S < (t)s).
then
A
V _s-r.e.
is
V _s-r.e.
then
A
via the canonical
~x,r>
< <y,t)
iff
E Vw.~_ W
where
ordering given by
x < y
W
S
or, if x = y,
on
46
Corollary:
For
each
i T 0'.
such
Proof:
That
that
Given
recursively
e-S the
from
is,
A
s-r.e,
given
is not
~-S
S
V
V
sets
S
f-r.e,
consider
we
[9],
p.
S"
A
is
V _S,, r.e.,
The
Theorem
Using
classes
I0 w e
canonical
~ ~ 6o6o.
Vl-r. e . . . .
V
are
out
...
hence
not
V
Vw. n
...
V 2-r.e ....
M.
0'
obtained
9 we
have
Then
V _ S r.e.
Using
e = ~-n
any
n.
for any
this.
V6o.n-r. e . . . .
vl
V 2
"'"
V 2
60
,n
6O
tT
V
A iT
V 6o-r.e.
6O
V
sets
as above.
classes
clarify
60
f-r.e.
for
t~
- -
~+I
canonically.
Vw-r. e . . . .
11
Vw
S'
the
V _ S.
By T h e o r e m
canonical
will
some
for
from
given
classes
A picture
Vn-r. e . . . .
11
obtained
~ ~ 6o
ii w e m a y n o w p i c k
where
f-r.e.
for
can g i v e
Theorem
= w.B
V
f ~
S'
206).
A E Vw.(~+I)_S,,-V~.~_S,,,
exhaust
find
for any
an
but not
can
a notation
(see R o g e r s
V +I_S,, r.e.
do not
~t
.
3
Lerman
.
V
.
has
6o
recently
communicated
to us that
he
and L. H a y
have
proved:
for
all
that
He
suggests
proof
shown
r.e.
the
that
of T h e o r e m
Also
R.
n ! 1
interval
the p r o o f
there
degree
b,
{b
are
two n+l-r.e,
: c < b < d}
is a not
difficult
degrees
~
has no n-r.e,
modification
< d
such
degrees.
of the
i.
Shore
that
there
and L. H a y
is a 2 r.e.
a
< b < O'
have
degree
communicated
a
< O'
to us
such
that
that
they have
there
is no
47
I.
Hay and Lerman have observed
prove that if
which is
a > 0
m+l-r.e,
is r.e.
that by a permitting
then for
and not m-r.e.,
Vm+l
2.
is f-r.e,
to
for any
n ! i.
from the fact that
And that
appears
Arch.
n-r.e,
b
A !tt 0'
iff
Math Logik 18(1976),
55-65.
the same as T h e o r e m 5 and 6(2) of
it fails for every
for every r.e.
[5]).
a
Vn
there is some
by Theorem 3 and the fact that
m < a
A
as Theorem 2,3 in
[6] (part II).
Actually
Epstein
f
by H. G. Carstens,
This proof is essentially
Ershov
A itt 0' .
but
for some recursive
"A~-Mengen"
4.
A iT 0'
A corollary
is
L. Hay has pointed out that Theorem 6 follows
there is an
3.
a
one can
there is some
a > b > 0.
Theorem 3 is that the same is true if
argument
of minimal
degree
(see
48
Bibliography
[I]
Addison, J., The method of alternating chains, in T h e o r y
M o d e l s , North-Holland, Amsterdam, 1965 (p. 1-16).
[2]
Cooper, S. B., Doctoral Dissertation, University of Leicester,
1971.
[3]
Epstein, Richard L., M i n i m a l
Full A p p r o x i m a t i o n
1975.
of
D e g r e e s of U n s o l v a b i l i t y
and the
Construction,
Memiors of the A.M.S., no. 162,
[4]
Epstein, Richard L., D e g r e e s of U n s o l v a b i l i t y : S t r u c t u r e and
T h e o r y , Lecture Notes in Mathematics no. 759, Springer-Verlag,
New York.
[5]
Epstein, Richard L., I n i t i a l
[6]
Ershov, A. Hierarchy of Sets I, II, III, A l g e b r a and L o g i c , VOI.
7, no. i, no. 4 (1968) and vol. 9, no. 1 (1970).
(English translation, Consultants Bureau, N.Y.).
[7]
Gold, Limiting recursion, J o u r n a l
no. I, p. 28-48, 1965.
[8]
Putnam, H., Trial and error predicates and the solution to a
problem of Mostowski, J o u r n a l of S y m b o l i c Logic, vol. 30, no. I,
p. 49-57, 1965.
[9]
Rogers, Hartley, T h e o r y of R e c u r s i v e
Computation,
McGraw-Hill, New York.
[I0]
[II]
segments
of Degrees
of Symbolic
Functions
< 0', to appear.
Logic,
and
VOI. 30,
Effective
Sacks, Gerald, D e g r e e s o f U n s o l v a b i l i t y , Annals of Math. Studies,
no. 55, Princeton, New Jersey, revised edition, 1965.
Shoenfield, J. R., On the degrees of unsolvability,
vol. 69, p. 644-653, 1959.
Annals
of
Mathematics,
[12]
Soare, Robert, Recursively enumerable sets and degrees, Bull.
VOI. 84, no. 6 (1978), p. 1149.
A.M.S.,
Iowa State University
Ames, lowa 50011
University of California
Berkeley, California
94720
Iowa State University
Ames, Iowa 50011