Models for Recursion Theory
by
Johan Moldestad and Dag Normann
~·
Several results in the theory of recursion in higher types
indicate that the effect of a higheJ:• type functional on the lower
types does not reflect the high type, i.e.
the same effect could
be obtained by functionals of relatively low type.
The two main
results here are :
Plus
Let
- 1 -theorem.
(G. Sacks [6]
F
of type
i.e.
the same subsets of
Plus
- 2- theorem.
= 1,
k + 1 such that
tp(k-1)
[7]
for
k > 1).
Then there exists
= k-
k- sc(F)
are x•ecursive in
F
and
sc(H),
H.
(L. Harrlngton [1].
H be a normal functional of type > k + 2.
a normal functional
i.e.
k
H be a normal functional of type> k + 1.
a normal functional
Let
for
F
of type
the same subsets of
k + 2
tp(k-1)
Then there exists
such that
= k-en(F),
k-en(H)
ax•e semirecursive in
F
and
H.
The results in this paper also indicate that higher types cannot
have too much influence on lower types.
Lowenheim theorem.
1.
Let
n < m.
Let
x E B
Among the results we mention
A c tp(n)t x tp (m)
~
The key is the Skolem-
be Kleene-semicomputable.
Vy E tp(m) <x,y> E A.
Then
B is
result may be relativized to a functional of type
2.
Let
ko
This
n + 1.
be the type-k-functional that is constant zero.
be a functional of type < k.
i-sc(F,kO)
= i-sc(F),
Then, for i < k-2
i-en(F,kO)
= Vtp(i)(i-enF)
Let
F
- 2 -
3.
Let
n,m
~
1.
such that for
4.
Then there is a functional
k ~ n,
F
of
type
n + 2
k + 1 - en(F) = TI~(tp(k)).
Tin-positive inductive definitions over
tp(n)
have
1
least fixedpoints.
All these results have relativized versions.
This paper includes results from Moldestad & Normann [5].
There we proved a relativized version of
derived
2
for
k = 3.
same ideas as in [5].
are from [5].
authors.
The proof of
4 for
3 from
n = 1, and
2
follows the
Also the discussions in§ 8 of this paper
These results and ideas are jointly due to both
The notion of recursion structures and theorem
1
are
due to the second author.
2.
Notation.
We will work with Kleene-recursion on objects of finite type,
and we assume familiarity with the contents of Kleene [3].
We
define the types as
tp(O) = w,
Let
X c tp(O)ko
We say that
Assume
6n
0
X
is
k
1
60
0
is defined.
n
vtp (n) ( E~),
nk+1 =
,.n
Tin
k = k
0
F,
n
't'n
"k '
tp(i+1) = tp(i)w.
x ••• , x tp(n) n.
if
X
Let
is Kleene-recursive.
Tin = En = 6 n •
0
0
0
n
tp(n)(Tin)
l:k+1= 3
k
u
kEw
Q, H, Q will denote formal symbols for functionals.
To each
symbol there is assigned a number indicating the type of the interpretations,
F, G, H, U will denote standard interpretations of the
symbols,
!F, 9 , 21/, 1v
other interpretations.
- 3 -
3.
Recursion structures.
-+
Let
be a list of functional symbols, a
F
the associated
list of type-indicators.
Let
i,j E
i < j.
~,
By a type i
1
j
1
a-recursion structure
we mean a structure
-+
= <A
0(.
k < i
i
i < k ~ j
0
,Ai, o o • ,Aj, ~,E>
-+
A
k-1 w
Ak S
-+
-+
~
ii
such that
= tp(k)
Ak
-+
, • • o
is an interpretation of
of type
iii
Each
iv
AC
v
E
then
k,
~
!7'
such that if
F
is in
is a symbol
F
Ak.
is closed under primitive recursive operations.
is satisfied in ot
is the evaluation-relation on
E(x,y,n) ._ x(y)
~
n
We will explain iv a bit :
~
Let
be a formula in
L(ot)
constants for all elements in
k2
~
j-
1•
t1tt=
A
0 '
(the 1.order language with
••• A .)
,
j
Let
k
1
< j,
-
Assume
Va E
3B € A~ <r(a,S)
Ak
2
Let
k
Ol:l=
3f3
= max(k1
€
,k +1).
2
Then
k-1
1
AkVa E Ak tp{a,A. x
tl(a,x)).
We assume here the existence of some standard coding of lower
functionals to higher functionals.
- 4 We say that a set
is an element of
A
=Ak
is in at if its characteristic function
Ak+ 1 •
ot is absolute if well-roundedness is absolute with respect
to
OL.
Now we are going to code some
of
tp(i+1)
Let
Let
+
F.
+
f
= <xi~,···,xj>
q
be a sequence of type (i+1)-elements.
be a list of elements from
tp(i) of the same length as
Define
Remark:
'Each a~
q.
z
There is a possibility that
valued function.
in
i,j,cr-structures as elements
ak
may be a many-
is given by a
6l+ 1 -statement
However,
is single-valued'
a~'s are
From now on we will, given q, assume that the
well-defined.
Now, given
q,
let
Ak
= tp(k)
z € tp(i)}
Define
Let
t1'(
~n
to be
= <A1
where
+
, • • •
,Aj, fF ,E>.
k
for
for
by
'structure'
i < k
~
j
is the type-number of
We say that
For the sake of simplicity we denote
structure'
< i
k
+
q,f
code
(}(_.
'i,j,a-recursion
when no ambiguity may arise.
Lemma 1 •
+
a
b
'q,f code
o,
= o,
a structure' is
+
If
i >
then 'q,f code
If
i
then this is
6i+1
1
•
an absolute structure' is
rrl
1
•
- 5 -
The language of ot. is ari thmetizable over
Proof.
thus 'truth in 0[•
~1
is
1
(Ot)-expressible.
over 0&:. is nothing more than a type
quantifiers in Ot. are
+
'q,f
in
q,f.
an
A set quantifier
i +1-quantifier, and 1 • order
tp(i)-quantifiers.
~
dt.l=
such that
and
Thus by standard coding:
is
AC'
~
i+1
1
+
(q,f),
uniformly
The rest of the properties of a structure are arithmetic
0'(,
over
code
OC,
and thus
This proves ..§!:•
in
To prove b note that ·0{.
VT E loti(T E (wf)
wellfounded relations.
Remark:
For
<q,f>.
wf
i >
~
ot
o,
T E wf), where
TI 1
is
is absolute if
for
1
i
= 0,
wf
denotes
~i+ 1
but
the
else.
0
we always assume that a structure is
absolute, since this does not affect the definability.
Moreover,
in our proofs, we do not need the full axiom of choice.
We may
give an upper estimate of the complexity of the formulas we need
AC
to hold for.
In that case
be
~i
n,
4.
n
for some
+
'<q,f>
codes a structure'
irrespectively of whether
i
=0
will
or
i
>
o.
Recursion in the structures.
The purpose with these structures is to simulate recursion in
a list of objects
+
F
over
tp(i+1).
+
that when the maximal type in
functionals, except those in
When
0
is a computation in
from
0
by replacing
By
Fn
by
F
is
We know from Kleene [ 3]
then no
j'
F, will occur in any subcomputation.
+
F, we let
n
in
0
0
1
2
be the list obtained
for each
the computation tree of a computation in
{<o-,cr->;
tp(j-1 )-
+
-+
F,
Fn
in
F.
we mean
o1 is an immediate subcomputation of
o2 ,
which
again either is a subcomputation of the given computation or is the
given computation}. The computation tree will then be a subset of
- 6 -
tp(j-2), and it will be well-founded.
formula
Moreover, there is a
nJ-1_
1
w such that
+
:1.s a computation tree_. Tis well-founded & w(T,F).
T
+
~,
Given
wat(T,ar)
has a natural interpretation, and we
have two possibilities in defining recursion in higher types over
ot:
1.
We use Kleene's inductive definition of recursion in higher
types, i.e. an outside definition.
8:
schema
Let
The only new there is
x1 E Ak
and let
be the index of schema 8.
e
We then say
« (x
· {e }
2.
1
o, xk )
~
n
c-c>
o o ,xk)
~
n
<-:>
, ..
xi ( >.. y
E
Ak _1 { e ' }Ot ( y , x
1
,
oo o , xk ))
Q£
n•
We define
· {e}Ol (x
1
, •
tree for
OJ_ ~ 3T
(T
is a computation
<e,\ ,o•o,xk,n>-) •
Lemma 2.
Let OL be .a structure.
3.
{e}
01
(x
1
,
o • • ,xk)
a~
or. I=
T
&
n
~
Then
1 •. is equivalent to
(3T E IO't I)
(T
is well founded
is a computation tree for
<e ,x ,••o,xk,n>-).
1
1
Proof.
3
~
1
follows by induction on rank
1
~
3
We prove this by induction on the length of the computation.
(T) •
The only nontrivial case is case 8.
- 7 ~
Let
,•••,xn
be given, each
x 1 € Ak 1
for some
k 1 < j-1.
-+
Let
Let
or let
e
x
0(
1
-+
-+
,f) = x(Ay
n
(x ,•••,x
A.y € Ak_ 2 {e' }oz: (x
1
By induction hypothesis
Assume that
Ak 2 {e'} (x ,•••,x ,?f)).
ffC 1
n
€
-+
, • • •
,xk,JZ)
Moreover,
$
is such that
<e' ,x
T
, o o •
1
,xn,n>
+
= m.
computation.
O(p llJ(T,JZ')
is the top of
T).
<e' ,x
, • • •
1
,xk,n >- and
are two computations where at least one is standard,
1
n
&
is unique (no standard computation
has nonstandard subcomputations), and if
<e',x ,•••,xk,m>
Ak_ 2 •
is total over
Vy € Ak_ 2 3T € Aj_ 1 3n (T is wellfound~d
&
then
Assume that
be the index such that
{e}
•
fF.
be from
This is proved by induction on the length of the standard
Then we may apply
AC
on
*
and obtain a well-founded
-+
computation tree for
<e,x
, ..
1
This ends the proof of
•,xk' f!:,n'>.
lemma 2.
Remark.
By this lemma, 1.and 2. are equivalent for absolute
structures, and 1. defines a stronger concept thant 2.
putations by 1. will be computations by 2.
i.e. all com-
When nothing else is
stated, we use 2. as our definition.
-+
6.
-+
F-structures and weak F-structures.
Definition.
Let
F
be a list of functionals.
Let
Ol be a
structure.
i
OL is an F-structure if whenever
We say that
x ,•••,xk € Ai+ 1
1
we have
-+
· {e } ()"'(. ( x
1
, • • • ,
xk ¥
)
Of
n ~ {e }K( x
1
-+
, • • • ,
xk , F )
Of
n •
- 8 -
Remark.
index
e.
{e}K
x
denotes the Kleene-computable function with
oo ,xk
, •
1
are all elements of type (i+1), so this
is meaningful.
+
or_ is a weak F-struc!ure if whenever x1
ii
have
{e}K(x ,•••,xk,F)
in
+
'q,f
we
1
+
code
an F-structure'
will be semicomputable
i+ 2E.
and
F
Of
1
Remark.
+
xJc € Ai +1
n • {e}t'!(x ,•••,xk,Jl) Of n.
, • • •,
Lemma 3.
Assume
j < i+3.
ot
If
+
is an F-structure, then Qr
+
is a weak F-structure.
+
Proof.
We prove that '{e}K(x ,•••,x ,F) Of n
1
n
.,.· {e}I'H" (x ,• •• ,x , ~) Of n by indUction on the length of the Kleenen
1
Vl.
computation.
The induction will be trivial except in case 8, and
then only recursion in some
F
of type
k
from
F
is interesting.
So assume
+
· {e} K( x
, • • • .,
1
+
xn, F) = F (X y {e ' }( y , \ , • • • , xn ,F))
Of
n .
+
By induction hypothesis
Xy E
Ak_ {e'}K(y,~
2
+
='AyE Ak_ 2{e'}t"l. (y,\ ,•••,~,/F) E Ak_ 1 •
Then
(Here we use that
f/F(Xy E Ak_ 2 {e'}ot (y,x ,•••,xn,cff) = m for some
1
.
v L
1
n
'ff)
Q:!
+
lemma.
type
and
+
The condition that
Assume the lemma holds for
j+4-functional.
mEw,
m.
01 is an F-structure, '{e}K(x1 ,•••,x n ,F)~ m.
Remark.
j < i+3).
+
{eL,..,. (x , .... ,x
Since
,••o,xn,F) =
Then
j
~
i+3
j = i+4.
j+2-en(F)
Then
n = m.
is essential for this
Let
F
be a normal
is closed under type (j+1)-
existential quantifiers.(MacQueen [4] or Harrington & MacQueen [2]).
- 9 By the lemma, however, and lemma 4, the following holds
F
is not a computation in
~
...
(q,f
3(q,f)
&
~
code
on type j+1
an i,i+4-F-structure
is not a computation in
ff).
By MacQueen'a theorem this would be semicomputable in
F.
Also note
that lemmas 3 and 4 give a new proof of MacQUeen's result that for a
functional
F
of
tp k+2, k+1-en F
is not closed under
tp(k)-existensial quantifier. Our proof works for all functionals
in which k+1E is recursive. MacQueen's proof is by a delicate
analysis of the subcomputation-relation and works for all functionals •
.....
Lemma 4.
Let
F
be a list of functionals of type
a ,•••,an E tp(i+1), where
1
from
a
1
tp(i+1)
, • • •,
an
E
Proof.
q,f
i,j ,o-F-structure tJt.. such that
Moreover 07..- will be absolute.
Regard the structure
ota
.....
= <x,tp(i+1),•••,tp(j),F,al ,•eo,an,E,=>xE tp(l),l < i
By Skolem-Lowenheim theorem, let OL'
such that
Then there are
4
coding an
Ai +1 •
i < j, j >max a.
a,
Ol'
and
OLare elementarily equivalent and
0
the same cardinality as
obtained from
be a substructure of
tp(i).
Since
oto
ot..
0
ot'
has
Let OL be the transitive structure
.....
is an
F-structure and
.....
OLand
or.0
are elementary equivalent, ot. will be an F-structure.
Now assume that
from a
T'
T'
~Y
T E Ak
is wellfounded in 0t •
T
comes
Mostowskis isomorphism, and by elementary equivalence,
will be well-founded in the real world.
will be mapped on a descending chain in
isomorphism.
Then
This proves .the lemma.
T'
A descending chain in
T
by the inverse Mostowski's
- 10 -
Lemma 5.
Let ()"& be a structure.
Then there exists a list
~
of functionals
F
~
such that
Ot is a weak F-structure.
Moreover if
~
S
is a type i +2-symbol in
interpretation of
S
F
and for some
in tJt ) ,
S,
S
I'
Ai +1
then we may interprete
S
= r.;
(the
by
S.
For technical reasons we cannot prove the theorem
Remark.
for more than one
s, for instance if
r.;
1
= r.; 2
while
S
1
*S
the
2
proof won't work.
Proof ••
x
€
Ak
We will define a function
~ ~(x)
€
{e}K
~=
ldtl
~
V such that
tp(k), and such that
(~(x
1
),•••,<v(xn))
ct
m-=> {e}ot.(x ,•••,xn)
Co!
1
The converse of this implication will not hold in general.
define
by induction on the type
~
k < i+1 ,
Else, assume that
X €
Ak+ 1 ,
If
y ~ ~"(A
some
(If
z
X
€
y
= r.;
Else, let
(If
x
~
~
k.
is defined for all elements in
Ak-1)
= q>( z) r
~" (Ak-1)
y € tp(i+1), then we are in this case if
= r.;(y) = S(y)
q>(x)(y)
y
~(x)
€
~(r.;)(y)
= S(y)
Uniqueness on type
k+1
y
€
Ai+ 1 •
= q>(y)
r ~"(Ak-1)
will then follow.
q>(x)(y)
=0
in this case.
Ak
r tt>"(~k-1)
for
here).
be anything you want, for example
we may choose
x E Ak'
Let
= x(z).
We must verify that there is no ambiguity here.
for
Ak.
tp(k).
€
and
= r.;,
We
be the identity.
(= q>-image of
k-1 )
Ak' then let ~(x)(y)
~(r.;)(y)
Thus
let
m.
-=>
x : y
We prove that
- 11 -
Let
x E Ak'
such that
x(z)
y E Ak'
* y(z).
~(x)(~(z))
X
* y.
Then there is a
z E Ak_ 1
But
= x(z)
= y(z).
~(y)(~(z))
and
Thus
Here we have used uniqueness of
on
~
Ak~
1
and
Ak.
Now we prove by induction on the length of the computation
{e}K(~(x ),·•·,~(xk)) ~
that
1
~(x )(~(xj))
1
Since
are trivial.
= xi(xj)
Assume
n
~
{e}
(/{_,
(x ,•••,xk)
1
~"A
n.
by definition, all cases except case 8
xi E Ak
and
By the induction hypothesis we have for all
q>(x) j
~
y E Ak_ 2
k-2
and
<.p(xi)(Ay{e}K(y,q>(x ),•••,c.p(xk))) = xi(x) = {e}CJ{ (x ,•••,xk).
1
1
This proves ·the claim.
-+
By letting
6.
F
-+
= <P( ff-),
the lemma also follows.
Applications on large quantifiers.
Theorem 1.
Let
Let
A
0 < i < j,
=tp (
X
E B
i) x tp ( j)
~
S
a functional of
type~
be semicomputable in
Vy E tp(j)<x,y> E A.
S.
i+1.
Define
B by
- 12 -
Then
B
is
ni(S)
in the following sense :
1
= recursive
6°(S)
0
s.
in
The rest is as in the definition
of
Proof.
Let
e
be a Kleene-index such that
<x,U> E A ~{e}(x,U,S)
X
E B ._ VU {e }( X , U, S )
~
c.t
O.
Thus
0 •
Claim.
~
V(q,u,s) E tp(i) (q,u,s codes
S n Ai
& I; =
&
X
<A ,•••,Aj,'UJ,z;,F>
0
E Ai+ 1 • {e}t'[ (x,rtu, z;;)
c.t
0).
Proof of claim.
Let
~
we find
q,u,s
U such that
{e}(x,U,S)
c.t
{ e}
( x ,'lll, z;;)
()1_
•
0,
c.t
be given satisfying the premise. By lemma 5
OL is a weak
and since
u,s-structure.
By assumption
ot is a weak U,S-structure,
0•
Let
U be given.
By lemma 4, let
0Z
be a U,S-structure
S n Ai+ 1 = z;;. By assumption
{e}Ot (x,'t4z;;)C¥ 0 , and since Ot is a U,S-structure,
containing
x.
Obviously
~
{e}K(x,U,S)
O.
By the claim, the theorem follows, since what is inside the
paranthesis is
6i(S).
0
Corollary 1.
computable and
B
If
A~
tp(i) xtp(j)
is defined by
x E B.-VUE tp(j) <x,U> E A,
then
B
is
ni.
1
is Kleene semi-
- 13 Corollary 2.
jo
functional.
Let
i.e.
S
be a type
jo
S
is normal, then
i+1-en(S, j 0)
· {e}(x,S,jO) ~ n ~
f
€ tp(j) ({f(e)}(x,S,U) ~ n).
vu
1.
note that when
i+1-sc(S), and
S is normal,
is closed under \ftp(j-2).
Corollar;y; 3.
m
=
Let
such that
c
n,m >
0~
i
To obtain the second part
< n.
i+1-en(F)
1+1-en(S,jO)
Then there exists a
over type (i).
n = m = 1, this is well known, let F = 2 .E.
1, n > 1, let F = <n+ 2o,n+ 1E>. By corollary 2,
Proof.
For
(S).
such that
To obtain the first part, use theorem
F
= IT 1i
is uniformly computable in any type-j-functional U,
there is a primitive recursive function
functional
i+1
the constant zero type j-functional.
Moreover, if
Proof.
Let
i < j-1.
For
However,
for
n > 1.
The corollary then follows
in this case.
For
complete
n > 1, m > 1, let
E~-1.-set.
n+1-en(n+ 2o,s)
Remark.
Again
is normal and since
It is not lcnown whether some
n
n+1-sc(S) c llm'
E~-sets are envelopes
However, we have
En
is never the
m
n+1-en F
n+1-en F c ll~.
S
be the characteristic function of a
= nn(S)
= nnm"
I
of functionals.
and if
S
n+1-envelope of any functional of type > n+2,
En for a functional of type n+2, then
m
This is seen by the result; of MacQueen [ 4] that says
c;:
- 14 There exists a set A which is semicomputable in F such that
<e,cr,k> is not a F-computation ~ 3x(x,e,cr,k) E A, where
cr is a tuple in tp(n) and x varies over tp(n).
1.
Skolern..:Lowenheim and inductive definitions.
In Moldestad & Normann [5] we proved a result on relativized
3
rr 1 -inductive definitions as a key to recursion in
1
n
= 1,
corollary 3 was proved.
lized to higher types.
0.
For
The proof in [5] may be genera-
We prove the theorem here, although we
have no applications of it.
Before we are able to formulate our result, we need a definition of
Tin
Let
type
~
relative to a set of higher type objects.
1
S
....
connectives
tp(n) x
is simple if
R(x,A,S)
n.
rJP (tp(n))
R is defined using the
and function symbols for all primitive recursive
tp(k)
We say that
for
k < n.
A occurs positively in
t E A occurs positively in
R,
where
r: q.>(tp(n))
rrn(S)
1
under
v,A
We define
by simple relative to
rrr(s)-operator
if
1
X E f(A)'
is
has the usual closure properties, i.e. closure
and quantifiers of lower type.
Theorem 2.
s.
A occurs positively.
ITn(S)
1
is a term.
tf(tp(n))
r is a positive
such that
Remark:
-+-
R if all subformulas
t
as before, when we replace
We say that
objects of
and I, evaluation in types, the €-relation in
v
operations on
Let
....
x
be a functional of tp(n+1), A= tp(n),
If r
is a positive
rrn(S)-operator over
1
\
tp(n),
- 15 -
Proof.
Let
X
€ r(A)
~ Vy~(x,y,A,S),
where
~
is
fin
0
and
A is positive in
Let
B c tp(n).
rB
Vy E
B~(x,y,A)
&
x E B.
will be a monotone operator.
We say that
B
Define
~
x E rB(A)
Note that
~.
B
is sufficiently closed if
includes all tp(n-1)-elements and is closed under
primitive recursion.
Note that if
x,y € B
tp(x,y,A,S)
a E tp(n).
Now let
is
~
and
B
is sufficiently closed, then
q>(x,y,An B,S) •
Define
Ba
= {t..z(a(z,y))
Claim 1
'Ba
is sufficiently closed' is
Proof.
Let
ay = A. z a ( z , y ) •
fin-definable, and equility is
1
~
x € r(A)
Va(B
~
;
y E tp(n-1)} •
Primitive recursion in
Then observe
nn-1 -definable.
1
is sufficiently closed
a
& x
E
B
a
x € fB (A)).
a
We obtain
x € rY ~ Va(B
Claim 2.
is sufficiently closed and
a
X E B => X E r BY) •
a
a
Proof
ordinals
Now let
by induction on
y
this
x E r(rY).
Ba
For
y
=0
and for limit
is trivial.
'rhen
Let
Since
y.
x € B
and let
a
is sufficiently closed
y E B
a
be arbitrary.
- 16 ~(x,y,rY A B~)
will hold.
rY
By induction hypothesis,
~(x,y,rBY)
Thus
A
=
B
a.
rY
B.
a.
•
holds by monotonicity, and
a.
and the claim is proved.
From the claim we derive
e r
x
(10
<>
Va.(B
is sufficiently closed
a.
B
a.
&
X €
a.
such that
(10
fB
X €
•
(J
By the next claim the converse will also hold.
Claim 3.
B
a.
x ~ reo, then there exists an
If
is sufficiently closed,
and
x
Let
K > !rl
and let
By Skolem-Lowenheim theorem
the same cardinality as
cardinality
tp(n-1)
=J(. n
Let
B
Now
(rco2k
absolute.
t p (n ) •
= rB
M
= <VK,tp(n-1),x,S,E>.
M has an elementary submodel of
tp((n-1).
Let~ be transitive, of
and isomorphic to the submodel of
Note that
s
:At.
=
s
f B, and that
Moreover
B
is sufficiently closed since
By cardinality there is an
a.
such that
f
00
reo
B
tp(n)
is in
= Ba. •
&
x E Ba.
is nothing more than an inductive definition over
rB
tp{n-1),
.4, 1= x ¢
By claims 2 and 3 we obtain
is sufficiently closed
Since
M.
since all lower-type quantifiers are made
Proof of the theorem.
Thus
¢ r; •
a.
Proof.
it.
x E Ba.
a.
rB
is
(10
a.
v1ill be
rrn(S).
1
Tin(S) uniformly in a., by standard p:r•oof.
1
- 17 It is still an open problem whether
TI~-positive inductive defini-
Tin-fixpoints (over tp(n)).
tions have
m
In claim
Remarks.
positive or
3 we did not use the fact that
r
was
Thus this claim holds for all definable
inductive definitions.
8.
Some properties of
Theorem 3.
assume
n > 0
jo.
Let s be a functional of type < n + 2.
n+2E is recursive in s.
Then
i
n + 2
sc(n+ko,s)
=n
ii
n + 2
en(n+ko,s)
1 ( s)
= Tin+
1
Proof.
ii
f, i.e.
{e}s,
n+ko
n+ko
+ 2 - sc(S)
n+k 0
'
for
is already verified,
If
i
k > 3
•
follows by a reindexing
(x) ~ n * {f(e)}s(x) ~ m.
can 'only check totality', so we replace recursion in
by the total zero function on indices.
\'le omit the details
here. See [ 5] •
The reflections that follow are done for recursion over tp(1).
Similar reflections may be done for I'ecursion in higher types.
Corollary 4.
2-en( 3 0)
2-sc ( 30)
= the
recursive sets
2-sc( 3 0, 2 E) =
For any functional
of
U over
tp(1)
U,
let
Th(U)
denote the Kleene theory
with associated length function.
We see that
- 18 Th( 2 E)
Th( 2 E, so)
and
have the same 2-envelopes and the same
2-sections. In both theories we have arbitrarily long countable
computations.
see that in
a tuple
However, if S is arbitrary of tp
Th(S, SO)
~
2
we shall
it is a 'quick' operation to check that
is a computation.
The set of computations in
Th(S' so)
is given by a TI 1 (S)1
expression.
a
is a
computation~
where
~
is simple.
a,cr,S
decide whether
aS-recursive function
and when
Let
~(a,n,a,S)
f
and
~(a,n,a,S)
a
we may effectively in
holds or not.
such that
f(a,a)
Thus there is
+ ~ 3n
~(a,n,a,S),
+ , the computation will be finite.
f(a,a)
g(a)
a,n
Given
Va 3n
= W(Xaf(a,cr)).
computation will be at most
Corollary 5.
Th{ 90,S)
If
g(cr)
+ the length of the
w.
is not p-normal, i.e.
we cannot
compare lengths.
Proof.
with
It is not hard to construct a computation in
length
greater than
w,
t
so
and which has a natural number
as argument. p-normality and the observation above would yield
that the set of computations were computable.
Thus
Th(2E)
and
TheE, 9 0)
are different, although they
have the same envelopes and the same sections.
This contrasts
that in the normal case, equality between evelopes gives
equivalence between theories.
Obser~e that 2-en(S, 9 0)
will always be closed under
3w.
- 19 -
However,
Let
Conjecture.
S
be an arbitrary type-2-functional.
In general, the functional
~
<p(A,a)
will not be
3
0
~
3n E w(<n,a> E A)
0 ,S-computable in the sense .of Moschovakis i.e.
there is no index
e
such that
+
· {e} w' s (e' ,x)
~
~
0
+)
3n{e} w' s< n,x
~
0
and
ll<e,e'x>llw 8 > inf{ll<e',n,~,O>IIw 8 ;
'
,
The conclusion is false when
S
nEw} •
= 2 E.
P. Aczel proved that the partial functional
~(f) ~
0 ._ 3nf(n) +
is not computable in any total functional.
Let
n11 (S)-ind =
1rr: (S)Idf
Problem.
Let
computations in
Remark.
{roo •
'
= Sup}lrl
IS, 3 0I
Th(S, 3 0).
r is a positive
IT1 (S)-operator} ,
r is a positive
IT 1 (S)-operator} •
1
1
denote the supremum of the lengths of
Will
?
If the conjecture above is disproved for arbitrary
S, we have a positive solution to the problem, by the first
recursion theorem.
!S, 3 0I
We will always have
the set of computations is given by a
~
IIT 1 (S)I, since
1
n (S)-inductive definition.
1
1
- 20 -
We end this note by the following observation
Observation.
Let
U € tp(n+2).
Then the following
statements are equivalent
a
U is normal
b
Th(U)
over
Proof.
tp(n)
is p-normal.
a • b
is well known. To prove
the following way of computing n+ 2E.
~ ~
a, regard
n+ 2E(Ax{e'}{x,y,U)
Compute
-+
OoU(Ax{e'}(x,y,U)).
AX{e'}(x,y,U)
To check if
functional
This converges if and only if
is total.
+
Yx{e'}(x,y,U)
-+
~({e'}(x,y,U))
where
~
0,
~(x)
When this computation converges,
length than the computation of
converges.
we may use
jO if X
:
OoU
0
on the
- ~ndefined o.w.
it will have shorter
-+
O•O•OoU(Ax{e'}(x,y,U)), which
By p-normality then, we may decide whether
~({e'}(x,y,U)) converges or not, i.e. compute n+ 2E.
- 21 -
Bibliography:
[1]
L. Harrington,
Contribution to recursion theory in higher
types, Ph.D. Thesis
M.I.T. 1973·
[2]
L. Harrington and D. B, MacQueen,
recursion theory,
[3]
S. C. Kleene,
[4]
D. B. MacQueen, Post's problem for recursion in higher
types, Ph.D. Thesis M.I.T. 1972.
(5]
J. Moldestad and D. Normann,
Recursive functionals and quantifiers of
finite type I, Trans.Amer. Math. Soc. 91
(1959) 1-52; and II 108 (1963) 106-142.
hierarchy.
Sack~,
[6]
G. F.
[7]
G. F. Sacks,
Selection in abstract
To appear.
2-envelopes and the analytic
Preprint Series Oslo No.21,1974.
The !-section of a type n object, in
J. E. Fenstad and P. Hinman (eds): Generalize~
Recursion Theory 81-93 North Holland 1974.
The k-section of a type
n
object, to appear.