Undecidability and 1-types in the recursively enumerable degrees

Annals of Pure and Applied North-Holland

Logic 63 (1993) 3-37

Undecidability and l-types in the recursively enumerable degrees Klaus Ambos-Spies* Mathematkches

Institut,

Universitiit Heidelberg,

D-6900 Heidelberg,

Germany

Richard A. Shore* Department

of Mathematics,

Cornell Universily,

Ithaca, NY 14853, USA

Communicated by A. Nerode Received 12 April 1992

Abstract Ambos-Spies, K. and R.A. Shore, Undecidability and l-types degrees, Annals of Pure and Applied Logic 63 (1993) 3-37.

in the recursively

enumerable

We show that the theory of the partial ordering of recursively enumerable Turing degrees is undecidable and has uncountably many l-types. In contrast to the original proof of the former which used a very complicated 0”’ argument our proof proceeds by a much simpler infinite injury argument. Moreover, it combines with the permitting technique to get similar results for any ideal of the r.e. degrees.

0. Introduction

Work on the structure P/I= (R , C, U ) of the recursively enumerable Turing degrees in the 50’s and early 60’s such as the embedding results of Friedberg [6,7] and Muchnik [ll] and Sacks’ [12] density theorem suggested that the structure was quite homogeneous. These results led to conjectures by Shoenfield [14] that it was countably categorical and indeed, in a precise but limited sense, saturated and by Sacks [13] that it was at least decidable. The first of these conjectures in its strongest form was quickly disproved by Lachlan [9] and Yates [18] who showed that there is a minimal pair of r.e. degrees: a, b > 0 with a fl b = 0. The second proved much more difficult. That the r.e. degrees are undecidable was finally announced in Harrington and Shelah [8]. Their proof was a very complicated Correspondence to: K. Ambos-Spies, Mathematisches Institut, Universitlt Heidelberg, D-6900 Heidelberg, Germany. * This research was done while the authors visited the Mathematical Sciences Research Institute, Berkeley. The authors were supported by the MSRI under NSF grant DMS-8505550, the second author in addition by NSF grant DMS-8912797. 0168~0072/93/$06.00

@ 1993 -

Elsevier

Science

Publishers

B.V. All rights reserved

4

K. Ambos-Spies,

R.A. Shore

monstrous (0”‘) injury argument. It underwent various changes and attempted simplifications over the years, particularly by Harrington and Slaman who proved that the theory of the structure is of the same degree as true first-order arithmetic. Due largely to the complexity of the constructions, no proofs of these results have yet appeared. A considerably simpler approach to these results has recently been developed by Slaman and Woodin. All of these proofs proceed by coding all recursive and indeed all A2 partial orderings into 3. The difficult part of the argument is producing a set which is definable from parameters on which one can define the desired partial ordering. Another view of the complexity of a structure can be seen in the number of its types. The realizability of countably many 3-types in the r.e. degrees was proven by Lerman, Shore and Soare [lo] at about the same time as the first proof of undecidability. Of course, this result implied that the theory is not countably categorical. It was extended to prove the existence of uncountably many 4-types in the theory and other nonhomogeneity results in Shore [Cl. Similar results on n-types in the theory also follow from the undecidability proofs mentioned above but the direct proofs were somewhat simpler and required fewer parameters. The problem of the number of l-types remained open until Ambos-Spies and Soare [2] proved that countably many l-types are realized. Like the proofs of undecidability, their argument was an extremely complicated monstrous injury one. Here we provide a new proof of the undecidability of the r.e. degrees which is relatively simple. It requires no more in the way of technique than the construction of branching and nonbranching degrees. Thus, it proceeds by a standard 0” tree argument. We also prove by the same method the existence of uncountably many l-types in the theory of the r.e. degrees. The arguments can easily be combined with permitting to get the same results of undecidability and the existence of uncountably many l-types in any ideal of the r.e. degrees. Of course, the major simplification comes as a result of finding sets of degrees which are more easily definable. One price we pay is that we can only define finite (rather than recursive or A,) partial orderings. This suffices to prove undecidability by appealing to the recursive inseparability of the theory of partial orderings rather than simply to its own undecidability. By compactness, it is also enough to prove the consistency of continuum many l-types. The shortcoming of this method is that there does not seem to be any way to extend it to prove that the theory of $8 is l-l equivalent to true first-order arithmetic. The outline of the paper is as follows. In Section 1 we state the main results. The proofs of the three main technical theorems are given in Sections 2-4. Finally, in Section 5 we consider the question which fragments of the theory of 9? are undecidable. In general, our notation is standard (as in Soare [17]). For the proof of the technical theorems we need in particular the following notation. For some fixed enumeration (w: i 3 0) of the r.e. sets, let W,[s] denote the part of w

Undecidability of l-types in the r.e. degrees

5

enumerated after s steps. Similarly for any r.e. set X we will construct below, let X[s] denote the part of X enumerated by the end of stage s of the construction. Instead of X[s](x) and {e}Tlsl we write X(x)[s] and {e}“(x)[s], respectively. We letX(m=Xn{O,... , m - l}. The use functions of {e}“(x) and {e}:(x) are denoted by u(X; e, x) and u(X; e, x, s), respectively. As usual we assume that {e>T(x)J 3

e, 4 4X;

e, x, s) <.s

(see Soare [17, 111.1.71). Moreover, conventions: + x < u(X; e, x)

HX(4L

and,

w.1.o.g. for any s,

(0.1) we adopt

the following

two

u(X; e, x, s) C u(X; e, x) (0.2)

and, for any set X under construction,

K4”Wbll~

HWb

+ 11# WY4bll

3 W”Wb + II?

(O-3)

(see Soare [17, VIII.11).

1. The main results

A set ,Z of sentences is called strongly undecidable if there is no recursive set R such that V n Z c_ R E 2, where V is the set of logically valid sentences. Ershov and Taitslin [3] have shown that the set of all sentences in the language L(s) which are true in all finite partial orderings is strongly undecidable. This easily implies that, for any structure Y in which all finite partial orderings are elementarily definable with parameters, the first-order theory Th(Y) of Y, i.e. the set of sentences true in Y, is (strongly) undecidable. So in particular for upper semilattices Y the following holds (see Section 5 for details). 1.1. Theorem. Let % = (U, s”, U) be an upper semilattice and let C#Ibe a formula in the language of partial orderings with k + 1 free variables x0, . . . , x~_~ and y. Moreover, assume that for any n 2 1 and for any partial ordering s(, on (0, . . * , n - l} there are elements a(), . . . , ak-l, bO, . . . , b,_, and c of U such that, for any i, j
3

bi # bi,

{bo, . ’ . , h-J

(1.0) = lb E 17: Q k 4%,,,_.., xk_,,ybo, . . . ,

iGOj @ bjGUbjUc. Then the first order theory of ( U, c ,)

ak-1,

bl),

(1.1)

(1.2) is undecidable.

To apply Theorem 1.1 to the u.s.1. 94 = (R, S, U) of the r.e. degrees, by (1.1) we need an elementary property of %! which for varying parameters q,,, . . . , 8k-1 defines finite sets of arbitrary size. We will get such a property in one parameter a by considering the branches of branching degrees.

K. Ambos-Spies,

6

R.A. Shore

1.2. Definition. (a) An r.e. degree a is branching if there are r.e. degrees b and c such that a < b, a < c and a = b n c. Otherwise, a is nonbranching. (b) Let a and b be r.e. degrees such that a s b. Then b is a-cappable if there is an r.e. degree c such that a < c and a = b n c. Otherwise, b is non-a-cuppable. A degree b is maximal-a-cappable if b is a-cappable and all degrees c > b are non-a-cappable. Lachlan [9] has shown that some but not all r.e. degrees are branching. In fact, both the branching degrees and the nonbranching degrees are dense subclasses of (R, S) (Slaman [16] and Fejer [5], respectively). Note that maximal-a-cappability is definable in the first-order language of partial orderings. 1.3. Proposition. There is a first order formula Q, in the language orderings with two free variables x and y such that 92 k &,[a,

b] e

of partial

b is maximal-a-cappable.

(1.3)

Ambos-Spies and Ding [l] have shown that there is an r.e. degree a which possesses exactly two maximal-a-cappable degrees. Our main technical theorem extends this result. 1.4. Theorem. Let Co be a partial ordering degrees a, bO,. . . , b,_, , c such that

There are r.e.

bi and bi are incomparable,

for i
bi is a-cappable

iSOj ifundonlyif

We will prove Theorem

(1.4)

(i
for any a-cappable degree d, for i, j < 12,

on (0, . . . , n - l}.

(1.5)

d 6 bi for some i < n,

and

(1.6)

bi
(1.7)

1.4 in Section 2.

1.5. Corollary (Ambos-Spies and Soare [2]). The first-order theory of the partial ordering (R, S) of the r.e. degrees has infinitely many l-types. Proof. By Proposition 1.3, for any n, there is a first-order formula language of partial orderings with one free variable x such that .$J? k ($&[a]

G there are exactly n maximal-a-cappable

I/J, in the

degrees.

Note that for a, bO,. . . , b,_I satisfying (1.4)-(1.6), the degrees bi are pairwise different and are exactly the maximal-a-cappable degrees. So, by Theorem 1.4, for any n 3 1 there is a degree a, such that 5%k (qln)x[a,,].Since, by definition, the types of the formulas qI, are pairwise disjoint, this implies the claim. Cl

Undecidability

of l-types in the r.e. degrees

1.6. Corollary (Harrington and Shelah [S]). The first-order ordering (R, S) of the r. e. degrees 13 undecidable.

7

theory of the partial

Proof. Fix # as in Proposition

1.3, let G0 be any partial ordering on (0, . . . , n l}, and, by Theorem 1.4, take r.e. degrees a, bo, . . . , b,_I, c satisfying (1.4)-(1.7). Th en, by (1.4) and (1.7), conditions (1.0) and (1.2) of Theorem 1.1 hold for b,, . . . , b,_l, c in Cii!.Moreover, by (1.5) and (1.6), {b,, . . . , b,_,} is the set of maximal-a-cappable degrees. So, by choice of #, {b,, . . . , b,_,} = {b~R:%k&,[a,b]}. H ence the claims follows from Theorem 1.1. Cl By Theorem 1.4 any finite partial ordering can be elementarily defined in with two parameters a and c. This implies that there are continuously many 2-types consistent with (R, G). As the next theorem shows, in certain cases the second parameter c can be defined from a. We will use this to show that there are actually continuously many l-types in (R, G). (R, S)

1.7. Theorem. Let +, be a partial ordering on (0, . . . , n - l} which has at least three minimal elements. Then there are r.e. degrees a, bo, . . . , b,_l, c satisfying (1.4)-(1.7) and

(1.8) The proof of Theorem

1.7 will be given in Section 3.

1.8. Corollary. The first-order degrees has 2” many l-types.

theory of the partial ordering

(R, C)

of the r.e.

Proof. It suffices to give a sequence

( qk: k > 0) of formulas with one free variable x such that for any nonempty finite set F of natural numbers there is an r.e. degree aF such that %L(qk)x[a,]

e

ke F.

(1.9)

Fix C$ as in Proposition 1.3 and define the formulas y = Y~,~, 6 = &, and qk = (v”)~ whose intended meanings are described below in (l.lO)-( 1.13) as follows: Y’3Y (G&VZ(Y f 2 & f#&[X, z] -+ u C 2) & VW [Vz (y # 2 & f&JX’ z]‘, w < .z)-, w < U]), 6~Vu(y~u~v)&Vt[Vu(y-,u~t)-,v~t] ?lk=33so,...

Y Sk

(hx,y[%

s,,]

&

and ’ * . &

d%x,y[x,

Sk]

&v~(6,,,~[S”~S,u~&‘.‘&Sk_,~Sku~ &Sl+s~~uv&*

..&Sk+k-_lb])

&vS,~(~~,,[~,S]&S#S”&‘.‘&S#Sk&~~,,--, [sps”uv&**

(where in qk the symbols f,

~&S~Sku~&S”~S~~&~“&Sk~Su~]))

U

and p should be expressed in terms of s).

K. Ambos-Spies,

8

R.A. Shore

Then, for any partial ordering so on (0, . . . , n - l} with at least three minimal elements and for r.e. degrees a, bO, . . . , b,_I, c as in Theorem 1.7, %k&,[a,e]

ec{bo,.

ifandonlyif

. . ,b,_I}

(1.10)

(by (1.5) and (1.6)), L%! L yx,Ja, f] if and only if f E n bj: i < n t j#i

I

(1.11)

(by (1.8) and (l.lO)), % k &,[a, (by (1.8) and (l.ll)),

g] if and only if g = c

(1.12)

and

%F(rlk)J a 1 1‘f an d only if the partial ordering co contains a maximal chain of length k + 1 and all numbers j not contained in the chain are $,-incomparable with all members of the chain, i.e., if there are numbers iO, . . . , ik such that i. Co il Co. . - Co ik and, for all j E (0, . . . , n - l} - {io, . . . , ik} and all p G k, j &, iP and iP =#0j. (1.13) (by (1.7), (1.10) and (1.12)). Now, for any nonempty finite set F = {m,, . . . , WI,,} of natural numbers, call a partial ordering co on (0, . . . , n - l} of chain type F, if it is the disjoint union of maximal <,-chains of length m, + 1 (1 sp) where members of different chains are +incomparable. Moreover, require that there are at least three chains of length m, + 1. (Since the bottom element of a chain is minimal this ensures that there are at least 3 <,-minimal elements.) Then, for any partial ordering so on r.e. degrees (0, . . * , n - l} of chain type F and for the corresponding . , b,_,, c as above, it follows from (1.13) that a, ho, . . %!k(q”)Ja]

e

kcF.

0

The proofs of Theorems 1.4 and 1.5 can be combined method to obtain degrees below any nonzero r.e. degree.

with the permitting

1.9. Theorem. Let e be any nonrecursive r.e. degree and let co be a partial ordering on (0, . . . , n - l}. Then there are r.e. degrees a, bo, . . . , b,_, , c below e such that (1.4)-(1.7) hold. If, moreover the partial ordering +, has at least 3 minimal elements, then (1.8) will hold too. For the proof of Theorem 1.9 see Section 4. By applying Theorem 1.9 in place of Theorems 1.4 and 1.7 we can extend Corollaries 1.5 and 1.8, respectively, to any nontrivial ideal of (R, c) . 1.10. Corollary. For any r.e. degree e > 0, the first-order theory of the partial ordering (R(se), S) of the r.e. degrees below e is undecidable.

Undecidability

of l-types

in the r.e. degrees

9

1.11. Corollary. For any r.e. degree e > 0, the first-order theory of the partial ordering (R(se), S) of the r.e. degrees below e has 2” many l-types.

We believe that Theorem 1.9 can be extended to intervals in place of ideals, so that for any interval [a, b] with a < b the theory of the r.e. degrees contained in [a, b] is undecidable and has 2” many l-types. (The undecidability part had been previously announced by Slaman and Woodin.) The proof of this extension, which requires a much more involved 0”’ argument, will appear elsewhere. Another interesting question is which fragments of the theory of the r.e. degrees are undecidable. We briefly address this question in the final Section 5.

2. Proof of Theorem

1.4

Note that, for rt = 1, the theorem holds if we let a = b. be any nonbranching degree and c = 0. So w.1.o.g. we may assume that n 2 2. The proof is an infinite injury argument and uses the framework of a priority tree to coordinate the strategies for meeting the individual requirements. For a general explanation of the tree method we refer to Soare [17, Chapter XIV]. The proof is based on variants of the nonbranching degree technique of Lachlan [9] and the branching degree technique of Fejer [4]. We construct r.e. sets A, Bi, C, and Dj,k,l (i < n; k, 1 < w) which will satisfy the following conditions (for i, j
3

Ci~TBi.

(2.0)

Ci $TA*

(2.1)

For any total function g, -K A @ Wk and Di,k,, ‘T

if g+A D;,k,, +A

IfWk~TA~BiandW,~TA~Bithen isOj

3

B;s,A@Bj@C,

@B; for all i
*

B;$TA@Bj@C.

(2.2) (2.3)

Di.k,l~TA.

whereC=@Ci. icn

i&j

then g+A.

(2.4)

(2.5) (2.6)

Then the degrees a = deg(A), b, = deg(A @ Bi), ci = deg(A @ Ci), and c = deg(C) have the required properties: By (2.0) and (2.1), a < ci < bj for i #j < n, while, by (2.0) and (2.2), a = bi fl c;. SO bi and bj are incomparable for i fj and bi is a-cappable. To show that (1.6) holds, for a contradiction assume that d is a-cappable but d + bi for all i a such that a = d II e. Since, by (2.2), a=b,,n... fl b,_l and since e > a, e #Gbi for at least one i
10

K. Ambos-Spies,

R.A. Shore

fse by (2.3), while, by (2.4), f=$a. So a # d fl e contrary to assumption. Thus (1.6) holds. Finally, (1.7) immediately follows from (2.5) and (2.6). The conditions (2.0)-(2.6) fall into three groups: First, conditions (2.0)-(2.2) which ensure that a is the common infimum b0 II - - - fl b,_i of the IZ degrees b 0,. . . , b,_i but also that a is not the infimum of any proper subset of these degrees. These conditions are satisfied using the branching degree strategy. Second, conditions (2.3) and (2.4) which ensure that the degrees not below any bi are not a-cappable. Here we use the nonbranching degree strategy. Third, conditions (2.5) and (2.6) which code so by bo, . . . , b,_r, c. These conditions are satisfied by direct coding and diagonalization, respectively. The conditions (2.1), (2.4) and (2.6) are broken up into the following finitary diagonalization requirements: li,e:

Ci

f

W,

N,k,l.e:

Oi,j.e:

{e}A~

&A

i +oj 3

@

Bi

and

W

&A

@

Bi

+

Die/c,/

f

{e>A,

Bi # {e}A@Bi@C.

These requirements are satisfied by the Friedberg-Muchnik strategy. E.g., for li,,, we reserve an infinite recursive set F(Zi,,) of numbers (i.e., a number x E F(Zi,,) will enter Ci only for the sake of Zi,e)*We then wait for a stage s such that, for some number x E F(Zi,,), {e}“(x)[s] = 0. Then we put x into Ci and, by a restraint on A, preserve the computation {e}“(x)[s] so that C,(X) = Ci(X)[S + l] = 1 # 0 = {e}“(x)[s] = {e}“(X). We will choose the sets reserved for the individual requirements F(Zi,J = g’i’el~

F(N~,~,].,) = gliPkJ~el,

as follows

F(o~,~,,) = W[n+i.j~el

(2.7)

where ~t~~,.~.+l= {(m,, (m,, . . . (m,, x)- * -)>:x ?=O}. The only numbers which will enter Ci are the ones put into Ci for the sake of some diagonalization requirement Ii,, (e 3 0). Similarly, Di,k,, will only contain attackers of requirements Ni,k,[,e (e 2 0). Hence, by (2.7), ci G &I.

(2.8)

To satisfy conditions (2.0), (2.2), (2.3) and (2.5) we use different sorts of coding strategies. (2.0) and (2.5) are satisfied by direct set coding. To ensure (2.0) we let B.I n &I=

Cj fl0G 0

if i #j, if i = j.

(2.9)

Since, by (2.8), Cj = Cj tl w lil, this will guarantee that Cj + Bi for i fj. To satisfy (2.5) we ensure that a number will enter Bi only for the sake of coding some Cj into Bi via (2.9) or for the sake of some diagonalization requirement Oi,j,e. In the latter case we put the attacker x not only into Bi but also into all Bk s.t. i so k,

Undecidability of l-types in the r.e. degrees

11

i.e., we insure iG,kII$

&no

ln+il = ~~ ”

&n+il

(2.10)

Obviously this will suffice to get (2.5). To satisfy (2.3) we use marker permitting. corresponding marker function. Definition.

We first define for any r.e. set a

The standard marker y,,, for the r.e. set W, is inductively defined by

ym(~)Pl = (km,

4 O>,

(2.11)

(l,m,x,s+l)

ifW,[s+l]Ix+lZW,[s]Ix+I,

(2.12)

otherwise.

The next two lemmas summarize some elementary markers. Lemma

1. (i) The

properties

of the standard

functionAm, x,

s. ym(x)[s] is total recursive. (ii) ym(.x)[s] E ePm*xl E cOll*mlc_ 01’1. So if (m, x) #(m’, x’) then y,,&)[s] #

Ym4m’l. (iii) Zfx GX’ and s ~8’ then ym(x)[s] c ym(x’)[s’]. (iv) lim, y&)[s] = sup, y&)[s] < w exists. Proof.

Straightforward.

q

In the following let y:(x) = lim, ym(x)[s] be the final position of ~/m(x). Lemma 2. yz CT W,. Proof. For s such that Wm[s] 1x + 1= W,, [ x + 1, yi@) = Y,&)[s].

0

Now to satisfy (2.3) we ensure that x E Di.,& + II-

&&I

3

~,h)bl,

~,(x)bl E 4s

Lemma 3. Assume that (2.13) holds. Then Di,k,/ + Proof. By symmetry

+ II- 4~1.

A @ W, and Di,k,, $-A

(2.13) CBW,.

it suffices to show Di,k,l STA @ Wk. By (2.13) and by

Lemma 1, A[s] ( yk*(X)+ 1 =A ) Yk*(X)+ 1 3

Di,k,l(X)[~Sl = Di,k,/(X).

Since, by Lemma 2, y:(x) can be computed recursive in the join of A and W,. 0

(2.14)

from W,, this implies that Di,k,l is

K. Ambos-Spies,

12

R.A. Shore

To show that marker permitting (2.13) is compatible with the FriedbergMuchnik strategies satisfying the diagonalization requirements Ni,k,,,e, we need the following property of standard markers. Lemma 4. Assume that W, & E, F is an infinite E-recursive

set, and g is a partial

E-recursive function such that F is contained in the domain of g. Then there is an infinite E-recursive

subset G of F such that, for any x E G, y:(x)

>g(x).

Proof. By Lemma 1, the set

H = {x E F: y:(x)

> g(x)}

= {x E F: 3s (y,(x)[s]

>g(x))}

is r.e. in E. So, since any infinite E-r.e. set contains an infinite E-recursive set, it suffices to show that H is infinite. For a contradiction assume that H is finite, say x0 = max(H). Then we prove that Wm =+ E contrary to assumption as follows. To compute Wm(x) from E for a given x, first find the least y >x such that y >x,, and y E F. Then y:(y) S g(y), whence by Lemma 1, .y??l(x)]sl=SYrn(Y)]Sls Y:(Y) cg(y) for all s > 0. By definition of ym this implies W,(x) = W,(x)[g(y)].

Cl

Lemma 4 will allow us to argue that if W, &A 63 Bi, W, =&A CI3Bi and {e}” is total then there will be infinitely many numbers x such that yk(x) and y!(x) outrun the use function u(A; e, x) (so that we can satisfy (2.13) and at the same time preserve the computation {e}“(x)[s]). Finally, to satisfy (2.2) we also use a marker technique. We split this condition into infinitely many (possibly) infinitary requirements M,:

{e}A@@I= {e}A@BI= . . . = {e}AeBn-I total 3

{e}AeB”+A.

By an observation of Posner (see Soare [17, 1X.1.41) the requirements A4, suffice to satisfy (2.2): If g sT A @ Bi for all i A@&= {ei}A@BI= . . . = {en_l}A@‘n-l

= g is total.

On the other hand the requirements Zi,, and the coding conditions (2.8) and (2.9) will ensure that there are finite sets E such that E is contained in Bj iff i = j. So if we let {e}“(x) = {e,}“(x) if (2y + 1: y E E} is contained in X and {e}x(x)f otherwise then A@&,= {e}A@B~= . . . = {e}A@L {e>

cg

whence ZVZ, will ensure that g + A. The length of agreement between the functions stage s is measured by

{e}A@Bi, i < n, at the end of

Z(e)[s] = max{x: Vy A@B”-‘(Y)[SIJ)].

(2.15)

Undecidability of l-types in the r.e. degrees

13

A stage s is called e-expansionary if Vt


As one can easily check,

(4 A@BII=

{e}AeBI = . . . = {e}A@Bn-l

total

iff

lim,

l(e)[s]

=

w

(2.16)

and limsup I(e)[s] = o iff there are infinitely many e-expansionary s

stages.

(2.17)

Now assume that (4

A@&= {e}A@Bl= . . . = {e}“@“n-1 = g is total.

(2.18)

Then, by (2.16) and (2.17), there are infinitely many e-expansionary stages. So g can be coded into A by a set of markers 6,(X) as follows: At the first stage s + 1 such that s is e-expansionary and I(e)[s] >X the marker &(X)[s + l] becomes defined to be a fresh number y >s, i.e., y $ A[s] and y is different from all previously defined marker positions. Now we will arrange the construction so that, if for all stages t Bs, {e}A@B’(x)[t] = {e}A@Bl(x)[s]

for at least one i
(2.19)

then the marker 6,(X) will not be moved after stage s + 1 and its final position &(X)[s + l] will never enter A. (Note that, by r(e)[s] >X, {e}A@Be(X)[s]i = {e}A@B1(x)[s]= * . . = {e}A@“n-‘(X)[s]. So (2.19) for all t > s implies that g(X) = {e}A@BO(x)[s].) On the other hand, if there is a t >s at which (2.19) fails, then 6,(X)[s + l] will be put into A at the least such stage t + 1 >s and 6,(x)[t + l] will become undefined. In this case we now repeat the above procedure. Now, by (2.18), there will eventually be an e-expansionary stage s such that &(X)[s + l] becomes defined at stage s + 1 and (2.19) holds for all t 5s. Then 6,(X)[s + l] will be the final position of 6,(X) and g(X) = {e}A@Bn(x)[s]. Moreover, since a position b,(x)[t] of 6,( x ) is final iff &(x)[t] $A, this stage s can be computed A-recursively. All this implies that g Gr A. This coding strategy for meeting M,, however, conflicts with the FriedbergMuchnik strategies to satisfy the diagonalization requirements. Suppose, for example, that at stage s + 1 we want to meet a diagonalization requirement I,,,, by putting a number x into Ci for which {e’}“(X)[s] = 0 and restraining A on u(A[s]; e’, x, s) thereby ensuring that Cj(X) = Ci(X)[S f l] = 1 # 0 = {e’}A(x)[s] = {e’}A(X). Then there might be a marker 6,(y)[s] for a meet requirement M, put down at stage t + 1, such that 6,(y)[s] < u(A[s]; e’, x, s), {e}A@Ei(y)[s] # {e}“@‘“a(y)[t] and putting x into Ci will, by (2.9), destroy all the computations {e}A@B(y)[S], i # i. So the M, strategy will put 6,(y)[s] into A thereby injuring the computation {e’}“(x)[s] which the I;,,, strategy wants to preserve.

K. Ambos-Spies,

14

R.A. Shore

This conflict is resolved as follows. If Z,,,. has higher priority than M,, then M, will be initialized at stage s + 1, i.e., all 6, markers will be lifted (without entering A). (Since the diagonalization strategies are finitary, this will only delay the coding strategy for M, by finitely many stages.) If M, has higher priority, then we distinguish two cases. First, if there are infinitely many e-expansionary stages, then the Z,,,. strategy will wait for an e-expansionary stage s to attack. Then, for any y such that for all i A@E”(y)[~]J = {e}“@“(y)[s] restraint of length s on A CDBi, the computation {e}“@“(y)[s] will be preserved, whence &(y)[s] will never enter A. (Note that, by (2.7) and (2.9), we can do this and at the same time allow x to enter Ci.) Since, by our convention (O.l), markers put down after stage s are too big to destroy computations existing at stage s, this allows the Z,,,. strategy to put x into Ci and at the same time preserve {e}“(x)[s]. On th e other hand, if there are only finitely many e-expansionary stages, then only finitely many 6, markers will ever be put down, whence a finite restraint r on A CT3B,G3BlCD*. * CT3 I?,_, will prevent these markers from entering A. So in this case I,,,. can be attacked at any stage, as long as the attacker x is bigger than r. The conflict between a meet requirement M, and a diagonalization requirement Oi,+’ can be resolved similarly. In case of a requirement Ni,k,,,e,, however, we need a somewhat different strategy. Since, by (2.13), Ni,k,l,e, will enumerate numbers into A, even at an e-expansionary stage, it can destroy all computations of an existing agreement {e} A@B”(y)[~]= {e}“@“(y)[s]J, for all i < 12. Hence the Ni,k,,,e*strategy will only attack via x if there is a number v such that u(A]sl; e’, x, s) c u s y&)]sl,

(2.20)

YG)]sI

and

vy (d,(y)[~]J


+

[u(A

CB B&l;

e, Y, s)<

v &

{e>“@“(~)]sl=

{e]A@B’(Y)]t]l

where t + 1 is the last stage G s at which 6,(y) has been put down]). (2.21) Note that, by (2.21), a &-marker less than v will enter A after stage s only if some other number less than v enters A or Bi. So (2.20) and (2.21) will allow Ni,k,[.e’ to diagonalize against {e’}A(x)[ s ] and at the same time protect this computation against M, by imposing a restraint of length v on A and Bi. Following Slaman [16] we call a number v satisfying (2.21) an e-i-configuration at stage s. We call a configuration permanent if it will be a configuration at all sufficiently large Stages. Now to meet Ni,k,,,e’in the presence of a higher priority infinitary meet requirement Me, it suffices that, for all numbers u, there is a permanent e-i-configuration v(u) 3 u. Then, since the function Au.v(u) is A G3&-recursively bounded, we will conclude from Lemma 4 that if the hypothesis of Ni,k,[,e’ holds and {e’}A is total then there will be infinitely many

Undecidability

of l-types in the r.e. degrees

15

numbers x reserved for Ni,k,,,e’ for which there will be a permanent e-iconfiguration v such that (2.20) holds for the final use of {e’}A(x). Note that, ifs is an e-expansionary stage, then s is an e-i-configuration at stage s. Moreover, by imposing a restraint of length s on A and Bi this configuration can be made permanent. Ni,k,[,e*itself cannot build configurations in this way, however, since if its hypothesis fails then (2.20) might never hold whence the restraint imposed by Ni,k,l,e, to preserve larger and larger configurations would go to infinity. Hence, we introduce requirements Ke,i., each of which will be responsible for building a permanent e-i-configuration Z=U.Note that any single such requirement will be finitary. To implement the above ideas for resolving the conflicts between the meet and diagonalization requirements, we will have different strategies for every requirement, one for each guess at which of the higher priority meet requirements are infinitary. These guesses are coded by binary strings where a 0 codes the infinitary and a 1 the finitary outcome. We assign one requirement of each type to each level of the tree T = (0, l}<, of binary strings. (The possible outcomes of the finitary diagonalization requirements will not be reflected by the tree.) Then the nodes on a fixed level n represent the different strategies for the requirements assigned to tz. We first recall some notation associated with T. In what follows, lower case Greek letters will denote elements of T. We let 1~~1denote the length of a and we let a E p ((Yc /3) denote that /? extends (properly extends) (Y. The empty string is denoted by A. For i < 2 we let (i) denote the string consisting of i and we let e$ denote the concatenation of a and /3. We say that a, is to the left of /3, and write a<, /3, if, for some y, y (0) is extended by (Yand y (1) is extended by p. Then a partial ordering s on T is defined by

As usual we write (Y< /3 if CY c /3 and cy# p. The set of infinite paths through T is denoted by [T], i.e., f E[T] iff f ( m E T for all m 3 0, where f ) m denotes the initial segment of f of length m. We say cx is on f (a c f) if cx= f 1m for some m 2 0. For f, g E[T] we say f is to the left of g (f CLg) if f 1n =LLg1n for some IZ. Now a-stages and cu-expansionary stages are defined by induction on 1~~1as follows. Definition. (a) O-expansionary. &-expansionary a-expansionary (b) Ifs is an

Any stage s is a A.-stage and s is L-expansionary if s is If s is an a-stage and 1~~1 0, then s is if I(lal)[ s ] > max{l(llYl)[t]: t
Lemma 5. For any stage s and any number length n such that s is an &,-stage. Moreover

n 6s

there is a unique string E,, of

E,, c E,+,.

16

K. Ambos-Spies,

Proof. Straightforward.

R.A. Shore

0

In the following let E[S] be the unique string of length s such that s is an c[s]-stage. Then (Yis accessible at stage s + 1 iff cxc E[s]. Definition. Let f E [T] be the leftmost path in [T] which extends E[S] infinitely often. I.e., for any n, there are infinitely many s such that f 1n G E[s], and, for any a such that ]CX] = it and a s,, lY+ &[S]. Note that f exists by Lemma 5. Lemma 6. For any number e the following hold. (a) There are infinitely many f 1e-stages. (b) There are only finitely many stages s such that E[S] CL f 1e.

I e)(0) .

1s on f iff th ere are infinitely many f I e-expansionary stages. (f (d) Zf {e}A@Bo = {e}A@B1 = . . * = {e}A@Bn-l is total then (f ) e) (0) is on f. (4

Proof. Straightforward.

0

By Lemma 6, f I e codes the correct outcome of the meet requirements e’ < e, and f I e is accessible infinitely often. Now we assign the requirements to the levels of the tree T as follows: ]LY]= n . e + i *

Z, is a strategy for Zi,,,

](Y]= n * (k, 1, e) + i +

N, is a strategy for Ni,k,,,e,

Ia] = (i, j, e) & i, j < n 3 10~1= e 3

M,,,

0,

is a strategy for O;,j,e,

AI, is a strategy for M,,

Ia] = n . (m, u) + i +

K, is a strategy for K,I~,~_,

where K m(m,i.u.

*

there is a permanent

((Y ) m)-i-configuration

A priority ordering on the strategies is obtained requirements by letting

v 3 u.

by first ordering

the types of

and then by letting R, have higher priority than R,$ if either LY< fi or (Y= p and R CR’.

In the following definition we state the precise rules for moving the coding markers 6, of the strategy M, for M,,,.

Undecidability of l-types in the r. e. degrees

17

Definition.

Let ( - > be a recursive bijection from T onto w. Then for cyE T with Ia]= e the marker positions 6,(x)[s] are inductively defined as follows. (60) For all x, S,(x)[O]T. (61) If E[S] < a/ or some requirement R, with /3 c cx acts at stage s + 1 or some marker GB(y)[s] with p (0) < ,_(Yenters A at stage s + 1, then 6,(x)[s + l]? for all x such that &(x)[s]J. In this case we say that M, is initialized at stage s + 1. (Note that in this case 6,(x)[s] will not enter A.) (62) If s is cr-expansionary, I(e)[s] >x and S,(x)[s]t then let 6,(x)[s + 111 = (0, (a), x, s + 1). In this case we say 6,(x) is put down at stage s + 1 (on (0, (a), x, s + 1)) and write 6,(x)[s + 1111 (= (0, (cY), x, s + 1)). (63) If &44bl~> Mmis not initialized at stage s + 1, t is the greatest stage u
Vi < n ( {e}A@Bg (x)[tl+ W”=w~l> (*) then put 6,(x)[s] into A at stage s + 1 and let 6,(x)[s + l]?. In this case we say 6,(x) is lifted a t st age s + 1 and write 6,(x)[s + l]??. (64) If not specified otherwise in (al), (62) or (63), then 6,(x)[s + l] =

4x(x)bl* The next three lemmas prove the properties of the markers 6, needed to show that the strategy Mfle will succeed in meeting M,. Lemma

7. Let ICYI= e.

(a) Zf 6,(x)[s]J

then &(x)[s] = 6,(x)[t + l] = (0, (cu), x, t + l),

the greatest stage u s s such that S,(x)[v]& a,(x)[s]

E &p’(dJl

(b) Ifs CS’, &(x)[s]J s SS’ Cs(‘,

(c) If

z

(g?(41

s

h3-e

t + 1 is

In particular, ()pl~

and S,(x)[s’]J

&44b1L

then &(x)[s] s S,(x)[s’]. ~&WIT and S,(x)[s”]J then

6,(x)[s]

<

4&m”1. Proof. Straightforward.

0

By Lemmas l(ii) and 7(a) the positions of the standard markers yrn are contained in wt’l while the positions of the markers 6, are contained in otol. So the coding strategies for (2.3) and (2.2) do not interfere with each other. Moreover, the only numbers which will enter A will be ym or 6, markers used for these codings. Lemma 8. Let 1~~1 = e and assume that M, is not initialized after stage sg. Then the following hold. (a) Assume that s >s, is cx-expansionary, 6,(x)[s + 111 and Vt>s Then s,(x)[t]

3i
({e} A@Bi(x)[t] = {e}“@“~(x)[s]). + l] for all t BS.

K. Ambos-Spies,

18

(b) Zf {e}“““(x)J

R.A. Shore

then lim, G,(x)[s] E w U {T} exists.

(c) Zf there are infinitely many

a-expansionary

S,(x)[s]J for infinitely many s. (d) Zf {e}A@Bo = {e}A@B1 =. . . = {e}A@Bn-l

stages then, for

is total then,

for

any x > 0, any

x 20,

lim, 6, I e(x)[s] = sup, S,I ,,(x)[s] e w exists. Proof. (a) Let U+ 1 be the last stage x, whence

Vi s 3i
CM = {e>A@BW[sl)

it follows that Vt > s 3i < n ({e}A@Bi

(XDI= -+>“~“‘wbJ1>-

Hence 6,(x) will not be lifted after stage s. By choice of s0 and since s > s0 this implies s,(x)[t] = b,(x)[s + l] for all t > s. (b) Assume that {e}A@B(x)J. Then there is a stage s1 >sO such that Now if there is an {e]“@“W]sl + 111 via an A CDB,-correct computation. a-expansionary stage sz > si such that 6,(x)[sz + 111 then, by (a), lim, 6,(x)[s] = 6,(x)[s2 + l] E w. Otherwise, 6,(x) will not be put down after stage sl, whence lim, &(x)[s] E w U { f } will exist. (c) This is immediate by (62). (d) If {e}A@BiI= {e}A@B1= . . . = {e}A@Bn-L is total then, by Lemma 6, there are infinitely many (f ) e)-expansionary stages. So, by (b) and (c), Finally, by Lemma 7(b), lim,afl h)bl = lim, afj &)bl E 0 exists. sups &I

&)bl.

rJ

If the limit lim, 6,(x)[s] E w exists then we denote it by 6:(x). Lemma 9. Assume that a number y E wlol enters A only according to rule (63). Moreover assume that Ial = e and M, is not initialized after stage sn. Then the following hold. (4

Ux)bl4 44.

(b) Zf s’ > s > s0 and S,(x)[s]J,

# 6,(x)[s’]

then 6,(x)[s]

E A[s’].

(c) Zf 62 is total then 6z+A.

(d) Zf {e}A@Bo = {e}A@BI = . . . = {e}A@Bn-I = g is total then g =+A.

Undecidability of l-types in the r.e. degrees

19

Proof. (a) and (b) are immediate by Lemma 7(a) and the rules (62) and (63). (c) follows from (a) and (b): If Sz is total then 6:(x) = ~,(x)[s] for the least s such that ~,(x)[s]~ and &(x)[s] $A. For a proof of (d) assume that {e}A@Bo= {e>A@BI= . . . = (4 AeBn-~ = g is total. Then, by Lemmas 6 and 8(a), there are

infinitely many (f ( e )- ex p ansionary stages and ST, e is total. Moreover, g sT $ e since g(x) = {e}A@BO(x)[s] for the least (f ( e )- ex p ansionary stage s such that S, 1&)[s] = ST1&). So the claim follows from (c). Cl By Lemma 9(d), the strategy M/ I e will succeed to meet M, provided that the requirements Rf 1er with e’ c e, act only finitely often. Definition.

A number u is an a-i-configuration

at stage s if

[u(A @BibI; IPLY, 8) < v & MWAe"i(~N4

= ~IPI~"~"~(YM~~

where t is the greatest stage w
+ l]JJ]).

(2.22)

if v is an cu-i-configuration at almost all stages,

Note that for any a-stage s and for any IZG 1~~1,s is an ((u 1n)-i-configuration at stage s. So by imposing a restraint on A G3Bi of length s, K, will try to make such a configuration s permanent. We will show that K, will succeed if cx is on the leftmost path J The construction

In the following let R, R’ E {I, K, N, 0) and S, S’ E {I, K, N, 0, M}. A strategy R, may act at certain stages. Like M, (see (61) above) R, is initialized at a stage s + 1 if E[S] < (Yor some strategy Rb of higher priority acts at stage s + 1 or some marker &(y)[s] with /3 (0) <,_ LYenters A at stage s + 1. We say R, is satisfied at stage s if R, has acted at some stage cs and has not been initialized since. The strategy R, requires attention at stage s + 1 if R, is not satisfied at stage s,

(2.23)

(Yis accessible at stage s + 1,

(2.24)

and, depending on the type of R, one of the following holds: R, = Z,,

Ial = n - e + i (i t, where t is the greatest stage GS at which R, has been initialized, and {e}“(x)[s] = 0. R,=K,,

lal=n*(m,u)+i.

(2.25.1) (2.25. K)

20

K. Ambos-Spies,

R, = N,,

R.A. Shore

Ial = n * (k, I, e) + i (i < n),

and there is a number x E c#~~~‘*~~(~)] such that x > t, where t is the greatest stage GS at which R, has been initialized,

{e>“(x)[sl = 0, Y&)[s] $4~1 and Y&)[sI4 4~1, and there is an o-i-configuration u at stage s such that s 3 v 2 u(A[s]; e, x, s) and y,&)[s] 2 ‘u and y&)[s] 2 V.

(2.25. N)

R, = O,, I(Y~= (i, j, e), i, j < n, i y$,j, and there is a number x E ~t~+~*j*~*(~)l such that x > t, where t is the greatest stage
(2.25.0)

Now the construction

is as follows.

Stage 0. All sets under construction

are empty and all strategies are initialized.

Stage s + 1. The stage consists of 3 steps. Step 1. Let R, be the highest priority

strategy which requires attention. (If there is no such strategy, step 1 is vacuous.) Depending on the type of R, distinguish the following cases. Case 1: R, = Z, and Ial = n - e + i (i < n). Put the least x as in (2.25.1) into Cj and Bj for all j
10. For i, j, k < n, (2.8), (2.9) and (2.10) hold.

Straightforward.

q

of I-types in the r.e. degrees

Undecidability

Lemma

11. For i #j < n, Ci +

Proof. Immediate Lemma

Lemma

B,. q

by Lemma 10.

12. Di,k,lQTA @ W, and D;,k,l +-A

Proof. By construction

@ W,.

(2.13) holds. So the claim follows from Lemma 3.

13. Let i c0 j. Then Bi +

Proof. By construction

21

0

Bi G3 C.

Bi c IJkczn &I.

Moreover,

by (2.9),

and, by (2.10),

since

Lemma

14. For any m 2 0 there is a stage s, such that f ) m < E[S] for all s > s,.

Proof. By Lemma 6(b).

0

Lemma

15. For any m 2 0 there is a stage t,,, 2 s, such that no marker 6,(y)[s] with P (0) -%f ) m enters A after stage t,.

Proof. If /3 (0) CL f 1m then, by choice of s,,

whence, by (62) no &-marker VP VY Vs ([B (0) -5f

no stage s 2 s, is P-expansionary, is put down after stage s,. Hence I m &s asrn & 41(~)]slll

+ Moreover, if SB(y)[s + 1141 then y < r(l/3l)[s]
P(0) %f

~~(Y)]Sl = %(Y%%zl).

s is P-expansionary,

whence

l/31
(2.26) and

1m and WY)]~~IJ>

is finite and, by (2.26), we can let t,,, be the least stage s as, such that all GB(y)[s,,J E M which will ever enter A have entered A before stage s. 0 Lemma 16. For any m 2 0 there is a stage u, 2 t,,, such that no strategy Rf 19, q G 172,requires attention after stage u,.

22

K. Ambos-Spies,

R.A. Shore

Proof. Fix Rf irn and, by inductive hypothesis, choose a stage s’ such that no strategy R; 14 with 4 Cm or ~7= m and R’ < R requires attention after stage s’. Then, by Lemmas 14 and 15, for s” = max(s’, t,), R, lrn will act whenever it requires attention after stage s” and it will never be initialized after this stage. So 0 RfIm will require attention after stage sn at most once. Lemma

17. Requirement

M, is met.

Proof. By Lemmas

14, 15 and 16, Mf I e will not be initialized after stage u,. Moreover, by Lemmas l(ii) and 7(a), a number y E &“] which enters A does so according to rule (63). So the claim follows from Lemma 9(d). Cl

Lemma

18. Ifs

is a-expansionary

then no marker 6,(x)[s]

Proof. For a contradiction

assume that s is a-expansionary l] - A[s]. Then, by (63), 6,(x)[s] = s,(x)[t + 11,

enters A at stage s + 1.

and S,(x)[s] E A[s +

Vi
(2.27)

3
(2.28)

but

(x)[tlJ = {e>AaBfx>b - 111,

where t is the greatest stage XI
By our convention impossible. Cl

(0.3) on redefining

We say that a strategy is permanently stages. Lemma

19.

If

corresponding

oracle computations,

however,

this is

satisfied if it is satisfied at almost all

the strategy Rf 1m is permanently

satisfied then the requirement

to it is met.

Proof. Fix m and assume that Rf 1m acts at stage s + 1 and is not initialized afterwards. Then, by the former assumption, all requirements M, with f ) m C (Y

are initialized at stage s + 1, whence Va Vy Vt ([f 1m c (~8~s 6 t &

6,(y)[tl EA[t + I]- A[t]l + 6,(y)[tl> s)

while, by the latter, VcuVy Vt ([a (0)

-+f ( m&s c t & &(yWlLl 3 Uy)Pl e A).

Hence vu vy

vt ([~,(Y)[s]EA - A[S]& 6,(y)[tl c ~1* (Y(0) C-f I ml.

(2.29)

Undecidabilityof l-types in the r.e. degrees

23

Similarly, since Rf 1m is not initialized after stage s + 1 and since all lower priority requirements are initialized by RY 1m at stage s + 1, only strategies initialized at stage s + 1 will act later. Hence, by definition of requiring attention, they will put only numbers x 3 s + 1 into A and Bi after stage s + 1, whence A~ll[s+l])s+l=A~ll~s+l&B~[~+l])s+l=B,)s+l

fori
(2.30)

(Note that, by definition of y, x s y&)[s] E o[‘] and only y-markers are enumerated into A by the diagonalization requirements.) Now to show that the requirement connected with the strategy Rf (m is met, we to the different types of distinguish the following cases corresponding requirements. Case 1: Rfl,=Z’l,

andm=n-e+i(i
i.e., Rf(,isastrafegyforZi,,.

By (2.251), there is a number x such that {e>“(x)[s] = 0 and, in step 1 of stage s + 1, x enters the set Ci and the sets Bj for i fj
e, x, s).

(2.31)

Since u(A[s]; e, x, s)
Bi[S]IS=B;Is.

and

A”‘[s] ( s = A”’ ( s

and

Bi[S] (S = Bi (S

(2.32)

is immediate by (2.30) since no number enters A[‘] or Bj at stage s + 1. It remains to show that A’“‘[s] ) s = A”” 1s.

For a contradiction assume that this is not the case. Then we can fix t and z minimal (in this order) such that z E A[“l[t + l] - A[($]

and

z < s < t.

BY (2.29),

z = 6,(y)[t]

for some y and (Ywith IX(0) s f ) m.

Let u + 1 be the greatest stage it at which 6,(y) is put down. Note that, by (2.24), f ( m is accessible at stage s + 1, so that, by a (0) sf I m, s is cu-expansionary. Hence, by definition of 6, and since z
z = UY)M = &(y)[sl {]LY]}A@Ei(y)[~]J= {]a]}“@“I(y)[s]

= UY)]~ + II for all j < IZ.

We will show that {(a(}A@Bc (y)[u] = {]~]}“@“(y)[t] stage t + 1 contrary to assumption. First, by (2.33), ~l4>“@“‘(Y)bl~ = ~I4~“~“~(Y>k4

and

so that S,(y)

(2.33) is not lifted at

24

K. Ambos-Spies,

R.A. Shore

Since u(A @ BJs]; Ial, y, s)
wl>A@“t(Y>[4L= wl~A@B’(YM = u4~A~B’(YN~l. Case 2: Rfl,,,= K f 1q.i,u.

&I,,,

and m=rz*

(q, u) +i

(i
i.e.,

Rflm is a strategy

for

By (2.24), f ( m is accessible at stage s + 1 and in step 1 of stage s + 1 no number enters any of the sets under construction. As in Case 1 we can argue that (2.31) holds. Moreover, by the accessibility of f 1m, q, u s m s s and s is an (f 1q)-i-config ura t ion. By (2.31) this configuration is permanent after stage s + 1. Case 3: Rfl,=

Nflm andm=n*(k,l,e)+i(i
i.e., Rfl,isastrategyfor

Ni,k,/,e.

By (2.25.N), such that

there is a number x and an (f 1m)-i-configuration

{e}“(x) = 0

u(A[s]; e, x, s) s v G min(y&)[s],

and

TVs s at stage s y,(x)[s])

and, in step 1 of stage s + 1, x enters Di,k,/ and yk(x)[s] and y&)[s] since u(A[s]; e, x, s) < IJ, it suffices to show that A[s] ) IJ= A ) u

enter A. So,

Bi[s] 1v = Bi ) IJ.

and

Since no number less than u enters A”’ at stage s + 1 and no number enters Bi at this stage, A”][s] 1u = AL1’ 1u

and

Bi[s] 1v = Bi ) U.

(2.34)

So as in Case 1 it suffices to show that A’“‘[s] 1IJ = A”” ( 21.

For a contradiction assume that this is not the case. Then-as fix t and z minimal (in this order) such that z E ALoft + l] - A’“][s]

and

in Case l-we

can

z < u G s G t.

BY (2.29),

z = 6,(y)[t]

for some y and a with a (0) Ef I m.

So if u + 1 is the greatest stage Al at which 6,(y) definition of 6, and since z < s, UGSGf

and

2 = UY)[~=

has been put down, then, by

UY)[SI = UYNUI.

Now, since (Y(0) c f I m and v is an (f I m)-i-configuration

{lal}Ae”fy)[s].l

= {Iaj}“@“~(y)[u]

and

44

(2.35)

at stage s,

@ I$; 14, Y, s) c v.

Undecidability

of l-types in the r.e. degrees

25

So, by (2.34) and minimality of t and z, A @ B;[s] ( u(A @ Bi[s]; Ial, Y, S) =A @ Bi[t] 1u(A @ Bi[s]; l&I, Y, S) whence {14~A@AfY)M~ = wl~A~Bi(Y)bl = {I4>“@“‘(Y)Pl. So, by (63), 6,(y)[t]

will not enter A at stage t + 1 contrary to assumption.

Case4:

and m = (i, j, e ) & i, j < n, i. e., Rf 1m is a strategy for Oi,j.e.

Rfl,,,=Oflm

By (2.25.0), there is a number x such that {e}“@“l@“(x)[s] = 0 and, in step 1 of stage s + 1, x enters Bk for all k
& is met.

assume that Ii,, is not met, i.e., (2.36)

Ci = {e}”

Fix m such that m = n - e + i. Then Z, 1,~is a strategy for Z,,,. Choose s, G t, s u, as in Lemmas 14, 15, and 16. Then Z, lrn is not initialized after stage u,. Moreover, Z, 1m doesn’t require attention after stage u,. So, since the row oli~‘~~1m)l has been reserved for Z, 1m, Cj(x)=O

forallxEWl’,e,(fIm)lsuchthatx~u,.

Hence, by (2.36), for any such x W"Wbl

=0

for all sufficiently large s and x is greater than all stages t at which I,, m is initialized. So, by Lemma 20, Z, 1m will require attention after stage U, contrary Cl to our choice of u,. Lemma 22. Requirement

Oi,i,e is met.

26

K. Ambos-Spies,

R.A.

Shore

Proof. For a contradiction assume that Oi,i,e is not met and let m = (i, j, e) so that O,I m is a strategy for Oi,i,e* Then, as in the proof of Lemma 21 one shows that, for any x E &,e,(f Irn)l such that x 2 u,, {e>“@%@“(x)[s] = 0 for all sufficiently large s and x is greater than all stages t at which Z, jrn is initialized. So, by Lemma 20, O,l m will require attention after stage U, contrary to our choice of u,. 0 Lemma 23. Requirement Kf 1e,i,u is met. Proof. For a contradiction assume that Kf 1e.i,u is not met and fix m such that m = n . (e, u) + i so that Kfl m is a strategy for Kf 1e,i,ue Then, by Lemma 20, Kf I m will require attention infinitely often. This contradicts Lemma 15. 0 Lemma

24. For

any m 2 0 and i < n there is u total (A @ Bi)-recursive

IJ,,~ such that v,,&)

2 x is a permanent

function

(f ) m)-i-configuration.

Proof. By Lemma 23, for any x there is a permanent (f ) m)-i-configuration 1x. Moreover we can find such a configuration V,,i(X) (A @ Bi)-recursively by letting v,,Jx) be the first (f ( m)-i-configuration v 2x existing at some stage s such that ACBB;[s])v=A~B,~v.

0

Lemma 25. Requirement Ni,k,[,e is met. Proof. For a contradiction Wk$TACBB;

assume that Ni,k,l,e is not met. Then

and

W,+TA@Bi

(2.37)

and D i.k.1

=

(2.38)

{e>“.

Let m = II . (k, 1, e) + i so that Nf I,,, is a strategy for Ni,k,,,e. Choose s, St, S u, as in Lemmas 14, 15, 16. Then Nf 1m is not initialized after stage u,. Moreover, Nfirn does not require attention after stage u,. So, since the row w[~,~,‘*~,@ Irn)l has been reserved for Nf I,,,, it follows from (2.38) that vX E WW.kV

1m)l(xz-u, + W= &,dX) = {e>“(x) &v,s(~&)bl, n(xNsl4 A)]).

Now let

F = {x: X e ,$.kJ.e.(f 1m)l ‘Qx 3 u,} and define the function g by g(x) = uj,,(u(A;

e7 x)).

(2.39)

Undecidability of l-types in the r.e. degrees

27

Note that if for some x,

x E F&g(x) =sY:(X), rf(x)

(2.40)

then, by (2.39), (2.25N) will hold for this x at all sufficiently large stages, whence, by Lemma 20, N, I,,, will require attention infinitely often contrary to Lemma 16. So it suffices to show that there is a number x satisfying (2.40). To get such a number x, we first observe that F is recursive and, by Lemma 24, g +A @ B;. So, since, by (2.37), W, $,-A G3B;, it follows from Lemma 4 that there is an infinite A G3B,-recursive subset F’ of F such that for all x E F’ l?(X) < Y/xX).

Since, by (2.37), W, &A G3Bi too, a second application infinite subset F” of F’ such that for all x E F”

of Lemma 4 yields an

i?(X) < Yi%). So, for x E F”, (2.40) holds.

0

Now, conditions (2.0)-(2.6) are satisfied by Lemmas 11, 21, 17, 12, 25, 13, and 22, respectively. This completes the proof of Theorem 1.4.

3. Proof of Theorem 1.7 The proof of Theorem 1.7 is similar to that of Theorem 1.4. So we describe only the necessary changes. As in the proof of Theorem 1.4 we construct r.e. sets A, Bj, Cj, C and Di,k,[ (i
{~}A@BI~ = . . . = {+A@B~-I = {e}A@Bt+I =...=

The requirements

69 A@Bn-I total 3

{e}A@Bo+A

G3 Ci.

fii,e ensure that

b(,fl . . . fl bi-1 II bi+l f~. * * fl b,_l =ci. Obviously, this implies (1.8). The strategy for the meet requirements fii,, is essentially the same as the one for the requirements M,. As there we define a set of correction markers. Markers for pi,,, however, will enter Ci not A. To make sure that there will be no collission with the strategy for the diagonalization requirements Ij,,, the set of possible attackers of the latter requirements is redefined to F(J,,) = ,j$.‘,er

(3.0)

K. Ambos-Spies,

28

R.A. Shore

(cf. (2.7)) and corrections markers for hi,= are taken from ol’*“l. Moreover, any marker which will enter Ci will also enter Bi for j # i at the same stage. So (2.8)-(2.10) will still hold. As for the previous meet requirements we have to guess whether a requirement Eu^l,,, is finitary or infinitary. So the even levels of the priority tree will be assigned to the requirements M, and the odd levels to the requirements hi,=. To be more precise, for a: with JayI= 2e, M, is a strategy for M, and, for (Y with 1~~1 = 2(n . e + i) + 1, &!l= is a strategy for &ii,,. (The strategies for the other requirements are assigned as before.) Accordingly, we have to modify the definitions of a-stages, the rules for the correction markers and the definition of a-i-configurations: In the definition of a-stages we say that a stage s is a-expansionary if ]a] = 2e

I(e)[s] > max{l(e)[t]: t
and

or if ]LY]= 2(n . e + i) + 1

and

T(i, e)[s] > max{P(i, e)[t]: t
where i(i, e)[s] = max{x: Vy A@Bg+l(y)

=.

. .=

{e)A@~n-l(y)J)).

Similarly, in the definition of the correction markers 6, for M,, 1al = e has to be replaced by ILY(= 2e and, in (al), M, will also be initialized if a marker &(y )[s] for a strategy I$ with /3 (0) cL a enters some Ci at stage s + 1. The rules for the correction markers 8i.e for Mi,, and 6, for fiW, )a] = 2(n *e + i) + 1, are essentially the same as the ones for the 6, and 6, markers. Only if 8,(x) is put down at stage s + 1 then we let S,(x)[s + l] = (i, 0, (a), x, s + 1) to ensure that 8,(x)[s + l] E c#~o1 (see above). Furthermore, the correction step has to be adapted to the changed hypothesis of the requirements: In the definition of 8;,, condition (2.19) has to be replaced by {e}“@“+)[t]

= {e}“@“i(x)[s]

for at least one j # i, j
Similarly, in the definition of 8, in case (63) we have to replace ( * ) by (*)

Vj E (0, . . . , i - 1, i + 1, . . . , II - l} ({e}A@B$x)[t]

Moreover, in either case, if necessary the correction

# {e}“@+(x)[s]).

marker is put into Ci not into

A.

For a requirement Vy (&,,(y)[s]L

< v 3

hi,, and for any j # i (j < n) we call a number ZJsatisfying [Q

63 Bj[s];

e, Y, s1-c IJ&

{e>“@%)[sl= {e)A@Bi(y)[tl~,

where t + 1 is the last stage ss at which 6i,e(y) has been put down])

Undecidability of l-types in the r.e. degrees

29

an &e-j-configuration at stage s. Note that, for such a configuration v, &, will not put a marker less than II into Ci after stage s unless some other requirement forces a number less than u into A 63Bj after stage s first. (Since, in contrast to the requirement M,, in the hypothesis of Z&, the function {e}A@Bi does not occur, here &e-i-configurations do not make sense, i.e., by restraining A and Bj we cannot prevent markers 8;,, from entering Ci.) To take into account the configurations for the &Zi,, requirements, in the definition of a-i-configurations, (2.22) has to be replaced by VVY ((P (0) G~&IPI=~~&&(Y)]~IJ<~ [u(A @ K[s]; e, y, s) < u &

3

{e>“@“~(y)[~l = {e>“@“‘(y)[tl~,

where t is the greatest stage w
+ l]J,J,])

and(p(0)c~&]PJ=2(n~e+j)+1&jfi&8P(y)[S]~
3

[u(A @ Bi[s]; c, y, s) < u & {e>“@“‘(y)[s] = {e]A@B(y)[tl~, where t is the greatest stage w
+ l]JJ,])).

(3.1)

As in the proof of Theorem 1.4, the above described marker systems will ensure that the meet requirements are met. It remains to show that the diagonalization requirements will succeed as well. In particular, we have to show that the restraints required by these requirements to protect certain computations will not be injured by the correction markers for the new & strategies. Since for the I, K and N requirements the arguments are very similar, we will consider only the cases of I and 0 requirements. First, fix a requirement Zi,,. Let Z, be the strategy for Zi,, on the true path, let t be a stage after which Z, is not initialized, and assume that Z, acts at stage s + 1 > t. Then, for some number X, {e}“(x)[s] = 0 and x is put into Ci (and all Bj with i #Z, ZA(x), after stage s. We will show that, in fact, no number less than s + 1 will enter A or Biafter stage s. By initialization and by our choice of t, neither a lower priority strategy nor a finitary higher priority strategy or a strategy to the left of the true path, respectively, will cause a number
K. Ambos-Spies, R.A. Shore

30

which 0, is not initialized, and assume that 0, acts at stage s + 1 > t. Then, for some number X, {e} A~B~~c(~)[s] = 0 an d x is put into Bi (and all Bk such that is0 k) at stage s + 1. Moreover, (Yis accessible at stage s + 1, i.e., s + 1 is an a-p-configuration for all p we can argue that besides 0, itself only infinitary strategies below cx might put numbers less than s + 1 into any set under construction. To show that this will not happen, we have to use the fact that the partial ordering co has at least three minimal elements. This allows us to fix a number q < n such that i PO q and j # q. We now will argue that for any infinitary strategy below cry,s + 1 is a configuration (for an appropriate BP) which is not injured by 0, at s + 1 and hence permanent. So fix /? with p (0) c_ LX If I/I] is even, say IpI = 2e’ then s + 1 is a permanent a-j-configuration, and hence a permanent p (0)-j-configuration for Mfi since, by assumption, i +,, j and since 0, puts x only into the sets Bk such that i cCJk. If [/II is odd, say IpI = 2( n . e’ + i’) + 1, then we distinguish two cases: If i’ #j then again s + 1 is a permanent /3 (0)-j-configuration for 6$. If i’ = j then a /3 (0)-j-configuration is of no use for freezing fif3 markers. Since q #j we can use the /I (0)-q-configuration s + 1, however, and since i +o q, 0, does not put a number into A or B, at stage s + 1. Hence this configuration is permanent.

4. Proof of Theorem

1.9

Here we show how the proof of Theorem 1.4 can be combined with the permitting method to obtain the first part of Theorem 1.9. We refer to the proof in Section 2 and give only the necessary changes. A proof of the second part of the theorem may be obtained in a similar way from the proof of Theorem 1.7. Fix an r.e. set E E e and let {E[s]: s 3 0} be a recursive enumeration of E. We will allow a diagonalization requirement to put a number x into any of the sets under construction only if x is permitted by E, i.e., if a smaller number enters E at the same stage. This gives E control over the action of the diagonalization requirements. As we will show this will also allow E to recognize whether a marker of a meet requirement will enter a set, so that the sets A, B;, Ci, C and Di,k,, will be recursive in E. In the previous construction, the action of an R, strategy (R E {I, N, 0)) was limited to a-stages s + 1. This ensured that s + 1 was an a-configuration which allowed us to protect the computations over which we wanted to diagonalize against the correction markers of higher priority meet requirements. Now, since we require E to permit, we have to allow R, to act also at stages s + 1 where cxis not accessible. As we have seen in case of the N, strategies already, this is possible as long as at stage s + 1 there is a sufficiently large configuration for protecting the corresponding computation. So we may overcome the difficulties arising from E-permitting by using

Undecidability of l-types in the r.e. degrees

31

followers.

A follower x for a diagonalization strategy R, is a potential diagonalization witness. Once a follower is appointed we wait till the computation we want to diagonalize against shows up and can be protected by some configuration. Then R, initializes all lower priority strategies to preserve the computation by making the current configuration permanent. We say the follower x becomes realized. (This negative action does not require E-permitting, so that realization can be restricted to the a-stages.) Then, at any later stage where x is permitted by E we complete the diagonalization action and call x successful. Since x might never be permitted, in general we need more than one follower for a single strategy. So whenever all current followers are realized but none of them is successful, we appoint a new follower. Since, for a strategy on the true path, the sequence of permanent followers will be recursive, by the nonrecursiveness of E, we can argue that eventually some follower will be permitted. We now formally describe the necessary changes in the construction of Section 2. These changes affect the definition of requiring attention and step 1 in the description of stage s + 1 of the construction. A K strategy requires attention as before. For R E {I, N, 0} we say R, is satisfied at stage s if it possesses a successful follower at the end of stage s. The strategy R, requires attention at stage s + 1 if R, is not satisfied at stage s

(4.0)

and one of the following three cases applies. Case 1: There is a realized follower x of R, at the end of stage s such that E[s] 1x # E[s + l] 1n. Case 2: There is an unrealized

follower x of R, at the end of stage s, a is accessible at stage s + 1, and, depending on the type of R, one of the following holds: If R, = Z,, (LYJ= n - e + i (i < n) then {e}“(x)[s] = 0. If R, = N,,

JaJ = rz * (k, I, e) + i

(i
(4.1.1)

then {e}“(x)[s] = 0 and

there is an cu-i-configuration v at stage s such that s 2 v 2 u(A[s]; e, x, s) and yk(x)[s] 1 u and n(x)[s] 2 U. If R, = O,,

/a( = (i,j,

e),

i,j
Case 3: There is no unrealized accessible at stage s + 1.

(4.1.N)

i=&)j then {e}“@‘“~@“(x)[s]=O.

(4.1.0)

follower of R, at the end of stage s and a is

In step 1 of stage s + 1 we let R, be the highest priority strategy attention. (If there is no such strategy, step 1 is vacuous.) We stage s + 1. If R, = K, then do nothing. Otherwise, depending on and depending on the case in the definition of requiring attention, following cases.

which requires say R, acts at the type of R, distinguish the

32

K. Ambos-Spies,

R.A. Shore

Case 1: R, requires attention via Case 1. For the least x as in Case 1, say x is successful and

put x into Ci and Bj for all j < n with i #j ifR,=Z,,

Ial=n-e+i(i
put x into Qk,[ and yk(x)[s] and y,(x)[s] into A if R, = N,,

Ial=n-(k,l,e)+i(i
put x into Bk for all k < IZwith i co k if R, = O,,

I4 = (i, i, e> (6 j
Case 2: R, does not require attention via Case 1 but via Case 2. For the least x as in Case 2 say x is realized. Case 3: Otherwise. Appoint the least number x > s such that

x E ,[i,e.(~)l

ifR,=Z,,

(al=n-e+i(i
x E ~~~~~~~~~~~~~~ if R, = N,, I aI = n . (k, 1, e) + i (i
as an R, follower. If R, (R E {Z, N, 0)) is initialized then all its current followers are cancelled. An R, follower which is never cancelled is called permanent. To show that the construction is correct we first note that Lemmas lo-15 hold by the previous proofs. For a proof of Lemma 16 we proceed by induction. Fix R, Jo. Then, by inductive hypothesis and by Lemmas 14 and 15 there is a stage s” after which Rf jrn will act whenever it requires attention and after which Rf I m will not be initialized. Now, for a contradiction assume that Rf I m requires attention infinitely often. Then R E {I, N, 0} and infinitely many Rf I m followers are appointed after stage s”, all permanent and realized but none successful. Let {x,: s 3 0} be an effective enumeration of these followers in order of appointment and let t, be the stage at which x, becomes realized. Since no x, becomes successful, we can argue that E 1x, = E[tJ ) xs, whence E is recursive contrary to assumption. Lemmas 17 and 18 and Lemma 19 for R = K follow as before. For R E {I, N, 0} a straightforward modification of the proof of Lemma 19 shows that if Rf 1m has a permanent successful follower than it is met. Then, to show that the R requirements are met, for R = K we proceed as before. For a requirement R of type I, N or 0, as in the proofs of Lemmas 21-22 and 24-25, we argue that if R is not met then the strategy Rf 1m for R on the true path will require attention infinitely often contrary to Lemma 16. To complete the proof it remains to show that the constructed sets A, B;, Ci, and Di,k,, are recursive in E. For the sets Bi, Ci and Di,k,[ this is immediate by permitting since these sets contain only followers. To show that A dT Ewe need the

Undecidability of l-types in the r.e. degrees

33

following definition. A stage s is called pure if no number enters any set under construction at stage s and no follower less than s enters any set after stage s. Lemma 26. (i) There are infinitely many pure stages.

(ii) {s: s pure} c,. E. (iii) Assume that 6,(x)[s

+ 111 $ A[s + 11 and s + 1 is pure.

Then 6,(x)[s

+ 11

does not enter A after stage s + 1.

(iv) A So E. Proof. For a proof of (i) fix t. We will show that there is a pure stage s + 1 > t. Fix m such that & 1m is not satisfied at stage t. By construction, any such strategy will be permanently satisfied. So fix s + 1 > t such that K, I ,,, acts at stage s + 1 and is not initialized afterwards. We claim that s + 1 is pure. Since Kr I m initializes all lower priority strategies, no such strategy puts a number x c s + 1 into a set after stage s. So, since Kf 1m is purely negative, it suffices to consider the higher priority strategies. Since Z$I m is not initialized after stage s, no diagonalization strategy of higher priority becomes active and no marker of a meet strategy MS with /3 (0) <,_f 1m is moved after stage s. So, by the former, no follower x
5. Fragments

of the theory

In this final section we address the question of which fragments of the theory of the r.e. degrees are undecidable. Let 2,, (Hn) be the set of sentences in prenex normal form where the block of quantifier starts with an existential (universal) quantifier and contains at most n - 1 alternations of quantifiers. For a structure 9’ over a language L the fl,-theory of 9’ is defined by n,-Th(Y)

= {C#J E X, n L: Yi= C/I}.

K. Ambos-Spies,

34

R.A. Shore

We will show that the results of Section 1 imply that the fls-theory of .S = (R, S) over the language L(S) of partial orderings is undecidable. For this we need the following theorem of Ershov and Taitslin. 5.1. Theorem ([3]). The set of all & sentences which are true in every finite graph is strongly undecidable.

By coding finite graphs into finite partial orderings corollary. 5.2. Corollary. The set of all & sentences ordering is strongly undecidable.

we obtain the following

which are true in every finite partial

Proof. Let G = (V, E) be a finite graph. We code G by a finite partial ordering 9 = (P, S) as follows. First we represent any vertex v E V by the middle element X, of a chain a,
are incomparable with the elements of the chain for v’. In addition, for any edge (v, v’) E E we add some element c,,,, such that c,,,, GX,, x,, and c,,,, is incomparable with all other elements of PP. In a dual fashion, for any pair v # v’ such that (v, v’) $ E we add some element d,,,. such that x,, x,, s d,,,. and d,,,. is incomparable with all other elements of 9’. Conversely, any partial ordering 9 of height ~3 with the property that any pair of incomparable elements X, x’, each of which possesses both a predecessor and a successor, have either a common lower bound or a common upper bound but not both code a graph in the above sense. Formally, this characteristic property of a partial ordering can be expressed by the sentence ?jJ=Ylxi,x2,

xg,

x4(x1
&b,~2([~1f.c&~al,

a2, 3

[3c(c
h,

bZ(al
3d(xl
For a partial ordering satisfying rj~we can extract the elements representing vertices of the coded graph by the formula 6=3a,

the

b(a
and the edge relation can be expressed in the following two ways: &~=x#y&3c(c
Note that $J is equivalent to a n2 sentence, 6 is a 2i formula and .sz and srr are equivalent to a _Zi formula and a 17i formula, respectively. By this coding, any sentence + in the language L(e) of graphs can be converted

Undecidability of l-types in the r.e. degrees

35

to the sentence t+!~+@’ in the language L(s), where GT is obtained from # by relativizing quantifiers to 6 and by replacing positive occurrences of the edge relation by .sh and negative ones by E= Then, as one can easily check, for a & sentence # the corresponding sentence r+~ + #T is equivalent to a & sentence, say r$*. Moreover, + is true in every finite graph iff @* is true in every finite partial ordering. Now let TFG and TFPO be the sets of sentences true in all finite graphs and all finite partial orderings, respectively, and for a contradiction assume that the corollary fails. Then there is a recursive set R such that Valid,,,,

fl & 5 R E TFPOn Z;.

But then, for the recursive set RPT = {C/IE L(e): @* E R}, Valid,,,, II & c ReT c TFGfl& contrary to Theorem 5.1.

q

Call a formula Y in the language L(G) with k + 1 free variables x0, . . . , xk_r, y a coding formula for a partial ordering Y = (S, cs) if, for any finite partial ordering 9 = (P, sP), there are elements a,, . . , , uk_] of S such that the partial ordering W ES: y!= Yx,,,...,+,,yt%

. . . > ak-1,

bl), 5)

is isomorphic to 9’ = (P, sP). 5.3. Lemma.

Let 9 = (S, Cs ) be a partial ordering and let y be a E,,, formula such that y is a coding formula for 9. Then II,,,+,-Th(Y) is undecidable. Proof. For any sentence @ E Z;, say $I = 3u0, . . . , u, Vq), . . , , v,, $I, where IJIis quantifier free, let 4’ be the sentence

@+ = Vxg, . . . , xk_1 ihi,, . . . , u, (Y&d”]&. ’ * & Y,,[Um] &Vu,,..., and let R = {$ E &: Yk $‘}. Valid,,,,

VP([YJVOI & . * * & YJ%ll+

VII))

Then

17E2 c R

since, if # E Z; is valid, it is true in any partial ordering W ES: YkYx ,,,.__, x*-,.&o,

. . . , ak-1,

bl),

%>,

whence Y L #‘. Moreover, RG Gron-?, since y is a coding formula for 9. So, by Corollary 5.2, R is not recursive. This q implies the claim since, for #J E .X2, @’ is equivalent to a II~,, sentence.

K. Ambos-Spies,

36

5.4. Theorem.

The &theory

R.A. Shore

of 92 = (R, G) is undecidable.

Proof. By Lemma 5.3, it suffices to show that there is a 2, coding formula

y for 9. Now, by Theorem 1.4, the formula y with free variables xn, x1 and y expressing that there is a maximal-x,,-cappable degree z such that y is the join of x1 and z is a coding formula for 92. Formally, y can be defined by y’32([x”~z&3v(x,,
&vu([x,~u&3v(x,,
&[x,, z cy&Vs

(x,, z ==s+y

=ss)])

which, as one can easily check, is equivalent to a & formula.

Cl

Theorem 5.4 is not optimal. The coding formula used by Harrington and Slaman (see Soare [13, Chapter XVI.31) is &, so that, by Lemma 5.3., &-Th(%!) is already undecidable. On the other hand, Sacks [13] has shown that every finite partial ordering is embeddable in 22 whence n,-Th(9?) is decidable. Whether &Th( 52) and &Th( 92) are decidable or not is not known.

References

[l] K. Ambos-Spies and D. Ding, Discontinuity of cappings in the recursively enumerable degrees and strongly nonbranching degrees (submitted). [2] K. Ambos-Spies and R.I. Soare, The recursively enumerable degrees have infinitely many one-types, Ann. Pure Appl. Logic 44 (1989) l-23. [3] Y. Ershov and M.A. Taitslin, The undecidability of certain theories, Algebra i Logika 2 (1963) 37-41 (in Russian). [4] P.A. Fejer, Branching degrees above low degrees, Trans. Amer. Math. Sot. 273 (1982) 157-180. [S] P.A. Fejer, The density of the nonbranching degrees, Ann. Pure Appl. Logic 24 (1983) 113-130. [6] R.M. Friedberg, Two recursively enumerable sets of incomparable degrees of unsolvability, Proc. Natl. Acad. Sci. USA 43 (1957) 236-238. [7] R.M. Friedberg, The fine structure of degrees of unsolvability of recursively enumerable sets, in: Summaries of Cornell University Summer Institute for Symbolic Logic, Communications Research Division, Inst. for Def. Anal., Princeton, NJ, 1957, 404-406. [S] L. Harrington and S. Shelah, The undecidability of the recursively enumerable degrees (research announcement), Bull. Amer. Math. Sot. (N.S.) 6 (1982) 79-80. [9] A.H. Lachlan, Lower bounds for pairs of recursively enumerable degrees, Proc. London Math. Sot. 16 (1966) 537-569. [lo] M. Lerman, R.A. Shore and R.I. Soare, The elementary theory of the recursively enumerable degrees is not X,-categorical, Adv. in Math. 53 (1984) 301-320. [ll] A.A. Muchnik, On the unsolvability of the problem of reducibility in the theory of algorithms, Dokl. Akad. Nauk SSSR (N.S.) 108 (1956) 194-197 (in Russian). [12] G.E. Sacks, The recursively enumerable degrees are dense, Ann. of Math. 80(2) (1964) 300-312. Ann. of Math. Studies 55 (Princeton [13] G.E. Sacks, Degrees of Unsolvability, revised edition, Univ. Press, Princeton, NJ, 1966).

Undecidability of l-types in the r.e. degrees [14] J.R. Shoenfield, Application of model theory to degrees of unsolvability, in: J.W. Addison eds., Symposium on the Theory of Models (North-Holland, Amsterdam, 1965) 359-363. [15] R.A. Shore, Finitely generated codings and the r.e. degrees in a degree d, Proc. Amer. Sot. 84 (1982) 256-263. [16] T.A. Slaman, The density of infima in the r.e. degrees, Ann. Pure Appl. Logic 52 155-179. [17] R.I. Soare, Recursively Enumerable Sets and Degrees (Springer, Berlin, 1987). [18] C.E.M. Yates, A minimal pair of recursively enumerable degrees, J. Symbolic Logic 31 159-168.

37 et al., Math. (1991)

(1966)