THE NON-ISOLATING DEGREES ARE UPWARDS DENSE IN THE COMPUTABLY ENUMERABLE DEGREES
S. BARRY COOPER, MATTHEW C. SALTS, AND GUOHUA WU
ABSTRACT. The existence of isolated degrees was proved by Cooper and Yi in 1995 in [6], where a d.c.e. degreedis isolated by a c.e. degree aifa<dis the greatest c.e. degree belowd. A computably enumerable degreecis non-isolating if no d.c.e. degree abovecis isolated byc. Ob-viously,0is a non-isolating degree. Cooper and Yi asked in [6] whether there is a nonzero non-isolating degree. Arslanov et al. showed in [3] that nonzero non-isolating degrees exist and that these degrees are down-wards dense in the c.e. degrees and can also occur in every jump class. In [10], Salts proved that there is an interval of computably enumerable degrees, each of which isolates a d.c.e. degree. Recently, Cenzer et al. [4] proved that such intervals are dense in the computably enumerable degrees, and hence the non-isolating degrees are nowhere dense in the computably enumerable degrees. In this paper, using a different type of construction to that of [3], we prove that the non-isolating degrees are upwards dense in the computably enumerable degrees. In the context of [4], this is the best possible such result.
1. INTRODUCTION
The existence of isolated degrees was proved by Cooper and Yi in 1995 in [6], where a d.c.e. degreedis isolated by a c.e. degreeaifa<dis the greatest c.e. degree belowd. Ding and Qian [7], LaForte [9] independently, proved that the isolated degrees, and hence the isolating degrees, are dense in the computably enumerable degrees. In [3], Arslanov, Lempp and Shore proved that the non-isolated degrees are also dense in the computably enu-merable degrees. In [12], Wu use the isolation phenomenon an alternative proof of Downey’s diamond embedding theorem. Ishmukhametov and Wu [11] proved that sometimes, the isolated degrees can be far from the corre-sponding isolating degrees. That is, there is a high d.c.e. degree isolated
This research was partially supported by a London Mathematical Society collaborative small grant. The first author was supported by EPSRC grant No. GR /S28730/01, and by the NSFC Grand International Joint Project, No. 60310213,New Directions in the Theory and Applications of Models of Computation, the second author by an EPSRC Research Studentship and the EU Human Capital and Mobility NetworkComplexity, Logic and Re-cursion Theory(COLORET), and the third author is partially supported by a research grant No. RG58/06 from Nanyang Technological University.
by a low c.e. degree. This too has had an interesting application (see [1]), related to Post’s problem for the d.c.e. degrees.
In this paper, we are mainly concerned with the non-isolating degrees, where a computably enumerable degree cis non-isolating if no d.c.e. de-gree abovecis isolated byc. Obviously,0is a non-isolating degree. Cooper and Yi asked in [6] whether there is a nonzero non-isolating degree. Ar-slanov et al. showed in [3] that nonzero non-isolating degrees exist and that these degrees are downwards dense in the c.e. degrees and can also occur in every jump class. Arslanov et al. actually proved a stronger result. They first pointed out that for any c.e. degree c and d.c.e. degreed > c, there is a degree a c.e. in c such that c < a < d, and then proved that there is a c.e. degree failing to isolate any 2-CEA degree in it. In [10], Salts proved that there is an interval of computably enumerable degrees, each of which isolates a d.c.e. degree. Recently, Cenzer et al. [4] proved that such intervals are dense in the computably enumerable degrees, and hence the non-isolating degrees are nowhere dense in the computably enumerable de-grees. In this paper, we prove that the non-isolating degrees are upwards dense in the computably enumerable degrees, so completing a near com-prehensive characterisation of the situation.
Theorem 1.1. For any incomplete c.e. degreea<00, there is an incomplete non-isolating degreecabovea.
Our construction of the non-isolating degrees is direct, and is different from the one given by Arslanov et al. in [3]. In section 2, we show how to construct a non-isolating degree. In section 3, we describe how to combine our construction with the upwards density to prove Theorem 1.1.
2. CONSTRUCTING A NON-ISOLATING DEGREE
In this section, we present a new construction of non-isolating degrees. We will construct c.e. setsA,Csatisfying the following requirements:
Pe: A6= Φe;
Qe: C 6= ΦAe;
Re: Dee= ΦAe ⇒ ∃Be ≤T De⊕A(Be 6≤T A) ∨ De ≤T A;
where{(De,Φe) : d ∈ ω}is an effective list of pairs(D,Φ), whereDis a
d.c.e. set andΦis a partial computable functional. Here,De is the Lachlan
set ofD, with respect to an effective (d.c.e.) approximation{Ds :s ∈ ω}.
That is:
e
Obviously,De ≤T Dand it is a c.e. set. From the approximation{Ds :s ∈ ω}, we have the following effective enumeration ofDe:hx, siis enumerated
intoDe at stagetiftis the least stage suchx6∈Dt.
The strategy for satisfying the P and Q requirements is the standard Friedberg-Muchnik one, and we assume the readers are familiar with it. TheR-requirements are non-isolating requirements. That is, for a d.c.e. set D, if Dis not reducible to A, then we want to find a c.e. set reducible to A⊕D, but not reducible toA, so thatAdoes not isolateA⊕D. This c.e. set mayeitherbe the natural candidateDe,orsome other,B, which we will
need to construct. On the other hand, ifDitself is reducible toA (where, of course,De is reducible toA), then we will need to show this fact.
To satisfyRe, we need to construct a c.e. set Be, and a p.c. functional Γe for which Be = ΓAe⊕D. At the same time, we also want to ensure that
Be 6≤T A, provided that D is not reducible to A. That is, the following
requirements should be also satisfied:
Se,i: Be6= ΦiAor there is a p.c. functional∆e,isuch thatD= ∆Ae,i.
AnRe strategy definesΓe at sufficiently large, that is “big”,
expansion-ary stages. Here we say that an expansionexpansion-ary stage is bigif the length of agreement between De and ΦAe is bigger than any number specified by a
substrategyS. Obviously, if there are infinitely many expansionary stages, there are also infinitely many big expansionary stages. AnRe strategy has
two outcomes:f for finitely many expansionary stages, and∞for infinitely many expansionary stages, with∞ <L f. Below outcome∞, we will list
substrategies Se,i, i ∈ ω, which will work together to constructBe, and to
satisfy requirementRe.
An Se,i-module consists of (infinitely many) steps, where each step n
tries to find a number xn such that either ΦAi (xn) 6= Be(n) or ∆Ae,i(n) is
defined. Stepnworks as follows: (1) Choosexnas a big number.
(2) Wait forΦA
i (xn)↓= 0.
(3) ForΦA
i (xn)↓= 0at stages–
We define∆A
e,i(n) = D(n)with use δe,i(n) =ϕi(xn). Here, again, when we seeΦA
i (xn)[s] ↓= 0, we do not put restraint onA to pre-serve this computation. Whenever A changes below δe,i(n), go
back to (2), in which case,∆A
e,i(n)is undefined by thisA-change.
[Here,ncan be inDsor not in inDs.
If n is currently not in Ds, then n can enterD later, and leave at a further stage. When n enters D, at stages0, we will use the as-sumption that De is equal to ΦAe, as at stage s0, hn, si is not in De,
and we want to restrainAfrom changing to preserve this computa-tion. If laternleavesD, thenhn, siwill enterDe, makingDeandΦAe
disagree athn, si.
If n is in Ds, we do nothing here, since if n leaves D later, this change will enable us to undefine ΓA⊕D(x
n), and allow us to put
xj,n intoBe. Note that axioms enumerated intoΓe before nenters
Dare all invalidated by theA-changes. Otherwise we will be in the situation described in the last paragraph, wherenleavingDcauses a disagreement betweenDe andΦAe.]
Wait forD(n)to change, and simultaneously start the stepn+ 1. (4) SayD(n)changes at staget > s. There are two possibilities.
(a) nentersDat staget. In this case, theD(n)change undefines
ΓAe⊕D(xn), and instead of putting xn into Be immediately at
this stage, we wait forDe andΦAe to agree on (all numbers≤) hn, ti.
[We delay the enumeration ofxnintoBemainly becausenmay leaveD later, making ΓA⊕D
e correct, forcing the enumeration of a number intoA to change ΓAe⊕D(xn), resulting in no real progress.]
(b) nleavesDat stage t. Thenhn, sientersDe at this stage. Since
the computation ΦA
e(hn, si) = 0 is preserved at stage s
0 by restrainingA, and we get a global win via
ΦAe(hn, si) = 0 = 1 =6 De(hn, si).
(5) At stage t0 > t, De and ΦAe agree on (all numbers ≤) hn, ti. We
enumeratexnintoBe, and put restraint onAto preserve both
com-putationsΦA
i (xn) = 0, andΦAe(hn, ti) = 0.
[Here, at staget,nentersD, which undefinesΓA⊕D
e (xn). From now on, we wait for the agreement between De andΦAe to exceedhn, ti,
and during this period, we do not defineΓA⊕D
e (xn). Thus, at stage
t0, we can enumeratexnintoBedirectly, asΓAe⊕D(xn)is undefined at this stage. We will see that in the whole construction, only P
strategies put numbers intoA. This explains why0is non-isolating and why the nonzero non-isolating degrees are downwards dense in the c.e. degrees. We will see in the next section that it will not be like this when we prove the upwards density, where a change of K
(6) At staget00> t0,nleavesD. Then as in 4(b),hn, tientersDe at this
stage. As the computation ΦA
e(hn, ti) = 0 is preserved at staget
0, by keeping this restraint onA, we get a global win via
ΦAe(hn, ti) = 0 = 1 =6 De(hn, ti).
We now consider the outcomes of Se,i, which can run finitely many or
infinitely many steps. Notice that if stepn2is in progress, and now we have
anA-change so that we come back to 2 of stepn1 < n2, then ∆Ae,i(n2)is
undefined automatically by thisA-change. After this, we need to start step n2 from 1. That is, we choose a newxn2.
If some step n passes 3, then we win either by 4(b) or 6, which is a global win viaΦA
e 6=De, where this disagreement is preserved forever. This
means that there will be no moreRe-expansionary stages, or we win by 5,
where we get aD-change atn, where thisnremains inD, ensuring that the previous axioms enumeratingxnintoΓA,D are invalid forever.
If no step passes 3, then notice that we put no restraint onA. It can happen that there is a leastnsuch thatΦAe(xn)does not converge to 0, or for each n,ΦAe(xn)converges to 0.Se,iis satisfied in both cases: In the former case,
Be(xn) = 0 6= ΦAe(xn); and in the latter case, each step remains at 3, and
hence∆Ae,i(n)is defined and equal toD(n). So∆Ae,i =D.
So thisSe,i module has two outcomes: w and s withw <L s. Here w
denotes the case in which no step passes 3, andsthe case in which some step passes 3. In the latter case, we put restraint onAto preserve computations.
3. UPWARDS DENSITY
We are now ready to present the strategy for constructing an incomplete non-isolating degree above any incomplete c.e. degree.
Fix U as an incomplete c.e. set. We will construct c.e. sets A and C satisfying the following requirements:
Pe: C 6= ΦA,Ue ;
Re: Dee= ΦA,Ue ⇒
(∃Be ≤T A⊕De⊕U)(Be 6≤T A⊕U) ∨ De≤T A⊕U;
where{(De,Φe) : d ∈ ω} is a standard list of pairs (D,Φ), whereD is a
d.c.e. set andΦis a partial computable functional, andDeis the Lachlan set
of D. Here we did not require that Ais not reducible to U, as can obtain this by applying Sacks’ density theorem first.
We apply the Sacks preservation strategy to satisfy the P requirements by running (infinitely many) cycles to threaten the assumption that U is incomplete via a p.c. functionalΘ. Cyclenbehaves as follows:
(1) Choose a big numberxn.
(2) Wait forΦA,U
e (xn)↓= 0.
(3) Say ΦA,U
e (xn) converges to 0 at stage s. Define ΘU(n) = Ks(n)
with useθ(n) =ϕ(xn). RestrainAfrom changing belowϕ(xn).
Wait for a change ofK(n)or a change ofU belowϕ(n), and simul-taneously start the next cycle.
(4) Say U changes below ϕ(xn) first. Then go back to 2. Note that
ΘU(n0)
, eachn0 ≥n, is undefined by thisU change.
(5) Say K(n) changes first. Then we put xn into C, and wait for a
change ofU belowϕ(xn).
[If there is no such aU-change, thenΘU(n)differs fromK(n). But as U below ϕ(xn) is fixed, we satisfy P because ΦA,Ue (xn) ↓= 0 is preserved forever, and C(xn) = 1. Otherwise, as above, a U -change undefines ΘU(n0), each n0 ≥ n, allowing us to redefine
ΘU(n) =K(n) = 1.]
(6) UndefineΘU(n0)forn0 ≥n, and redefineΘU(n) =K(n) = 1with
use 0. Start the next cycle.
TheP module has the following outcomes:
hn, wi: cyclenstops at 2. P is satisfied since ΦA,U
e (xn)does not converge
to 0 andC(xn) = 0.
hn, si: cyclen stops at 2. P is satisfied sinceΦA,Ue (xn)converges to 0 and
C(xn) = 1.
hn, ui: cyclenruns through the loop 2-3-4-2 infinitely often.P is satisfied sinceΦA,Ue (xn)does not converge at all, andC(xn) = 0.
These outcomes are ordered:
h0, ui<Lh0, wi<L h0, si<L h1, ui<Lh1, wi<Lh1, si<L h2, ui
<Lh2, wi<Lh2, si<L · · ·<L hn, ui<Lhn, wi<Lhn, si<L· · · .
AsU is incomplete, there is a leastn such that one ofhn, ui,hn, wi,hn, si
is the true outcome forP, since otherwise,ΘA will be totally defined, and
will computeKcorrectly, which is impossible.
We now consider how to satisfy the R-requirements, with U included. Again, for any d.c.e. setD, ifDis not reducible toA⊕U, we need to find a c.e. set reducible toA⊕D⊕U, but not reducible toA⊕U, so thatA⊕U does not isolateA⊕D⊕U. Again, this c.e. set can be De, or another set B which we will construct. On the other hand, if D itself is reducible to A⊕U, then we need to construct a p.c. functional∆to reduceDtoA⊕U. The following are the details.
We will construct a c.e. set Be and a p.c. functionalΓe such that Be = ΓA⊕D⊕U
e . At the same time, ifD is not reducible to A ⊕U, we need to
ensure thatBe 6≤T A⊕U. That is, the following requirements should also
be satisfied:
Se,i: Be6= ΦAi⊕U or there is a p.c. functional∆e,isuch thatD= ∆Ae,i⊕U.
As in section 2, aRestrategy definesΓe at big expansionary stages, and
has two outcomes: f for finitely many expansionary stages, and ∞ for infinitely many expansionary stages, with ∞ <L f. Below outcome ∞,
we will list substrategiesSe,i,i∈ω.
An Se,i-module consists of infinitely many cycles and each cycle
con-sists of infinitely many steps. All the cycles are devoted to defining a p.c. functional Θe,i, and each cycle j tries to find some some x such
that Be(x) 6= ΦiA⊕U(x), or to define a p.c. functional ∆e,i,j such that
D = ∆Ae,i,j⊕U, or to define Θe,iA⊕U(j) = K(j). This task will be realized by cyclej’s steps, where as in section 2, each stepnof cyclej, denoted by
hj, ni, tries to find a numberxj,n such that either ΦAi (xj,n) 6= Be(xj,n)or ∆A
e,i,j(n)is defined. Stephj, niproceeds as follows:
(1) Choosexj,nas a big number.
(2) Wait forΦA,Ui (xj,n)↓= 0.
(3) IfΦA,Ui (xj,n)↓= 0at stages—
We define∆A,Ue,i (n) = D(n)with useδe,i(n) =ϕi(xn). Here, again, when we seeΦA,Ui (xn)[s]↓= 0, we do not put restraint onAto pre-serve this computation. WheneverAorU changes below δe,i(n),
we go back to 2, in which case,∆A,Ue,i (n)is undefined. [Note that here,ncan be inDsor not inDs.
If n is currently not in Ds, then n can enter D later, and leave at a further stage. When n enters D, at stage s0, we will use the assumption thatDe is equal toΦeA⊕U, as thenhn, siis not inDe, and
we want to restrainAfrom changing to retain this computation. Ifn
leavesDlater (ΓA⊕D⊕U(xj,n)reverts to a previous value, which is equal to 0), thenhn, siwill enter De, makingDe andΦAe⊕U disagree
at hn, si. This disagreement can now fail only when U changes, which will undefineΓA⊕D⊕U(xj,n).
If n is inDs, then we do nothing here, as ifn leaves D later, this change will allow us to undefineΓA⊕D⊕U(xn), and we can putxj,n intoBe. Notice that axioms enumerated into Γe beforen entersD are all invalid due to the AorU-changes. Since otherwise we will
be in the situation described in the last paragraph, wherenleaving
Dwill cause a disagreement betweenDe andΦAe⊕U.]
Wait forD(n)to change, and simultaneously commence the step
hj, n+ 1i.
(4) SayD(n)changes at staget > s. There are two possibilities: (a) n enters D at stage t. In this case, the D(n) change makes
ΓA⊕D⊕U
e (xj,n) undefined, and instead of putting xj,n into Be
immediately at this stage, we wait forDe and ΦAe to agree on
(all numbers≤)hn, ti.
(b) nleavesDat staget. Then thisD(n)change necessarily results inΓA⊕D⊕U
e (xj,n)becoming undefined, and we defineΘU(j) =
K(j)with useθ(j) = ϕi(xj,n), and wait forU to change below
θ(j) or K(n) to change. Of course, a restraint is put onA to preserve the computationΦA⊕U
e (xj,n).
IfU changes first, go back to (2), and the restraint is canceled. IfK(n)changes first, go to (6).
(5) At staget0 > t,De andΦAe agree on (all numbers≤)hn, ti. We put
restraint on A to preserve both computations ΦA,Ui (xj,n) = 0, and ΦA,Ue (hn, ti) = 0. Create a link between Se,i and Re, the mother
node. This link can be traveled and canceled when we find at Re thatUhas changes belowϕe(hn, ti). If there is no such aU-change, then this link can survive for ever.
We defineΘU(j) =K(j)with useθ(j) = max{ϕ
i(xj,n), φe(hx, si)},
and wait for U to change below θ(j), or K(n) to change, or n to leaveD. Simultaneously, start cyclej+ 1.
IfK(j)changes first, then go to (6). IfnleavesDfirst, go to (7). If U changes first, see whether U has a change below ϕi(xj,n). If
yes, then go back to (2). Of course, the restraint onAestablished at staget0 will be canceled. Otherwise, go back to (4) and wait forDe
andΦA⊕D
e to agree on (all numbers≤)hn, ti.
(6) K(j)changes.
We enumerate xj,n intoBe, and the current γe(xj,n)into A. Note that this new γe(xj,n) is bigger thanθ(j), since it is defined after staget0.
Wait forU to change below θ(j), and also wait fornto leaveD. If nleavesDfirst, then go to (7). IfU changes first, go to (8).
(7) At staget00 > t0, nleavesD. Thenhn, tientersDe at this stage. As
the computation ΦA
e(hn, ti) = 0 is preserved at stage t
0, we get a global win via
ΦA,Ue (hn, ti) = 0= 1 =6 De(hn, ti),
provided thatU does not change change belowϕe(hn, ti).
Wait forU to change belowθ(j).
(8) Put γe(xj,n) intoA and redefineΘU(j) = K(j), with use 0. Start
cyclej + 1.
The outcomes forSe,iare rather more complicated. As in theP-module,
we will have outcomes for each cycle, and each cycle has outcomes different from those given in section 2. Below, we only describe the outcome for a particular cyclej.
hj, wi: Cyclejruns many steps, and each one stays at (2) or (3), or returns to (2) infinitely often from (3) or (4b). In this case, no restraint is put on A, andSe,i is satisfied by either ΦA,Ue (xj,n) not converging
for somexj,n (some step stops at (2) or loops between (2) and (3), or (4b)), or by ∆A,Ue,i becoming totally defined and computing D correctly. [Note that if this outcome is true, a step can return to (2) from other point beyond (5) at most once, via aU-change.]
No restraint is put onAin this outcome.
hj, n, ui: Step n of cycle j reaches (5) and later returns to (4) to wait for
e
D and ΠA⊕U
e to agree on number ≤ hn, ti, infinitely often by U
-changes, wheretis the stage at whichnentersD. That is, infinitely many links to the mother node Re are created and canceled in the
construction. If this outcome is true, then ΦA⊕U
e (hn, ti) diverges,
leading to a global win forRe.
No restraint is put onAin this outcome.
hj, n, di: Stepnof cyclej reaches (7) because after (5),n leavesD, no mat-ter whether K(j) has changed or not. As we put a restraint on A at (5), if U does not change below ϕe(hn, ti), then we will have ΦAe⊕D(hn, ti) = 0andhn, ti ∈De — again, a global win forRe. If
this outcome applies, then there are only finitely many expansionary stages. That is, Re will havef as its outcome, and we do not put
this outcome belowSe,i.
[Notice that if this outcome applies,ΓA⊕D⊕U
e (xj,n)can be wrong in the case that stepnreaches(6), andxj,n is put intoBewhennis in
D. If latern leavesD, thenΓA⊕D⊕U
one beforenentersD, which is currently defined as 0. If so, then ei-ther cyclejreaches (7) and stays at (7), or it eventually reaches (8). If the former case, as indicated above, we have a global win forRe. If cyclej eventually reaches (8), then cyclej also wins asΘU(j)is defined, and is equal toK(j). To keepΓA⊕D⊕U
e (xj,n)correct, at (8), we also putγe(xj,n)(the old one) intoA. This enumeration does not injure cyclej.]
hj, si: Some step n of cycle j reaches and stays at (6). Then Se,i is
sat-isfied, via Be(xj,n) = 1 6= 0 = ΦiA⊕U(xj,n). Notice that since n
remains inD(otherwise stepnwill go to (7), which cannot be true by our assumption), the enumeration ofγe(xj,n)(the new one) into
Awill not change the computationΦAi ⊕U(xj,n). The enumeration of
thisγe(xj,n)makesΓeA⊕D⊕U well-defined and correct atxj,n. Also
note that U will not change so as to injure this computation, since otherwise stepnwill go to (8), which again cannot be true.
We arrange the outcomes of cyclej as:
hj, wi<L hj,0, ui<Lhj,1, ui<L· · ·<Lhj, n, ui<L· · ·<Lhj, si,
where the outcomes forj1 are always to the left of those forj2, whenever j1 < j2. Again, sinceU is incomplete, there is a leastj such that one of the
outcomes for cyclejis the true outcome relative toSe,i.
The whole construction turns out to be a0000argument, in a quite standard way. The details will appear in [5].
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