CC BY
22
Tom 154, kii. 2
УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА
Физико-математические пауки
2012
UDK 510.532
ISOLATION: MOTIVATIONS AND APPLICATIONS
G. Wu, MM. Yamaleev
Abstract
In this paper, we briefly review the origins of the isolation phenomenon and its applications. We discuss a stronger notion of double bubbles. We also show recent achievements in the study of lattice embeddings with the help of the isolation property.
Key words: Turing degrees, Ersliov hierarchy, isolated degrees, lattice embeddings.
Introduction
An n + 1-c.e. degree d is isolated by an n-c.e. degree a if a < d is the greatest n-c.e. degree below d. The existence of such isolated n + 1-c.e. degrees, for n > 1, can
n
in different levels of the Ershov hierarchy are considered. The case when n =1 was first proposed explicitly by Cooper and Yi in their paper fl], and the general case, i.e. when n > 1
Kalinmllin and Lcriipp in [3] also gives rise to the existence of isolated degrees, and it is attempting to extend such bubble constructions to show that different (finite) levels in the Ershov hierarchy are not elementary equivalent.
In this paper, we briefly review the origins of the isolation phenomenon and how variants of this phenomenon can be applied to study local and global properties of the Ershov hierarchy. In Section 1, we first show how to obtain isolated degrees from Ivaddah's thesis, and then give a brief description of early development in this area. In Section 2, we consider isolation from side, a property that was nsed by Yang and Yu to show that the c.e. degrees is not a Si substructure of d.c.e. degrees. This phenomenon n
with those nonisolated degrees that can be isolated from side, which are nontrivial extensions of Cooper and Yi's isolated degrees. In Section 3, we give a direct construction of a bubble, a work of Arslanov, Kalinmllin and Lcriipp. That is, we will provide a construction of a d.c.e. degree d and a c.e. degree a < d such that every d.c.e. degree e below d is comparable with a. Obviously, d is isolated by a. We believe that, like the isolation phenomenon, the bubble phenomenon is an important tool for studying the structural properties of the Ersliov hierarchy. In Section 4, we show recent development of applications of isolation to lattice embeddings. This project was initiated by Wu in his thesis [5] and has come to a highlight in a recent work of Fang, Lin and Wu, who proved in fC] that any nonzero cappable c.e. degree can have a d.c.e. degree with almost universal clipping property as its complement.
Our notation and terminology are standard and generally follow Soare [7]. We suggest the readers to refer Cooper's paper [8] and Arslanov's paper [9] for the general idea on local degree theory.
1. Kaddah's work and isolation
A Turing degree is properly d.c.e. if it contains a d.c.e. set. but no c.e. set. Cooper proved in [10] the existence of properly d.c.e. degrees, and Lachlan observed that any-nonzero d.c.e. degree bounds a nonzero c.e. degree. That is, given a d.c.e. set D with an effective approximation {Ds : s e i}, the associated set
L(D) = {(x,s) : x e Ds - D}
is c.e., and is Turing reducible to D, while D is c.e. in L(D). If D is c.e. and {Ds : s e i} is an effective enumeration of D, then L(D) is empty. On the other
D L(D) L(D)
Lachlan set of D with reference to the enumeration {Ds : s e i}. Lachlan's observation shows that the d.c.e. degrees are downwards dense, which is also true for the c.e. degrees. However, the d.c.e. degrees are not dense, and hence these two degree structures are not elementarily equivalent.
Theorem 1 (Nondensity Theorem for D2 [XI])- There exists a maximal d.c.e. degree d < 0', and hence the d.c.e. degrees are not dense.
The fact that these structures are not elementarily equivalent was first proved by Arslanov [12] and Downey [13], who proved that any nonzero d.c.e. degree is cnppable, and that the diamond lattice can be embedded into the d.c.e. degrees preserving 0 and 1, respectively.
In contrast to this nondensity theorem, Islinmkliametov [14] and, independently, Cooper and Yi [1] proved that any nonempty interval [a, d], with a c.e., contains infinitely many d.c.e. degrees, a weak density theorem of d.c.e. degrees.
Theorem 2 [1, 14]. If d is a d.c.e. degree and a < d is a c.e. degree, then there is a d.c.e. degree c between a and d.
Here we cannot require the degree c above be c.e., as there are a c.e. degree a and a d.c.e. degree d > a such that no c.e. degree is between ^md d. This can be obtained from the following theorem of Kaddah.
Theorem 3 [15]. Every low c.e. degree is branching in the d.c.e. degrees.
Let a be a low, nonbranching, c.e. degree, and let d, e be two d.c.e. degrees above a such that ^^^^^^mum of ^d e in the d.c.e. degrees. Then one of the intervals (a, d), (a, e) contains no c.e. degrees, as a is assumed to be nonbranching.
Cooper and Yi first noticed this structural phenomenon and proposed the notion of isolation explicitly in their paper [1].
Definition 1 [1]. A d.c.e. degree d is isolated % a c.e. degree a if a < d is the dd
a
After showing the existence of the isolated degrees, Cooper and Yi continued to
d
dd
of a properly d.c.e. degree as a minimal upper bound of a uniformly c.e. sequence of degrees, an even stronger result. These two kinds of degrees are proved to be dense in the c.e. degrees.
Theorem 4 [16—18]. Doth the isolated d.c.e. degrees and the nonisolated d.c.e. degrees are dense in the c.e. degrees.
Theorem 4 says that the isolated degrees could be as close to the isolating degrees as wanted. Ishnmkhanietov and Wu proved that in terms of the high/low hierarchy, the isolated d.c.e. degree and the isolating degree can be quite far from each other.
Theorem 5 [19, 20]. There is a high d.c.e. degree d isolated by a low c.e. degree c. Such a c.e. degree c can be found below any nonzero c.e. degree a.
Cooper [21] proved in 1974 that any high c.e. degree bounds a minimal pair, and hence no high c.e. can be nonbounding. However, there do exist high d.c.e. nonbounding degrees, as first constructed by Chong, Li and Yang in [22] by a fairly complicated 0'''
a
c d c
d
In [18]. Arslanov, Lempp and Shore showed the existence of the nonisolating degrees, and proved that these degrees are downwards dense in the c.e. degrees, and can occur in every jump class. In contrast to this. Cooper. Salts and Wu proved in [23] that the nonisolating degrees are upwards dense in the c.e. degrees. Furthermore, Salts proved in [24] that the nonisolating degrees are not dense in the c.e. degrees.
Theorem 6 [24]. There is an interval of c.e. degrees, [a, c], each of which isolates a d.c.e. degree.
Recent work of Wu and Yamaleev1 shows that such an interval can be large. That
ca
Lachlan [25] proved in 1966 that the infinmm of two c.e. degrees in the c.e. degrees and the infimum of two c.e. degrees in the A2 degrees coincide. In contrast to this, in [15], Kaddah proved that the infima of n-c.e. degrees in the n-c.e. degrees can be different from that of these two n-c.e. degrees in the (n + 1)-c.e. degrees.
Theorem 7 [15]. For each n > 2, there are n-c.e. degrees d,e such that they have f as infimum in the n-c.e. degrees, and there is an (n + 1)-c.e. degree x with f < x < d,e.
xd
and e do not have f as their infimum in the (n + 1)-c.e. degrees. Following Cooper and Yi, we can say that x is isolated by f in the n-c.e. degrees. Liu, Wang and Wu2
xf
result was proved previously by LaForte in [2] by a different approach.
2. Variants of isolation
Arslanov's cupping theorem shows that the structures of the c.e degrees and the d.c.e degrees differ at £3 level, and Cooper et al.'s proof of the existence of incomplete maximal d.c.e. degrees, and Downey's diamond theorem show that these two structures differ at £2 level. It becomes interesting to consider whether two structures differ at £1 level.
Say that nonzero c.e. degrees a, b, c form a Slaman triple if b > c, and for any-nonzero c.e. degree w < a w v b > c. It is easy to check that a Mid b above form a minimal pair, and Shore and Slaman proved in 1993 that every high c.e. degree bounds a Slaman triple.
In 1983, Slaman proved the following strengthened version of Slaman triples.
1A large interval of isolating degrees, in preparation.
2An alternative approach of isolated (n + 1)-c.e. degrees, in preparation.
Theorem 8 (Slaman 1983). There are e.e. degrees a, b, c and a A2j degree d with 0 < d < a such that (1) a, b, c form a Slaman triple, (2) d V b > c.
Theorem 8 says that no nonzero c.e. degree w below a has the property that c < w V b, while there is a nonzero A2 degree d below a that has this property. This is a Ei property, which provides a Ei difference between the c.e. degrees and the A2 degrees.
In [26], Yang and Yu proved that the c.e. degrees and the d.c.e. degrees also differ
Ei
dd
Theorem 9 [26]. There are c.e. degrees a,b, c, e and a d.c.e. degree d < a such that d V b > c, d < e, and for any c.e. degree w < a, either w V b > c or w < e.
This proof was recently extended by Cai, Shore and Slaman to prove that for any m < n, the m-c.e. degrees is not a E1-substructure of the n-c.e. degrees.
Theorem 10 [4]. There are c.e. degrees a, b, c, e and an (n +1) -c.e. degree d < a such that d V b > c, d < e, and for any n-c.e. degree v < a, either v V b > c or v < e.
nd
e
such an e below d (d is isolated by e in the n-c.e. degrees) or e cannot be below d (d is nonisolated). We consider the case when n =1, and in the latter case, we say dd e d d
e
d. That is, it is hard to make d isolated. For Cai, Shore and Slaman's result, for n > 2, if we really want to move e below d, then e cannot be c.e. anymore, and we will make n
In the following, we show how to construct a d.c.e. degree isolated from side nontriv-ially. We will build d.c.e. sets B, D and a c.e. set C satisfying the following requirements:
G : B <t C,
PD : D = if,
PeB : B = $We V We = *f,
Qe : = We ^ (3 c.e. U <t B)(Vi)(Ue = ),
Re : = We ^ 3re(rf = We),
where |($e, We) : e G w} is a standard enumeration of all ($, W} for which ^ ^^e partial ^^^^^teble functional and W is a c.e. set.
Let b, c, d be the Turing degrees of B, C and B © D, respectively. By the Q-requi-rement, b < c. All the P^^^^^^^^^^^^ ensure that b is a properly d.c.e. degree, and hence b < c. By the Q-strategies, b is nonisolated. The R-strategies ensure that all the c.e. degrees below d are also below b, hence below c. According to Wu [5], d is pseudo-isolated.
For the G-requirement, we construct a p.c. functional A such th at B = AC, and for a number x, if we put a number x into B, ot extract it from B, at a stage s, we always put A(x)[s] into C automatically. Obviously, A is totally defined.
A PD-strategy is a standard Friedberg - Muchnik strategy, and a PB-strategy is
R
to an isolation strategy. That is, we check at every expansionary stage whether rf and We agree, and if not, suppose they differ at x, then we extract relevant numbers out of D to recover a computation (x) to a previous one, which has value 0. This
creates a disagreement between and We, and the requirement is satisfied.
A Qe-strategy, Z say, attempts to construct a c.e. set Ue such that if We = , then Ue <T B and for all i G w, Ue = . Define Z-expansionary stages in a standard way, and Z has two outcomes: 0 Mid 1, where 0 stands for the case that there are infinitely many Z-expansionary stages, and 1 for the other case.
Suppose that Z has outcome 0. The construction of Ue will be carried out by Qe's substrategies, Seji, i G w, which are arranged in the cone below Z ^ (0):
Se,i : Ue = .
Let p be an Seji -strategy. Then p tries to figure out a disagreement between Ue and or between W^d .
(1) Choose x fresh number.
(2) Wait for a stage s such that
^W5" (x) 1= 0 and We,s \ Vi,s(x) = \ ^,s(x).
(If this never happens, then x is a witness to the success of Seji.)
(3) Put x into Ue and B. Protect B \ s from other strategies.
(4) Wait for a stage s' such that
(x) 1= 1 and We,s' \ Vi,s(x) = r ^,s(x).
(If this never happens, then again x is a witness to the success of Seji. If it happens, then the change in (x) between stages s and, s' can only be brought about by a change in We r ^ilS(x), which is irreversible since We is a c.e. set.)
(5) Remove x from B and protect B r s from other strategies. (Now x is a permanent witness to the success of Seji because
$f r ^i,s(x) = r V>i,s(x) = We,s r ^i,s(x) = We r V>i,s(x).
x B Q e Ue
Bx
Ue
In the construction, since we are constructing B d.c.e., the R-strategy is a little more complicated than the standard isolation strategy. Suppose that rf (x) gets defined at stage s^d a PB (or an S )-strategy £ with lower priority enumerates a number z < (x) into B at a stage si > so. This enumeration forces (and allows) us to lift Yn(x) to a larger number, Yn(x)[s1]. Later, at stage s2, to get a disagreement, £, or its mother node, takes z out, and thus Yn (x) returns to its definition at stage s0. Such a variation does no harm to the disagreement strategy of R-strategy, n say-Suppose that n observes at some stage between s1 and s2 that rf (x) is incorrect and performs the disagreement strategy; then £ will be initialized, and so £ has no chance to take z out. In this case, s2 does not exist. In case that s2 does exist, and n finds
an incorrectness of (x) at stage s3 > S2, since at stage S2, Yn (x) returns to that of stage so, we have
Bs2 \ ¥e,s0 = BS0 \ ¥e,s0 ■ Now by the fact that Yn (x)[s3] = s0, we have
Bs3 \ Pe,80 = Bs0 \ Pe,80 ■
This guarantees the success of n's disagreement strategy.
d above is called a pseudo-isolated degree, as a d.c.e. degree b < d bounds all the c.e. degrees below d. Note that d is nonisolated, as if a is ct c.e. deg d
a is also below b. As b is nonisolated, there is a c.e. degree e below b (hence below d) but not below a. Wu proved in [27] that the pseudo-isolated d.c.e. degrees are dense in the c.e. degrees.
3. Double bubbles: a stronger notion
In this section, we consider a phenomenon called bubbles, which was discovered by Arslanov, Kalinmllin and Lempp in their work [3]. A basic fact about isolation is that a isolates d if and only if a Mid d have the same lower cones in the c.e. degrees.
d
a
Fix a d.c.e. degree d. Let L(d) be the collection of all Lachlan sets L(D), where D is a d.c.e. set in d. It's easy to see that d is isolated by a if and only if each X e L(d) has its degree deg(X) < a. In [28], Ishmukhametov proved that there exist a c.e. degree a and a d.c.e. degree d such that L(d) C a, and called such degrees d exact degrees. Obviously, all exact degrees are isolated by the degree of Lachlan sets. Ishmukhametov also proved in [28] that there exist isolated non-exact degrees.
Say that two nonzero d.c.e. degrees d and a (together with 0) form a bubble if all da interval (a, d) and a also form a bubble. Arslanov, Kalimullin and Lempp in their recent
a
must be c.e. The construction has special difficulties and it is still unknown whether such a structural phenomenon can be combined with other properties (in a similar way like isolated degrees, see, e.g., recent work of Wu [29] and his joint works with Fang and Lin [6] and [30]).
Arslanov, Kalinmllin and Lempp also proved in [3] the following important theorem.
Theorem 11 [3]. Let D and A be d.c.e. sets with D A, and X be a c.e. set such that X <T D,A X, and both D and A are c.e. in X. Then there exists a d.c.e. set U with X <T U <T D, and U and A are Turing incomparable.
Theorem 11 implies that for any bubble pair a < d, the c.e. degrees x such that d is relatively enumerable in and above x should be above or equal to a. However,
ax
strictly above a. That is, x and a are the same, and d is an exact degree, and hence isolated. The proof of the existence of exact d.c.e. degrees is simpler than the one for bubbles.
One property of bubbles is that the splittings of the top d.c.e. degree of a bubble a < d a
first proposed by Cooper and Li in [31].
a< d
(1) a splitting x0, xi of d is nontrivial if d ^ Xj for i = 0,1;
(2) d is splittable above a if there exist a nontrivial splitting of d = x0 U xi such that a < Xj for i = 0,1;
da
d = xo U x1 such that a ^ Xj for i = 0,1.
Note that if d is nonsplittable avoiding the upper cone of a, then d is also non-splittable avoiding the upper cone of any degree e below a. On the other hand, in [32], ad
da
This result is interesting due to the following reason. Assume that both a < d are d.c.e. degrees and each d.c.e. degree in the interval (a, d] is nonsplittable avoiding the a
if a < d form a bubble, then (a, d] is a nonsplitting interval.) Sacks' splitting theorem (avoiding upper cones) implies that no c.e. degree is in this interval. By Yamaleev's result mentioned above, a is c.e., and hence d is isolated by a.
Below we provide a sketch of the proof of a bubble, which contains all features of constructions of d.c.e. degrees which are nonsplittable avoiding upper cones and also constructions of nonsplitting intervals.
ad
0 < a < d and any d.c.e. degree u < d is comparable with a.
Sketch of proof. In the sketch, we will provide the basic idea of constructing c.e. set A and d.c.e. set D, including individual strategies and the interactions between these strategies. A and D are constructed to meet the following requirements:
Pe : a =*e,
Se : D = ©A,
Re : Ue = ^ (Ue = Г^ V A = Деи),
where {(Ф^ Oe)}eew is an effective enumeration of all p.c. functionals, and {Ue}e£w
AD
quirements, then the degrees of A and A © D form a bubble, as wanted. Later we will often omit indices.
A P-requirement and an S-requirement can be satisfied by the Friedberg - Muchnik strategy (or a variant of it). For an R-strategy, we assume that there are infinitely many expansionary stages (approximating U = ФА®В), and we try first to build a p.c. functional Г at these stages. It can happen that some S-strategy below it enumerates a number ж into D, and this enumeration can change сomputation ФА®-° (y), hence allowing U to change at y. We could not change rA(y) since it requires changes of A on small numbers. In the construction of isolated degrees, we just need to extract ж from D, making U(y) = ФА®В (y). In our construction, if U(y) changes from 1 to 0, i.e. y leaves U, then we act in the same way: extract ж from D, making U(y) = 0 = 1 = ФА®В (y), and hence satisfy this R-requirement. However, if U(y) changes from 0 to 1, i.e. y U ж D U(y)
U(y)
instead of making diagonalization immediately, we will turn to extend the definition of Ди on more arguments, z say, with y less than the ¿-use. If later we want to enumerate z into A, we will need to force y out of U to undefine Ди(z), and we make it by extracting ж from D to recover ФА®В (y) = 0, and now either y keeps in U (we get U(y) = 1 = 0 = ФА®В (y)) or y leaves U and we have Ди (z) undefined, and we zA
An P-strategy has three ou tcomes A, r, fin with ord er A <L r <L fin. If there are only finitely many expansionary stages, then fin is the correct outcome. Otherwise, there are infinitely many expansionary stages, and if from some stage on, rA keeps correct in the remainder of the construction, then r is the correct outcome. If each version of rA appears incorrect, then AU will be defined infinitely many times, and AA
current version of rA becomes invalid, and we will start a new version of rA.
For convenience, we use x as witnesses for S-strategies (can be enumerated into D and perhaps removed out later), z as witnesses for P-strategies (can be enumerated into A) and y for elements of U (can be in or not in U).
As described before, if an S-strategy, a say, is working below the outcome r of an P-strategy, t say, then a works in a standard Friedberg-Muchnik manner, trying to find a witness x to satisfy the requirement. If the enumeration of x causes U(y) different from rA(y), without loss of generality, we assume that U(y) changes from 0 to 1, then t will have outcome A, i.e. t tries to extend AU on more arguments.
A P-strategy, P say, below the out come A of an P-strategy t, tries to enumerate its witness z into A. p cannot enumerate z into A immediately if AU (z) is defined, as 0, at the moment. Here is the point: when p chooses z as its witness, t has outcome A, which is caused % an enumeration of an S-strategy, say a (below outcome r), at a stage s, which makes U(y) to change from 0 to 1. Obviously, when z is selected, z is selected much bigger, and especially z > s. Now if p wants to enumerate z into A, then the action is to extract x out of D and also enumerate z into A simultaneously. Of course, we can enumerate z later, after we see that U(y) changes back to 0. We prefer to enumerate z into A at the same time when we extract x out of D, as z > s, which is bigger than the use in (y)[s]. If U(y) does not change back to 0, then
we win as (x) = 0=1 = U(y), and the P-requirement is satisfied. Otherwise,
AU ( z) (and also AU(s)) is undefined, which makes p's enumerations into A consistent.
As U is assumed to be d.c.e., after y leaves U, it can never come back. Due to this, once p enumerates z into A, z remains in A, as AU(z) will be redefined as 1 later
U
in [3].
We now consider more complicated interactions among several strategies.
• P below A-outcome of P2 below r-outcome of Pi.
A generic case is that after we put a number x2 into D, rA is defined at some point y1, and extracting x2 from D may now change U1(y1), and the action described
D
x2 D z
A A s x2 D
z A P s A
undefine rA(y), which are defined after stage s. This idea is exactly the same as that in
P
We use s(x) to denote the stage at which x is enumerated into D. It is a routine to show that for a particular n, rA(n) can be undefined in this way by at most finitely many times, which ensures that if (n) converges, then rA(n) is defined.
• P A P2 P1
For simplicity, we use A1 and A2 to denote the A-outcomes of P-strategies t1 and t2 , respectively, where t2 is below outc ome A1 .Let p be a P-strategy be low t2's outcome A2. We now describe how p works below these two A-outcomes.
Recall that for any ^-strategy, when it turns to have outcome A, it is caused by an enumeration of some S-strategy below out come r. Here is the idea: wh en an S-strategy a2 below t2 's outcome r2 sees that (x2) converges to 0, at stage s1 say, instead, of enumerating x2 into D immediately, it waits for the next time when a2 is visited again, which will actually show that an S-strategy, a1 say, below t1 's outcome r 1, already enumerates a witness x1, being selected after stage s1. Note that at this stage, s(x^ is bigger than the uses of all computations seeing at stage s1. Now assume that a2 is visited again at stage s2, then at this stage, a2 enumerates x2 into D, so s(x2) = s2. Without loss of generality, suppose that this enumeration leads t2 to have outcome A2, and p selects a number z as its witness. We then associate z with x^d x2, which means that enumerating z into A and extracting x1 and x2 out of D should happen at the same time. Thus, we have x2 < x1 < s(x1) < s(x2) < z. Assume later that p wants to enumerate z into A; it will do so and at the same time extract both x2,x1 from D and enumerate s(x1), s(x2) into A. As discussed before, if we have a new n-expansionary stage, then U1 should have a change on the associated number, y1 say, which undefines AU (s(x1)^ AU (s(i2)^d AU (z). Also, if we have a new t2-expansionary stage later, then U2 has a change on y2 say, which undefines AU(s(x2)), A^2(s(i1)^d AU(z). This nested procedure is the core part of the construction of bubbles, and the idea can be generalized to a case when p is working below A-outcome of several ^-strategies. □
In [3], Arslanov, Kalimullin and Lenipp actually proved the existence of 3-bubbles, a generalization of double bubbles.
Definition 3. Let d, e, f be 3-c.e. degrees with 0 < d < e < f. Say that these degrees form a S-bubble in D3 if any 3-c.e. degree u < f is comparable with e and d. Say that these degrees form a weak S-bubble in D3 if any 3-c.e. degree u < f is either ed
Theorem 11 implies that weak 3-bubbles do not exist in D2.
f, e, d
weak 3-bubbles can be 3-c.e., d.c.e. and c.e., respectively. In the following, we show that such weak 3-bubbles are actually 3-bubbles, so the construction given in [3] produces a 3-bubble.
First, we show that all d.c.e. degrees below f are comparable with e and d. Suppose not, and let g be a d.c.e. degree below f, but not comparable with e and ^^en g U d would be d.c.e. and gUd > d, which would imply that gUd > e, as g is not below e. By assumption that f, e, d form a weak 3-bubble in D3, we know that g U d, e, d also form a weak 3-bubble in D3, which is also a weak 3-bubble in D2, a contradiction.
We now assume that h is a 3-c.e. degree below f but incomparable with e and d. Then h is a properly 3-c.e. degree, and degrees of Lachlan sets of those 3-c.e. sets in h
ed
Let u be a degree of the Lachlan set of a 3-c.e. set in ^^en u is not above d, as h d u
d. Now consider hU d, which is 3-c.e and relative enumerable in and above d. As here, d
and Slanian in [33] that the class of the d.c.e. degrees coincides with the intersection of the class of the w-c.e. degrees and the class of the 2-REA degrees, we know that h U d is d.c.e. Note that h U d > d, and hence h U d is comparable with ^sh itself is incomparable with e, h U d is above ^^us, h U d, e, d form a weak 3-bub ble in D3, which is also a weak 3-bubble in D2. A contradiction again. This completes the proof.
4. Isolation, cupping and diamond embeddings
In this section, we show how to use isolation phenomenon to provide alternative proofs of several known results of cupping properties and diamond embeddings.
In [11]. Cooper. Harrington. Lachlan, Lempp and Soare proved the existence of
dd cups every c.e. degree not below it to 0'. In contrast, Li, Song and Wu proved in [34] the existence of an incomplete w-r.e. degree cupping every nonzero r.e. degree to 0'. These degrees are said to have the universal cupping property. In terms of the Ershov hierarchy, Li, Song and Wu's result is optimal.
In [30], Liu and Wu proposed a cupping property for the d.c.e. degrees, where a d.c.e. d
0'
property. However, compared to the construction of incomplete maximal d.r.e. degrees, the construction of d.c.e. degrees with almost universal cupping property is much easier.
d
such that d is also isolated by a c.e. degree b < d.
d
b < d b d
DB
such that (i) D <T B, (ii) for any c.e. set We, either We <T B or B ©D © We =T 0'. That is, the constructed sets need to satisfy the isolation requirements and also the following cupping requirements:
Re: K = rf'D'We V We = Af, where r^d Ae are p.c. functional constructed by us.
Here K is a fixed creative set. Note that the R-requirements ensure that B © D has the almost universal cupping property.
Let P be an Re -strategy. For convenience, we write r^ for W^ for We(a).
P will construct a partial computable (p.c.) functional r^ such that K = rf'D'W, and if P fails, due to the actions of the isolation strategies, then an isolation strategy will show that Wa <T B •
r^ is constructed as follows:
A. At a stage s, define rf'D'W (z)[s] = Ks(z) for those z < s with rf'D'W (z)[s] not defined, and the use Ya (z)[s] is selected as a fresh number.
B. If rf'D'W (z)[s] | = Ks(z), then we put Ya(z)[s] into D to undefine the current rf'D'W(z) for the least z < s.
In the construction, to correct r, p may enumerate uses Ya(z) into D for infinitely many z. These enumerations can cause direct conflicts between p and those isolation strategies, n say, below P, which want to preserve computations. This type of interaction is an important component of the whole construction.
Let n be an isolation strategy. The basic idea of n is to construct a p.c. functional ©n at expansionary stages to ensu re that if is total, th en ©n is well-defined
and computes Wn correctly. If later, at an n-expansionary stage, we see that ©f (y) and Wn (y) differ at an argument y say, we will then force a disagreement between (y) = Wn (y).
n has three outcomes f ^d to, with priority to <l / <l d fere / denotes
nn
disagreement, and to denotes that there are infinitely many n-expansionary stages, d denotes the outcome that n succeeds in creating a disagreement between $B®D and Wn.
A crucial action of n is that when a number, z say, is removed from D, then another number, for example, the stage when z is put into D, is enumerated into B simultaneously. This action can ensure that all the isolation strategies work consistently.
We now consider the interaction between one isolation strategy and one R-strategy. Let p and n be an R-strategy and an isolation strategy respectively, with p c n- As mentioned in a single R-strategy, a disagreement created by n could be destroyed by P's enumerations into D. Also, when n has to as its outcome, p may enumerate Y(n) into D as n enters ^^ow n may see an opportunity to diagonalize by extracting Y(n) from ^^d n cannot do this as p would be injured by this extraction.
To avoid this, when n sees a computation $B®D (y) and wants to preserve it, n needs to make this computation clear of the -uses, by applying the "capricious method',
n
is first visited, it picks a number kn as its threshold, and whenever a number k < kn enters K, we enumerate the current Yg(k)-use into D toundefine r^' ' n (k), and also reset n % cancelling all the parameters associated to n > except for the parameter kn.
n aims to define a p.c. functional A^g with the purpose that if n cannot satisfy the
associated isolation requirement, then A^g should be total and computes Wg correctly. Rg
Suppose that after stage s, n is not reset and suppose that at a stage t > s, n sees
a potential witness y for its disagreement argument, then n puts Yg(kn)[t] into D first, p
ABg \ Yg(kn)[t]= Wg,t \ Yg)[t]
t
If Wg changes be low Yg (kn )[t] after stage t, at an ^^^^^^^ary stage t' > t say, nD Yg (kn )[t^, ^o recover computation ^B0D(y)[s']' e s' is the stage at
which ©B(y) is defined, as indicated above. This Wg-change lifts the value of Yg(z) for those z > k^, and hence, after stage t', the enumeration of the Yg-uses will not affect the computation (x). That is, the atta^ is completed at stage t', and n
passes the threshold kn for p.
On the other hand, if Wg has no changes below Yg(kn)[t] after stage t, then the attack associated with Yg (kn )[t] keeps active until a new attack is activated. If infinitely many such attacks are started, then ABg is defined as a total function and computes Wg correctly, and hence We is computable in B. n
f : There are only finitely many n-expansionary stages, d : n passes its threshold kn for p, and a disagreement is created.
to : There are infinitely many n-expansionary stages, and only finitely many attacks are started. In this case, ©B is total and computes Wn correctly.
gg : Infinitely many 3,"tt9.cks n
threshold kn for p. ABg is total and com putes Wg correctly. T he Rg -requirement
is satisfied. In this case, (pn) diverges.
Let £ be any strategy below the outcome ge,then £ knows that Yebuses goes to infinity, and we say that a computation Ф^ (y) at a stage s is £-believable if
Ye (pn) [s] is bigger than the use ^ (y)[s]. If £ is a back-up strategy for ц, then by using ££
standard way, as after £ sees at £-believable computations, e's further enumerations into D will not affect these computations.
This basic idea can be generalized to the situation when one isolation strategy is working below several R-strategies, where an attack of n needs to pass several thresholds. Please refer to [30] for further development. In [30], Lin and Wu also proved that b can be cappable. This implies that any d.c.e. degree below b and any d.c.e. degree above d, together with 0 and 0', form a diamond.
This isolation feature allows Fang, Lin and Wu to improve a result of Downey, Li and Wu in [35]. Fang, Lin and Wu proved recently that for any nonzero cappable c.e. degree c, there is a d.c.e. degree d with almost universal cupping property and a c.e. degree b < d such that b isolates d and that c and b form a minimal pair. By applying this result twice, first to c and then to b, we have d and b first, and then e and a such that e has almost universal cupping property and a < e isolates e, and a
and b form a minimal pair. Now for any nonzero c.e. degree w, w cups either e or d, 0'
Both authors are supported by NTU grant RG37/09, M52110101. The second author is supported by RFBR (Projects 09-01-97010,10-01-00399), ADTP "Development of the Scientific Potential of Higher School" of the Russian Federal Agency of Education (Grant 2.1.1/5367), Federal Target Grant ''Scientific and Educational Personnel of Innovation of Russia" (Government contract No. П 269).
Резюме
Г. By, M.M. Ямалеео. Изолированность: обоснования и приложения. В статье рассматриваются феномен изолированной степени и его приложения к исследованию свойств степеней их иерархии Ершова. Анализируются степени, образующие «восьмерку» (более сильный вариант изолированной степени), а также демонстрируются последние достижения в изучении вложимости решеток при помощи свойства изолированности.
Ключевые слова: тыорипговые степени, иерархия Ершова, изолированные степени, вложения решеток.
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Поступила в редакцию 01.02.12
Wu, Guohua PliD in Mathematics, Associate Professor, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore.
By, Гохуа доктор математических паук, адъюнкт-профессор Школы физических и математических паук Напьяпского технологического университета, Сингапур.
E-mail: guohuaentu.edu. ад
Yamaleev, Mars Mansurovich Candidate of Physical and Mathematical Sciences, Research Fellow, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore: Research Fellow, Lobaclievsky Institute of Mathematics and Mechanics, Kazan Federal University, Kazan, Russia.
Ямалеев Марс Мансурович кандидат физико-математических паук, научный сотрудник Школы физических и математических паук Напьяпского технологического университета, Сингапур: паучпый сотрудник Института математики и механики им. Н.И. Лобачевского Казанского (Приволжского) федерального университета, г. Казань, Россия.
E-mail: ymarsQntu.edu.sg. marsiam.2eyandex.ru
