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УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА
Физико-математические пауки
UDK 510.532
RELATIVE ENUMERABILITY AND THE D-C. E. DEGREES
M.M. Arslanov
Abstract
We study the relationship between the relative enumerability and the d-c. e. degrees. We prove that the degree of the halting problem is splitt.able into two c. e. degrees such that
d
Key words: Turing degrees, comput.ably enumerable degrees, relative computable enumerability, splitting, definability.
The A2 de groos of unsolvability arc basic objects of study in classical compntability theory, since they are the degrees of those sets whose characteristic functions are limits of computable functions. A natural tool for understanding the Turing degrees is the introduction of hierarchies to classify various kinds of complexity. The most common such hierarchy, the arithmetical hierarchy, is itself not of much use in the classification of the A2 degrees, since it is far too coarse. This fact has led to the introduction of hierarchies based on finer distinctions than quantifier alternation. Two such hierarchies are by now well-established. One. the CEA hierarchy independently defined by Arslanov fl] and Jocknsch and Shore [2. 3]. is. like the arithmetical hierarchy, based on the complexity of the definitions of the involved sets, replacing the alternation of quantifiers with the iteration of computable ermmerability in and above 3. S6t, 3. SI gnificantly less powerful procedure. The second snch hierarchy, the difference hierarchy duo to Putnam [4] and Ershov [5. G], is built up by starting with the computable enumerable sets as a base, and then iterating the operation of taking set-theoretic differences, thereby classifying sets on the basis of the difficulty of their construction in comparison with c. e. sets. Analysis of the relationship of the CEA hierarchy to the difference hierarchy is therefore a natural means of comparing the definability of sets to the inherent difficulty of their construction.
We took the first steps toward this analysis in fl]: we proved that there is a A2j 2-CEA set which is not of n-c. e. degree for any n < u. Generalizing this result Jockusch and Shore [3] proved that for any computable ordinal a < ft, there is a fl-c. e. degree which is not a-CEA, while, on the other hand, that for every uniformly given class of A2j degrees, there is a A2j 2-CEA degree which is not in this class. From the latter result, it follows that for each a < u, there is a 2-CEA set which is not of a-c. e. degree. This result is more interesting when a > 1, sinee d-c. e. sets, and hence 2
In [7] we took a further step towards analyzing the relationship of this second level of the CEA hierarchy to the difference proving that any u-c. e. degree which is 2-CEA is 2
limit on possible extensions of this result: there exists a d-c. e. set C such that for every n > 3, there exists a set A which is simultaneously C-CEA and (n + 1)-c. e., n
Further results in this direction are obtained in [8]. Let u and v be c. e. degrees such that v < u. Then there is a d-c. e. degree d such that v < d < u and d is not v
This result naturally raises the following problem, which has a long history. Let a < b be non-computable c. e. degrees. Is there a CEA in a d-c. e. degree d < b such that b is not c. e.?
Below we list all so far known results on this question:
1. [9] Let a be a non-computable c. e. degree such that a' = 0'. Then there is a non-c. e. but c. e. in a degree b > a. Moreover,
2. [10] Let c < h be c. e. degrees such that c is low and h is high. Then there is a degree a < h such that a is CEA in c degree.
3. [10] For all high c. e. degrees h < g, there is a properly d-c. e. degree a such that h < a < g and a c. e. in h.
4. [10] There is a c. e. degree a, 0 < a < 0' such that for any c. e. in a degree b > a, if b ^ 0' then b is c. e.
a > 0 d
d > a d a
(A set A is called superlow if A' =tt 0'. A degree is superlow if it contains a super low set.)
(A set A is called superlow if A' =tt 0'. A degree is superlow if it contains a superlow
These results allow us to formulate the following well-known hypothesis which is still can be considered as an open problem.
a > 0 a d
b
Recent investigations have shown that this problem is closely related to another
d
d
may be useful.
a > 0 d b
b < a. Since a is c. e. in some c. e. degree a0 < a, it follows from the Sacks Splitting Theorem, relativized to a0 U b < a, that a is splittable into two 2-CEA degrees which are above b, i.e. there are 2-CEA-degrees co and ci such that c0 U ci = a and b < c0 < a, b < c1 < a. Moreover, Arslanov, Cooper and Li [12, 13] proved that
dd 0' v0 vi
d-c. e. degree d and each i < 1, if Vj < d then d is c. e. in v1_i.
We adopt the usual notational conventions found, for instance, in [14]. In particular, we write [s] after functional and formulas to indicate that every functional or parameter therein is evaluated at stage s. In particular, for an oracle X and c. e. functional $(X; y, s) means only that at most s steps are allowed for the computation from oracle X to converge, whereas $(X; y)[s] means also that the approximation Xs is used as the oracle, and may mean as well that some function-value y(s) is being used as the argument for the computation. When using a c. e. oracle, we adopt the common practice of taking the use function to be nondecreasing in the stage.
Theorem 1. There exists a splitting of 0' into c. e. degrees v0 and v1 such that for every d-c. e. degree d and each i < 1, if vi < d then d is c. e. in v1_j.
Proof. We will construct c. e. sets V0 and V1 so that the degrees vi = deg Vj have the desired properties. We also construct auxiliary c. e. sets U0, U1. This is ensured by the following two types of requirements. To ensure that 0' <T Vj, we satisfy requirements
• Pj: U = ©^ (f°r each partial computable functional ©jje).
To ensure that for all d-c. e. sets D, if Vj <T D then D is of degree c. e. in V1_i, we satisfy requirements.
• Rj: = AV;®Q: & Qj <T Veach d-c. e.
d-c. e. set Qj c. e. in V1_i and a partial computable functional Aije). The condition Qj <T V © Dj will be met by the usual permitting argument.
To ensure that Qj is c. e. in V1_i we use a common method which works as follows. When an integer x is enumerated into Qj at stage s we appoint a certain marker a(x). Then we allow x to be removed from Qj clt 9. Iclt6r Stcl ge t only if V1_i \ a(x) [t — 1] = = V1_i \ a(x)[t].
The condition 0' <T V0 © V1 will follow from the construction directly. P
and Mnclmik:
x
(< i, w > for i < 1), which is larger than all higher-priority restraints, and keep U
(2) Wait for ©V(x) |= 0.
(3) Put x into U and protect V \ (0(x) + 1).
The basic strategy for R-requirements in isolation is to build AV, ensuring that it is total and computes D correctly. Since we build the set V during the construction,
V
While the strategies for the requirements in isolation are thus very simple, there are
R
VP
Pi
Sj -requirements below one Re-strategy.
Basic module for the Rj -strategy above P -requirements.
We use an w-sequence of "cycles", where each cycle k proceeds as follows:
(1) At a stage s set A^®^(k) = D^k) with a use ^^(k) > all P^- and P 1_i-restraints, and ^^(k) > ^^(k — 1) and start cycle k + 1 to run simultaneously k
Dj(k) t
(3) (i) Enumerate ¿¿^^(k) into Qj,
(ii) set A^®^ (k) = Djji(k) with a new use ¿^^(k) > all Pi-restraints, and ^^(k) > ¿^(k), and
(iii) appoint the marker »¿(^^(k)) as the first integer y such that y > ¿^t (k) and y = < 2, l > for some l.
(4) Wait for De(k) to change back (at a stage u, say).
(5) We need
- to keep Qe below V® © De (at stage t k enters De, and we put Si,e,s(k) into Q^,- Now k leaves De).
- to correct the axiom A(V ®Qe)(k) = De(k) We have two possibilities to achieve this:
- either by enumerating Si,e,s (k) into Vi
- or by removing ¿iie,s(k) from Qe (in this case we need to enumerate ai(Jiie,s(k)) into Vi_i).
The crucial point here is that our choice between these two possibilities depends upon the priority ordering of requirements P® and P1-i that may be injured:
a) If the highest-priority strategy which would be injured by this correction is
Pi
enumerate ai(Jiie,s(k)) into V1_i and remove ¿iiejS(k) from Qle.
b) Otherwise, put ¿iie,s(k) into V% and set AV/e®Qe(k) = De,„(k).
Set AVe®Qe(k) = De,u(k) with the same use ¿iie,«(k) = 5ije,f(k). In both cases start cycle k + 1 to run simultaneously.
We now give the construction. We say that the axiom AVe®Qe (k) = De(k) requires correction at stage s if at a stage t < s we set AV/e®Qe (k) = De,t(k) with a use
5i,e,t(k),De,s(k) = De,t(k),and (Vi © Qe)t t ¿i,e,t(k) = (Vi © Ql)s t ¿e,t(k).
Stage s = 0. Set Uo = Vo = Vi = 0[0]. ^o =< 0, e >, x1,0 =< 1, e >. Stage s > 0. Fix e such that s = (e, m) for some m.
Peo
a) If ©V°e(x0)[s] |= 0 Mid x0,s-1 ^ U0,s, then enumerate x0,s-1 into U0,s, and protect V0 t ^(xe^) with priority P°.
b) If QV°e(x0)[s] |= U0,s(x0,s_1) = 1, then define
x0,s = (Mx)[(3y)(Vj)(x = (0, y) A x > all Rj-uses assigned so far]. Otherwise, set x°,s = x0,s-1. Pe1
(©0, V), U0, x0, 00,^ by ©j1, V1, U1, x^01,e accordingly).
Substage 3. Let z be the greatest integer such that for any k < z there exists
y-®0,-i e)
a stage s' < s such that at stage s' the axiom A®^ (k) = De(k)[s'] k<z
A(iei®Qi'e)(k) = De(k) requires correction at stage s. Let t be a stage at which the axiom A(V?®Qi'e)(k) = = De(k) was set. We consider two cases.
Case 1) De,s(k) = 1. In this case we proceed as in step (3) of the Basic Module: (i) enumerate ¿ije,i(k) into Qije,
(ii) set AV®Qi'e(k) = Des(k) with a new use 6ies(k) > all P-restraints, and Si,e,s(k) > Si,e,t(k), and
(iii) appoint the marker ai(Si,e,s(k)) as the first integer y such that y > iiiejS(fc^d y =< 2,1 > for some I.
Case 2) De,s(k) = 0. Therefore, there is a stage u < s such that De,u(k) = 1, and at stage u we (re)set the axiom AVe®Qi'e(k) = De(k). It follows also that at stage u we enumerated Sp,e,t(k) into Qi}e. In this case we proceed as in step (5) of the Basic Module:
a) if the highest-priority strategy which would be injured by the Qi,e($i,e,t(k))— or Vi(Si,e,t(k))- correction is a P-strategy (or there is no strategy at all that would be injured), then enumerate ai(Si,e,t(k)) into Vi_i and remove 6ije,t(k) from Qije.
b) Otherwise, put 6ije,t(k) into V;.
Set AV,;®Qi>e (k) = De,s(k).
Substage 4. If none of the axioms AVe®Qi'e (k) = De(k) for k < z requires correction at stage s, then set the new axiom De>s(z) = AV®Qi'e (z) with a use Si,e,s(z) > all P-, R-restraints.
Substage 5. Go to stage s + 1.
End of the construction.
Verification.
Let vi=deg(V^), i < 1. Lemma 1. Qie <T V © De.
Proof. To V © De-computably compute whether x e Qie, first find a stage u at which a new axiom De(y) = A^® Qi'e(y) with a use 5ieu(y) > x is settled. Obviously, u
Suppose now that x = 5ies(k) was chosen as a use for some AVe®Qi'e(k) at a stage s < u (otherwise, x e Qi,e)- Find a st age v > u at which Viv \ x = V \ x and Dev (x) = De(x) .Now x e Qie if and only if x e Q%eev ■
□
Lemma 2. If De = V then De <T V © Qe.
Proof. It follows immediately by construction. □
Lemma 3. Qie is c. e. in V1_i.
Proof. It follows immediately from the construction. □
Lemma 4. For each i < 1 mid e e w, requirements P£, are eventually satisfied.
Proof. Fix e and assume % induction that the Lemma holds for all j < e. Choose s minimal so that no P,®-restraints may be injured by some R-requirement.
Pei
into V to protect the V^-restraint of higher priority. But beginning at some stage s we take witnesses for A-uses greater than the V1_i -restraints, after that we meet the Pei
□
Lemma 5. 0' = vo U vi.
Proof. Suppose, in contrary, that v0 U v1 < 0'. Then by [8, Theorem 4.1] there exists a d-c. e. degree d such that v0 U v1 < d and d is not c. e. in v0 U v1, and
therefore it is not c. e. in vi. We have v0 < d and d is not c. e. in v1, a contradiction.
□
The author was partially supported by the Russian Foundation for Basic Research (Grant No. 10-01-00399-a) and the Presidential Program for Support of Leading Scientific Schools (Grant No. 5383.2012.1).
Резюме
M.M. Арсланов. Относительная перечислимость и d-вычислимо перечислимые степени.
В работе изучается связь между относительной перечислимостью и d -вычислимо перечислимыми степенями. Доказано, что тыорипговая степень проблемы остановки машины Тыоринга разложима па такие две вычислимо перечислимые степени, что верхний
d
перечислимы относительно второй степени.
Ключевые слова: тыорипговые степени, вычислимо перечислимые степени, относительная перечислимость, разложение, определимость.
References
1. Arslanov M.M. On a hierarchy of the degrees of unsolvability // Ver. Metody i Kiber-netika. - 1982. - No 18. - P. 10-17 (in Russian).
2. Jockusch C.G., Jr., Shore R.A. Pseudo jump operators I: The R.E. case // Trans. Amer. Math. Soc. - 1983. - V. 275, No 2. - P. 599-609.
3. Jockusch C.G., Jr., Shore R.A. Pseudo jump operators II: Transfinite iterations, hierarchies, and minimal covers //J. Symb. Logic. - 1984. - V. 49, No 4. - P. 1205-1236.
4. Putnam H. Trial and error predicates and the solution to a problem of Mostowski // J. Symb. Logic. - 1965. - V. 30, No 1. - P. 49-57.
5. Ershov Y. On a hierarchy of sets I // Algebra i Logika. - 1968. - V. 7, No 1. - P. 47-73 (in Russian).
6. Ershov Y. On a hierarchy of sets II // Algebra i Logika. - 1968. - V. 7, No 4. - P. 15-47 (in Russian).
7. Arslanov M.M., LaForte G.L., Slaman T.A. Relative enumerability in the difference hierarchy // J. Symb. Logic. - 1998. - V. 63, No 2. - P. 411-420.
8. Arslanov M.M., Lempp L., Shore R.A. On isolating c. e. and isolated d-c. e. degrees // S.B. Cooper, T.A. Slaman, S.S. Wainer (Eds.) Computability, enumerability, unsolvability (London Math. Soc. Lect. Note Series). - Cambridge: Cambridge Univ. Press, 1996. -No 224. - P. 61-80.
9. Soare R.I., Stob M. Relative recursive enumerability // Proc. Herbrand Symposium, Logic Colloquium. - 1981. - P. 299-324.
10. Arslanov M.M., Lempp S., Shore R.A. Interpolating d-r.e. and REA degrees between r.e. degrees // Ann. Pure Appl. Logic. - 1995. - V. 78. - P. 29-56.
11. Arslanov M.M., Cooper S.B., Li A. There is no low maximal d.c.e. degree - Corrigendum // Math. Logic Quart. - 2004. - V. 50, No 6. - P. 628-636.
12. Arslanov M.M. The Ershov Hierarchy // Cooper S.B., Sorbi A. (Eds.) Computability in Context: Computation and Logic in the Real World. - Imperial College Press/World Scientific, 2011. - P. 49-100.
13. Arslanov M.M., Cooper S.B., Li A. There is no low maximal d.c.e. degree // Math. Logic Quart. - 2000. - No 46. - P. 409-416.
14. Soare R.I. Recursively Enumerable Sets and Degrees: A Study of Computable Functions and Computably Generated Sets (Perspectives in Mathematical Logic). - Berlin: Springer-Verlag, 1987. - 437 p.
Поступила в редакцию 15.12.10
Arslanov, Marat Mirzaevich Doctor of Physics and Mathematics, Professor, Head of the Department, of Algebra and Mathematical Logic, Kazan Federal University, Kazan, Russia.
Арсланов Марат Мирзаевич доктор физико-математических паук, профессор, заведующий кафедрой алгебры и математической логики Казанского (Приволжского) федерального университета.
E-mail: Marat.ArslanovQksu.ru
