Index sets of universal codes

Transcription

1 Index sets of universal codes Konstantinos Beros (UNT) joint with Achilles Beros (University of Hawai i)

2 Medvedev reductions Definition Suppose A, B are subsets of 2 ω or families of functions on ω. Say A is Medvedev reducible to B if there is a single algorithm (i.e., Turing functional) Φ such that every member of B computes a member of A, via the algorithm Φ. Write ( s for strongly ) A s B Idea The problem of finding a member of A is reducible to the problem of finding a member of B.

3 Medvedev reductions Definitions {e} is the Turing machine coded by e W e is the domain of {e} f : ω ω is DNC if e f (e) {e}(e) IM = family of immune sets, i.e., sets that contain no infinite c.e. sets a weak genericity/randomness property BI = family of all bi-immune sets Examples IM s DNC (Folklore) BI s DNC (Jockusch-Lewis)

4 Many-one reductions Definition Suppose A, B ω. Say A is many-one reducible to B if there is a computable map f : ω ω such that A = f 1 (B). If F P(ω), a set B F is F-complete if every A F is many-one reducible to B. Idea The many-one hierarchy measures the effective complexity of subsets of ω. Analogous to Wadge reductions for subsets of Polish spaces.

5 Definable subsets of ω Suppose A ω. A is Σ 0 1 if A = W e, for some e A is Π 0 1 if ω \ A is Σ0 1 A is Σ 0 2 if there is a Π0 1 set B ω ω such that A = {x : y (x, y) B} A is Π 0 2 if ω \ A is Σ0 2 etc... A is Π 1 1 if there is a computable tree T on ω ω such that A = {x : y (x, y) is not an infinite branch of T }.

6 Examples Examples FIN = {e : W e is finite} is Σ 0 2 -complete INF = {e : W e is infinite} is Π 0 2 -complete TOT = {e : W e = ω} is Π 0 2 -complete COF = {e : W e is cofinite} is Σ 0 3 -complete WF = {e : {e} is the char. function of a wf subtree of ω <ω } is Π 1 1 -complete

7 Medvedev codes Definition Suppose A, B F(ω). Say e ω is a B-universal A-code if, for every Y B, the algorithm coded by e computes a member of A from Y. Let B[A] be the set of all B-universal A-codes. Remark B[A] is the set of codes for Medvedev reductions of A to B. Question What is the many-one complexity of B[A] for various families A and B?

8 Basic results A. Beros and K. Beros INF[TOT] is Π 0 3-complete (note that INF and TOT are only Σ 0 2 and Π0 2, respectively) 0 2 [FIN] is Π0 4 -complete ( 0 2 = the set of limit computable functions = the set of functions computable in 0 )

9 DNC functions Recall IM s DNC, i.e., DNC[IM]. A. Beros and K. Beros DNC[IM] is Π 1 1 -complete.

10 Main result More generally Suppose A and B are hyperarithmetic with A s B. Suppose also that A is a nontrivial tail set, and that there is a 0 2 embedding of ω ω into B, with relatively closed range. Then B[A] is Π 1 1 -complete. Definition A 0 2 embedding f : ωω B is a continuous injection such that there exists a limit computable -embedding f : ω <ω ω <ω with x ω ω f (x) = lim n f (x n) -embedding means f preserves both and

11 Proof Fix a Medvedev reduction Φ : ω <ω 2 <ω of A to B, i.e., Φ is computable and, for each x B, lim Φ(x n) n exists and is a member of A.

12 Proof Fix a 0 2 embedding f : ωω B, with relatively closed range. Let f : ω <ω 2 <ω be a limit computable -embedding such that for all x ω ω. f (x) = lim n f (x n) Let (f s ) s ω be a uniformly computable sequence of -embeddings such that f (α) = lim s f s(α) for all α ω <ω.

13 Proof Let WF = {x ω : {e} is the char. fn of a wf tree T e } If e codes a tree T e, let T e,s be this tree, computed up to s computation stages. Hence, T e = s T e,s. Recall WF is a Π 1 1-complete set. Goal Define effectively a computable map Φ e : ω <ω 2 <ω such that Φ e is a Medvedev reduction of A to B iff e WF. This will show that B[A] is Π 1 1 complete.

14 Proof For e ω and σ ω <ω, let m e,σ = max{m : {e} len(σ) m <m is the char. fn of a tree} Lemma m e,σ, as len(σ), iff e codes the characteristic function of a tree.

15 For e ω and σ ω <ω, let Proof n e,σ = max { min{s, len(β)} : α T e,len(σ) s.t. β f s (α) & σ β } Lemma 1 If x [T e ], and y = f (x), then lim n e,y k = k Lemma 2 If [T e ] =, then, for all y B {n e,y k : k ω is bounded}

16 Proof Now define Φ e (σ) = ( ) Φ(σ) [0, n e,σ ] [0, m e,σ ] Suppose e codes a tree T e. Otherwise, lim k Φ e (y k) is finite and hence not in A. [T e ] x lim k n e,f (x) k = and lim k Φ e (f (x) k) = ω / A [T e ] = For all y B, n e,y k is bounded by Lemma 2 and lim k Φ e (y k) = lim k Φ(y k) A Hence lim k Φ e (y k) A also, since A is a tail set.

17 Proof Aside from verifying Lemmas 1 and 2, this completes the proof.

18 Another Medvedev reduction Definition A function f : ω n is DNC n if f (e) {e}(e), for all e. Every DNC n is DNC. Hence, DNC n [IM]. But the previous theorem cannot be used to show DNC n [IM] is Π 1 1, since there is no 0 2 embedding of ωω into DNC n, with relatively closed range. Such a map would be a homeomorphism between ω ω and a compact set, since DNC n is closed.

19 Another theorem A. Beros and K. Beros Suppose A and B are hyperarithmetic with A s B. Suppose also that A is a nontrivial tail set, and that there is a 0 2 embedding of ω ω into B, with relatively Π 0 2 range. Then B[A] is Π1 1 -complete. In the case of DNC n, the range of the embedding looks something like {f DNC n : i s.t. f (x i ) = 1}, where (x i ) i ω is a sequence of natural numbers such that {x i }(x i ) does not converge. (This theorem is neither a generalization of the last one, nor vice versa. Not every closed set is Π 0 2 and not every Π0 2 is closed.)