Lowness for weakly 1-generic and Kurtz random

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Lowness for weakly 1-generic and Kurtz random Frank Stephan and Liang Yu National University of Singapore

Lowness for weakly 1-generic and Kurtz random – p. 1/1

1-Generic and Weakly 1-Generic Generic sets are recursion-theoretic counterpart to forcing. Every set above the halting problem is Turing-equivalent to Jump of a 1-generic set. Definition x ∈ {0, 1}ω is 1-generic iff for every Σ01 class S not containing x there is an n such that no y ∈ S conincides with x below n. x ∈ {0, 1}ω is weakly 1-generic iff every dense Σ01 class S contains x. 1-generic sets are weakly 1-generic but not vice versa. A Turing degree contains a weakly 1-generic set iff it is hyperimmune.

Lowness for weakly 1-generic and Kurtz random – p. 2/1

Randomness and Kolmogorov complexity Definition Kolmogorov complexity of σ is shortest p with U(p) = σ . C(σ): U is universal plain machine. H(σ): U is universal prefix-free machine. Randomness x ∈ {0, 1}ω is Martin-Löf random iff • there are no uniformly Σ01 classes S0 , S1 , . . . with x ∈ Sn ∧ µ(Sn ) ≤ 2−n for all n; • H(x(0)x(1) . . . x(n)) ≥ n for almost all n;

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Lowness Notions Definition x is low (for the jump) ⇔ ∀y, y can compute the halting problem relative to x iff y can compute the unrelativized halting problem. x is low for Martin-Löf random ⇔ ∀y, y is Martin-Löf random relative to x iff y is Martin-Löf random unrelativized. Characterization of H-Trivial [Nies 2002] The following notions are equivalent for a set x. • x is H-trivial [Chaitin, Solovay 1975]; • H relative to x equals H [Muchnik 1999]; • x is low for random [Zambella 1990; Kuˇcera, Terwijn 1999]; • x is limit-recursive and Chaitin’s Ω is Martin-Löf random relative to x. This coincidence motivated the study of other lowness notions.

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Hyperimmune-freeness Hyperimmune free x is hyperimmune-free if for each function f ≤T x, there is a recursive function g dominating f . Recursively traceable A set x is recursively traceable iff for every f ≤T x there is a recursive function g such that, for all n, |Dg(n) | ≤ n and f (n) ∈ Dg(n) . I.O. Recursively traceable A set x is infinitely often recursively traceable iff for every f ≤T x there is a recursive function g such that, for all n, |Dg(n) | ≤ n and there are infinitely many n, f (n) ∈ Dg(n) . Observation If x is recursively traceable then x has hyperimmune-free Turing degree.

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Low for Schnorr Random Recursively Random x is recursively random if no recursive martingale succeeds on x. That is,lim sup M(x(0)x(1) . . . x(n)) = ∞ where M(τ ) = 12 (M(τ 0) + M(τ 1)) for all τ and {(σ, q) : q < M(σ)} is an r.e. set. Theorem [Nies, 2005] x is low for recursively random iff x is recursive. Schnorr Random x is Schorr random if there are no uniformly Σ01 classes T {Sn }n with x ∈ n Sn and µ(Sn ) = 2−n . Theorem [Terwijn and Zambella, 2001; Kjos-Hanssen, Nies and Stephan, 2006] x is low for Schnorr random iff x is recursively traceable.

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Low for 1-Generic Theorem [Greenberg, Miller and Yu, 2006] A set is low for 1-generic iff it is recursive. Proof Let x be given and nonrecursive. There is an 1-generic set y such that y ⊕ x ≡T x′ . [Posner, Robinson 1981; Slaman, Steel 1989]. Then (y ⊕ x)′ ≡T x′′ . If y would be 1-generic relative to x then x′′ ≡T (y ⊕ x)′ ≡T y ⊕ x′ ≡T x′ .

Thus y is not 1-generic relative to x and x is not low for 1-generic.

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Low for Weakly 1-Generic Theorem [Downey, Griffiths and Reid, 2004] If a set is recursively traceable then it is low for weakly 1-Kurtz random. Conjecture [Downey, Griffiths, Miller, Nies, Reid, Yu] A set is low for weakly 1-generic iff it is recursively traceable. This conjecture will be refuted.

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Recursively Traceable Degrees DNR x is DNR if for there is a function f ≡T x so that for all n, f (n) 6≃ Φn (n). Theorem [Kjos-Hanssen, Merkle, Stephan 2006] Let the set x be not high, the following are equivalent: x is not dnr. For every function f ≤T x there is a recursive function g such that ∃∞ n [f (n) = g(n)]. x is not autocomplex, that is, there is no f ≤T x such that H(x ↾ m) ≥ n whenever m ≥ f (n).

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Subclasses of hyperimmune-free degrees Proposition There is a partial-recursive {0, 1}-valued function ψ such that every extension is neither dnr nor recursively traceable. But some extension has hyperimmune-free degree. Three types of hyperimmune-free degrees recursively traceable ⊂ dnr ∩ hyperimmune-free ⊂ hyperimmune-free.

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Diagonally Non-Recursive Degrees Theorem Let x be dnr. Then x is not low for weakly 1-generic. Proof There is f ≤T x such that H(x ↾ f (n)) ≥ n querying x only below f (n). Let [ Tn = {y : ∀m ≤ f (n) [y(m + n) = x(m)]}, Sn = T . m≥n m There is a constant c such that for all n and y ∈ Tn , H(y(0)y(1) . . . y(n + f (n))) ≥ n − c.

All Sn are dense Σ01 (x)-classes. No H-trivial set is in all classes Sn . Thus there are H-trivial 1-generic sets which are not contained in all Sn . So x is not low for weakly 1-generic.

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Neither dnr nor hyperimmune Theorem Let x be of hyperimmune-free and non-dnr Turing degree. Given a dense Σ01 (x) class S there is T ⊆ S such that • T is dense; • T is a Σ01 class (without an oracle); • if µ(S) = 1 then µ(T) = 1.

Thus x is low for weakly 1-generic. Corollary If x is of hyperimmune-free and non-dnr Turing degree then x is low for Kurtz random. This refutes the conjecture of Downey, Griffiths, Miller, Nies, Reid and Yu.

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Construction of T S is in Σ01 (x), x has hyperimmune-free and non-dnr Turing degree. There are a recursive function f and an x-recursive function g such that, for all n, • g(n) ⊆ {0, 1}f (n+1) ; • ∀σ ∈ {0, 1}f (n) ∃τ σ[τ ∈ g(n)]; • g(n) · {0, 1}ω ⊆ S. ˜ such that for all n, There are recursive functions h, h • h(n) ⊆ {0, 1}f (n+1) ; • ∀σ ∈ {0, 1}f (n) ∃τ σ[τ ∈ h(n)]; ˜ • ∃m ∈ {n, n + 1, . . . , h(n)}[g(m) = h(m)].

Now T contains x iff x(0)x(1) . . . x(f (m + 1) − 1) ∈ h(m) for ˜ some n and all m ∈ {n, n + 1, . . . , h(n)} .

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Kurtz Random S is in Σ01 (x), µ(S) = 1, x has hyperimmune-free and non-dnr Turing degree. There are a recursive function f and an x-recursive function g such that, for all n, • g(n) ⊆ {0, 1}f (n+1) . . . (as before) • µ(g(n) · {0, 1}ω ) ≥ 1 − 2−n ; ˜ such that for all n, There are recursive functions h, h • h(n) ⊆ {0, 1}f (n+1) . . . (as before) • µ(h(n) · {0, 1}ω ) ≥ 1 − 2−n ;

Now T contains x iff x(0)x(1) . . . x(f (m + 1) − 1) ∈ h(m) for ˜ . some n and all m ∈ {n, n + 1, . . . , h(n)}

µ(T) = 1 since, for every n, µ({x : ∀m > n [x(0)x(1)...x(f (m + 1) − 1) ∈ h(m)]}) ≥ 1 − 2−n

Lowness for weakly 1-generic and Kurtz random – p. 14/1

Summary Lowness Notions in Randomness quite popular after Nies showed that H-Trivial is equivalent to Low for Martin-Löf random. Four notions considered: 1. Recursively traceable; 2. Neither dnr nor hyperimmune; 3. Low for weakly 1-generic; 4. Low for Kurtz random. The conjecture that (1) ⇔ (3) ⇔ (4) has been refuted. The implication (1) ⇒ (2) is proper and (2) ⇔ (3) ⇒ (4). Unknown whether (3) ⇐ (4).

Lowness for weakly 1-generic and Kurtz random – p. 15/1