## Kolmogorov complexity and the Recursion Theorem. Manuscript, submitted for publication (2005)

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@MISC{Kjos-hanssen05kolmogorovcomplexity,

author = {Bjørn Kjos-hanssen and Wolfgang Merkle and Frank Stephan},

title = {Kolmogorov complexity and the Recursion Theorem. Manuscript, submitted for publication},

year = {2005}

}

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### Abstract

Abstract. Several classes of diagonally non-recursive (DNR) functions are characterized in terms of Kolmogorov complexity. In particular, a set of natural numbers A can wtt-compute a DNR function iff there is a nontrivial recursive lower bound on the Kolmogorov complexity of the initial segments of A. Furthermore, A can Turing compute a DNR function iff there is a nontrivial A-recursive lower bound on the Kolmogorov complexity of the initial segements of A. A is PA-complete, that is, A can compute a {0, 1}-valued DNR function, iff A can compute a function F such that F (n) is a string of length n and maximal C-complexity among the strings of length n. A ≥T K iff A can compute a function F such that F (n) is a string of length n and maximal H-complexity among the strings of length n. Further characterizations for these classes are given. The existence of a DNR function in a Turing degree is equivalent to the failure of the Recursion Theorem for this degree; thus the provided results characterize those Turing degrees in terms of Kolmogorov complexity which do no longer permit the usage of the Recursion Theorem. 1.

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Citation Context ... if and only if it wtt-computes a DNRfunction. A set A = {a0, a1, . . .} with a0 < a1 < . . . is called hyperimmune if A is infinite and there is no recursive function h such that an ≤ h(n) for all n =-=[11]-=-. A further characterization is that there is no recursive function f such that A intersects almost all sets of the form {n + 1, n + 2, . . . , f(n)}. Intuitively speaking, hyperimmune sets have large... |

117 |
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Citation Context ... n+1 elements falling into the rectangle {0, 1, ..., h(n)} × {0, 1, ..., g(n)}. Now one knows that f(A(0)A(1)...A(g(n))) ≥ 2 n+1 and thus C(A(0)A(1)...A(g(n))) ≥ n. So A is complex. ⊓⊔ Remark 11 Post =-=[12]-=- introduced the notion of hyperimmune sets and demonstrated that every r.e. set with a hyperimmune complement is wtt-incomplete and that there are such sets, that is, hyperimmunity was used as a tool ... |

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Citation Context ...g conditions are equivalent. (1) The set A is complex. (2) There is a recursive function h such that for all n, C(A↾h(n)) ≥ n. (3) The set A tt-computes a function f such that for all n, C(f(n)) ≥ n. =-=(4)-=- The set A wtt-computes a function f such that for all n, C(f(n)) ≥ n. We omit the proof of Proposition 4, which is very similar to the proof of Proposition 3, where now when proving the implication (... |

23 |
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Citation Context ...is known that a set can compute such a function iff it can compute a diagonally nonrecursive (DNR) function g, that is, a function g such that for all e the value ϕe(e), if defined, differs from g(e) =-=[6]-=-. By a celebrated result of Schnorr, a set is Martin-Löf random if and only if the length n prefixes of its characteristic sequence have prefix-free Kolmogorov complexity H of at least n − c for some ... |

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Citation Context ...in the following wellknown characterizations of T- and wtt-completeness [11, Theorem III.1.5 and Proposition III.8.17], where the latter characterization is known as Arslanov’s completeness criterion =-=[1, 6]-=-. Corollary 9 An r.e. set is Turing complete if and only if it computes a DNRfunction. An r.e. set is wtt-complete if and only if it wtt-computes a DNRfunction. A set A = {a0, a1, . . .} with a0 < a1 ... |

12 | Lowness for the class of Schnorr random reals
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Citation Context ...)) for almost all n. On the other hand, H(x) ≤ Hf(n)(F (n)) for all n, so the statement of the theorem is satisfied in the case that A has high Turing degree. □ 6. R.e. traceable sets It was shown in =-=[8]-=- that a set A is r.e. traceable if and only if every Martin-Löf random set is Schnorr random relative to A. We remind the reader of the definitions. (Recall that Wn is the nth r.e. set, and Dn the fin... |

11 |
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Citation Context ...-Spies, Kjos-Hanssen, Lempp and Slaman [2], we obtain in Proposition 7 that there is 2sa complex set that does not compute a Martin-Löf random set, thus partially answering an open problem by Reimann =-=[13]-=- about extracting randomness. Theorem 8 states that recursively enumerable (r.e.) sets are complete if and only if it they are autocomplex, and are wtt-complete if and only if they are complex. Arslan... |

8 |
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Citation Context ...ven acceptable numbering of all r.e. sets: ∃e [We = W f(e)]. Second one can say that every total recursive function f coincides at some places with the diagonal function: ∃e [ϕe(e)↓ = f(e)]. Jockusch =-=[5]-=- showed that these two variants of the Recursion ∗ Department of Mathematics, University of Hawai‘i at Mānoa, 2565 McCarthy Mall, Honolulu, HI 96822, U.S.A., bjoern@math.hawaii.edu. Partially supporte... |

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Citation Context ...R function is eventually different from every partial-recursive function and every function eventually different from all partial-recursive ones is DNR. Theorem 5.1 has been applied by Stephan and Yu =-=[12]-=-, and Greenberg and Miller [4]. Theorem 5.1. The following statements are equivalent: (1) A computes a function f that is eventually always different from each recursive function. (2) A computes a fun... |

5 |
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Citation Context ...ent from every partial-recursive function and every function eventually different from all partial-recursive ones is DNR. Theorem 5.1 has been applied by Stephan and Yu [13], and Greenberg and Miller =-=[4]-=-. Theorem 5.1. The following statements are equivalent:12 KJOS-HANSSEN, MERKLE, AND STEPHAN (1) A computes a function f that is eventually always different from each recursive function. (2) A compute... |

5 |
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Citation Context ...ntil today, but the above theorem gives a partial answer to this question as it shows that K ≤T A is true for all sets A of maximal H-complexity which contain at least one string of each length. Nies =-=[12]-=- pointed out to the authors that one might study the analogue of incompressible strings in the sense that one looks at functions F producing strings of length n and approximate complexity n + H(n). Mo... |

4 |
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Citation Context ...x, defined in terms of monotonic complexity, are the same as being wtt-complete and T-complete in the special case of recursively enumerable sets. 3. Hyperavoidable and effectively immune sets Miller =-=[11]-=- introduced the notion of hyperavoidable set. A set A is hyperavoidable iff it differs from all characteristic functions of recursive sets within a length computable from a program of that recursive s... |

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2 |
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Citation Context ... and 8, which in summary yields simplified proofs for these criteria. Theorem 10 asserts that the complex sets can be characterized as the sets that are not wtt-reducible to a hyperimmune set. Miller =-=[8]-=- demonstrated that the latter property characterizes the hyperavoidable sets, thus we obtain as corollary that a set is complex if and only if it is hyperavoidable. In the characterization of the auto... |

2 |
personal communication
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(Show Context)
Citation Context ... containing H(n). By a result of Beigel, Buhrman, Fejer, Fortnow, Grabowski, Longpré, Muchnik, Stephan and Torenvliet [2], such an A-recursive algorithm can only exist if K ≤T A. □ Remark 4.4. Calude =-=[3]-=- had circulated the following question: If A is an infinite set of strings of maximal H-complexity, that is, if A satisfies ∀x ∈ A ∀y [|y| = |x| ⇒ H(y) ≤ H(x)], is then K ≤T A? The question remains op... |

2 |
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Citation Context ...R function is eventually different from every partial-recursive function and every function eventually different from all partial-recursive ones is DNR. Theorem 5.1 has been applied by Stephan and Yu =-=[13]-=-, and Greenberg and Miller [4]. Theorem 5.1. The following statements are equivalent:12 KJOS-HANSSEN, MERKLE, AND STEPHAN (1) A computes a function f that is eventually always different from each rec... |

2 | Torenvliet. Enumerations of the Kolmogorov function
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Citation Context ...above algorithm computes relative to A for input n a set of up to 4 k − 1 elements containing H(n). By a result of Beigel, Buhrman, Fejer, Fortnow, Grabowski, Longpré, Muchnik, Stephan and Torenvliet =-=[2]-=-, such an A-recursive algorithm can only exist if K ≤T A. Remark 4.4. Calude [3] had circulated the following question: If A is an infinite set of strings of maximal H-complexity, that is, if A satisf... |

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1 |
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Citation Context ...dom set, that is, in general it is not possible to compute a set of roughly maximum complexity from a set that has a certain minimum and effectively specified amount of randomness. Reimann and Slaman =-=[14]-=- have independently announced a proof of Theorem 7 by means of a direct construction. Theorem 7 There is a complex set that does not compute a Martin-Löf random set. Proof. Ambos-Spies, Kjos-Hanssen, ... |

1 |
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Citation Context ...iven acceptable numbering of all r.e. sets: ∃e [We = Wf(e)]. Second one can say that every total recursive function f coincides at some places with the diagonal function: ∃e [ϕe(e)↓ = f(e)]. Jockusch =-=[5]-=- showed that these two variants of the Recursion Theorem are also equivalent relative to any oracle A: Every function f ≤T A admits a fixed-point iff every function g ≤T A coincides with the diagonal ... |

1 |
The complexity of the enumeration and solvability of predicates, Dokl
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Citation Context ... y) ≤ C(A ↾ y). • for every n the set of strings σ ∈ {0, 1} ∗ with h(σ) ≤ n is finite. Similarly A is autocomplex iff there is a function h ≤T A with these same properties. Remark 2.10. M.I. Kanovich =-=[6,7]-=- (see Li and Vitanyi [10], Exercise 2.7.12, p. 184) states a result to the effect that the notions of being complex and autocomplex, defined in terms of monotonic complexity, are the same as being wtt... |