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Another Example of Higher Order Randomness
 FUNDAMENTA INFORMATICAE
, 2002
"... We consider the notion of algorithmic randomness relative to an oracle. We prove that the probability # that a program for infinite computations (a program that never halts) outputs a cofinite set is random in the second jump of the halting problem. Indeed, we prove that # is exactly as random as ..."
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We consider the notion of algorithmic randomness relative to an oracle. We prove that the probability # that a program for infinite computations (a program that never halts) outputs a cofinite set is random in the second jump of the halting problem. Indeed, we prove that # is exactly as random as the halting probability of a universal machine equipped with an oracle for the second jump of the halting problem, in spite of the fact that # is defined without considering oracles.
Kolmogorov complexity for possibly infinite computations
 Journal of Logic, Language and Information
, 2005
"... In this paper we study the Kolmogorov complexity for noneffective computations, that is, either halting or nonhalting computations on Turing machines. This complexity function is defined as the length of the shortest inputs that produce a desired output via a possibly nonhalting computation. Clea ..."
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In this paper we study the Kolmogorov complexity for noneffective computations, that is, either halting or nonhalting computations on Turing machines. This complexity function is defined as the length of the shortest inputs that produce a desired output via a possibly nonhalting computation. Clearly this function gives a lower bound of the classical Kolmogorov complexity. In particular, if the machine is allowed to overwrite its output, this complexity coincides with the classical Kolmogorov complexity for halting computations relative to the first jump of the halting problem. However, on machines that cannot erase their output –called monotone machines–, we prove that our complexity for non effective computations and the classical Kolmogorov complexity separate as much as we want. We also consider the prefixfree complexity for possibly infinite computations. We study several properties of the graph of these complexity functions and specially their oscillations with respect to the complexities for effective computations. 1
Program size complexity for possibly infinite computations
"... We define a program size complexity function H ∞ as a variant of the prefixfree Kolmogorov complexity, based on Turing monotone machines performing possibly unending computations. We consider definitions of randomness and triviality for sequences in {0, 1} ω relative to the H ∞ complexity. We prove ..."
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We define a program size complexity function H ∞ as a variant of the prefixfree Kolmogorov complexity, based on Turing monotone machines performing possibly unending computations. We consider definitions of randomness and triviality for sequences in {0, 1} ω relative to the H ∞ complexity. We prove that the classes of MartinLöf random sequences and H ∞random sequences coincide, and that the H ∞trivial sequences are exactly the recursive ones. We also study some properties of H ∞ and compare it with other complexity functions. In particular, H ∞ is different from H A, the prefixfree complexity of monotone machines with oracle A. 1
Kolmogorov complexities K max, K min on computable partially ordered sets
, 801
"... We introduce a machine free mathematical framework to get a natural formalization of some general notions of infinite computation in the context of Kolmogorov complexity. Namely, the classes MaxX→D PR and MaxX→D Rec of functions X → D which are pointwise maximum of partial or total computable sequen ..."
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We introduce a machine free mathematical framework to get a natural formalization of some general notions of infinite computation in the context of Kolmogorov complexity. Namely, the classes MaxX→D PR and MaxX→D Rec of functions X → D which are pointwise maximum of partial or total computable sequences of functions where D = (D,<) is some computable partially ordered set. The enumeration theorem and the invariance theorem always hold for MaxX→D PR, leading to a variant KD max of Kolmogorov complexity. We characterize the orders D such that the enumeration theorem (resp. the invariance theorem) also holds for MaxX→D Rec. It turns out that MaxX→D Rec may satisfy the invariance theorem but not the enumeration theorem. Also, when MaxX→D Rec satisfies the invariance theorem then the Kolmogorov