Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefix-free Kolmogorov complexity. Such H-trivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an H-trivial real. We also analyze various computability-theoretic properties of the H-trivial reals, showing for example that no H-trivial real can compute the halting problem. Therefore, our construction of an H-trivial computably enumerable set is an easy, injury-free construction of an incomplete computably enumerable set. Finally, we relate the H-trivials to other classes of "highly nonrandom " reals that have been previously studied.

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How fast can one approximate a real by a computable sequence of rationals? We show that the answer to this question depends very much on the information content in the finite prefixes of the binary expansion of the real. Computable reals, whose binary expansions haveavery low information content, can be approximated (very fast) with a computable convergence rate. Random reals, whose binary expansions contain very much information in their prefixes, can be approximated only very slowly by computable sequences of rationals (this is the case, for example, for Chaitin's \Omega numbers) if they can be computably approximated at all. We show that one can computably approximate any computable real also very slowly, with a convergence rate slower than any computable function. However, there is still a large gap between computable reals and random reals: any computable sequence of rationals which converges (monotonically) to a random real converges slower than any computable sequence of rat...

Abstract We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefix-free set of binary strings.

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Abstract. No algorithm can of course solve the Halting Problem, that is, decide within finite time always correctly whether a given program halts on a certain given input. It might however be able to give correct answers for ‘most ’ instances and thus solve it at least approximately. Whether and how well such approximations are feasible highly depends on the underlying encodings and in particular the Gödelization (programming system) which in practice usually arises from some programming language. We consider BrainF*ck (BF), a simple yet Turing-complete real-world programming language over an eight letter alphabet, and prove that the natural enumeration of its syntactically correct sources codes induces a both efficient and dense Gödelization in the sense of [Jakoby&Schindelhauer’99]. It follows that any algorithm M approximating the Halting Problem for BF errs on at least a constant fraction εM> 0 of all instances of size n for infinitely many n. Next we improve this result by showing that, in every dense Gödelization, this constant lower bound ε to be independent of M; while, the other hand, the Halting Problem does admit approximation up to arbitrary fraction δ> 0byan appropriate algorithm M δ handling instances of size n for infinitely many n. The last two results complement work by [Lynch’74]. 1

We prove that any Chaitin # number (i.e., the halting probability of a universal self-delimiting Turing machine) is wtt-complete, but not tt-complete. In this way we obtain a whole class of natural examples of wtt-complete but not tt-complete r.e. sets. The proof is direct and elementary. 1 Introduction Kucera [8] has used Arslanov's completeness criterion 1 to show that all random sets of r.e. T-degree are in fact T-complete. Hence, every Chaitin # number is T-complete. In this paper we will strengthen this result by proving that every Chaitin # number is weak truth-table complete. However, no Chaitin # number can be tt-complete as, because of a result stated by Bennett [1] (see Juedes, Lathrop, and Lutz [9] for a proof), there is no random sequence x such that K # tt x. 2 Notice that in this way we obtain a whole class of natural examples of wtt-complete but not tt-complete r.e. sets (a fairly complicated construction of such a set was given by Lachlan [10]). # The first has...