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There in no sw-complete c.e. real. (English) Zbl 1070.03028
A real number $$\alpha$$ is called random if for every $$n$$, the sequence $$\alpha_{| n}$$ of its first $$n$$ binary digits cannot be compressed, i.e., if $$K(\alpha_{| n})\geq n-O(1)$$, where $$K(z)$$ is a prefix-free Kolmogorov complexity of the sequence $$z$$ (i.e., crudely speaking, the shortest length of a program that computes a binary sequence starting with $$z$$).
For computably enumerable (c.e.) real numbers, we can define when one of them is “more random” than another: $$\alpha\leq _K \beta$$ if and only if $$K(\alpha_{| n})\leq K(\beta_{| n})+O(1)$$. It is known that Chaitin’s $$\Omega$$ (the halting probability of a universal prefix-free Turing machine) is random and c.e. and it is “more random” than any other c.e. real $$\beta$$ in the sense that $$\beta\leq _K \Omega$$.
It is difficult to compute Kolmogorov complexity, so a new definition of sw-reducibility (strongly weak truth table reducibility) $$\leq_{\text{sw}}$$ was proposed. It is known that sw-reducibility implies $$\beta \leq_K\alpha$$. Because of this, it was conjectured that there exists an sw-complete c.e. real number, i.e., a c.e. real number $$\alpha$$ such that $$\beta \leq_{\text{sw}}\alpha$$ for all c.e. real numbers $$\beta$$. The authors prove that there is no such number; moreover, they prove that there exist two non-random c.e. reals $$\beta_0$$ and $$\beta_1$$ for which no c.e. real $$\alpha$$ exists for which $$\beta_0\leq_{\text{sw}}\alpha$$ and $$\beta_1\leq _{\text{sw}}\alpha$$.

##### MSC:
 03D80 Applications of computability and recursion theory 68Q30 Algorithmic information theory (Kolmogorov complexity, etc.)
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