# Algorithmic randomness and stochastic selection function

Date:
Friday 6th July 2012 - 12:00 to 12:30
Venue:
INI Seminar Room 1
Abstract:
For $x=x_1x_2\cdots,y=y_1y_2\cdots\in\{0,1\}^\infty$, let $x^n:=x_1\cdots x_n$ and $x/y:=x_{\tau(1)}x_{\tau(2)}\cdots$ where $\{i\mid y_i=1\}=\{\tau(1) The following two statements are equivalent. 1.$x\in {\cal R}^u$, where$u$is the uniform measure on$\{0,1\}^\infty$. 2.$\exists \mbox{ computable }w\ \ x\in{\cal R}^w \mbox{ and }x/y_i(w, x)\in{\cal R}^u\mbox{ for }i=1,2,\ldots, 6, \mbox{ where }\\ \{y_1(w, x),\ldots, y_6(w, x )\} \mbox{ consists of non-trivial selection functions and depends on } w\mbox{ and }x.$The author do not know whether we can drop the assumption that$x$is random w.r.t. some computable probability in (ii), i.e., whether we can replace (ii) with$x/y\in R^u$for$y\in Y^x$where$Y^x$consists of non-trivial selection functions and depends on$x$. We also have a similar algorithmic analogy for Steinhause theorem [1]. Let$w$be a computable probability such that 1.$\forall y\in{\cal R}^w,\ \lim_n K(y^n)/n=0$, (b)$\lim_n \sum_{1\leq i\leq n} y_i/n$exists for$y\in{\cal R}^w$, and 2.$\forall \epsilon >0 \exists y\in{\cal R}^w \lim_n \sum_{1\leq i\leq n} y_i/n>1-\epsilon$. Then the following two statements are equivalent. 1.$\lim_{n\to\infty} \frac{1}{n}K(x^n)=1$. 2.$\lim_{n\to\infty} \frac{1}{| x^n/y^n|}K(x^n/y^n)=1$for$y\in{\cal R}^w$, where$K$is the prefix-complexity. For example,$w:=\int P_\rho d\rho$, where$P_\rho$is a probability derived from irrational rotation with parameter$\rho\$, satisfies the condition of Prop. 2, see [2]. There are similar algorithmic analogies for Kamae's theorem [4], see [3].

[1] H. Steinhaus. "Les probabilités dénombrables et leur rapport à la théorie de la meésure." Fund. Math., 4:286–310, 1922.

[2] H. Takahashi and K. Aihara. "Algorithmic analysis of irrational rotations in a sigle neuron model." J. Complexity, 19:132–152, 2003.

[3] H. Takahashi. "Algorithmic analogies to Kamae-Weiss theorem on normal numbers." In Solomonoff 85th memorial conference. To appear in LNAI.

[4] T. Kamae. "Subsequences of normal numbers." Israel J. Math., 16:121–149, 1973.

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