skip to content

Propagation of partial randomness

Presented by: 
S Simpson Pennsylvania State University
Tuesday 3rd July 2012 - 09:00 to 10:00
INI Seminar Room 1
Let $X$ be an infinite sequence of $0$'s and $1$'s, i.e., $X\in\{0,1\}^\mathbb{N}$. Even if $X$ is not Martin-Löf random, we can nevertheless quantify the amount of partial randomness which is inherent in $X$. Many researchers including Calude, Hudelson, Kjos-Hanssen, Merkle, Miller, Reimann, Staiger, Tadaki, and Terwijn have studied partial randomness. We now present some new results due to Higuchi, Hudelson, Simpson and Yokoyama concerning propagation of partial randomness. Our results say that if $X$ has a specific amount of partial randomness, then $X$ has an equal amount of partial randomness relative to certain Turing oracles. More precisely, let $\mathrm{KA}$ denote a priori Kolmogorov complexity, i.e., $\mathrm{KA}(\sigma)=-\log_2m(\sigma)$ where $m$ is Levin's universal left-r.e. semimeasure. Note that $\mathrm{KA}$ is similar but not identical to the more familiar prefix-free Kolmogorov complexity. Given a computable function $f:\{0,1\}^*\to[0,\infty)$, we say that $X\in\{0,1\}^\mathbb{N}$ is strongly $f$-random if $\exists c\,\forall n\,(\mathrm{KA}(X{\upharpoonright}\{1,\ldots,n\})>f(X{\upharpoonright}\{1,\ldots,n\})-c)$. Two of our results read as follows.

Theorem 1. Assume that $X$ is strongly $f$-random and Turing reducible to $Y$ where $Y$ is Martin-Löf random relative to $Z$. Then $X$ is strongly $f$-random relative to $Z$.

Theorem 2. Assume that $\forall i\,(X_i$ is strongly $f_i$-random$)$. Then, we can find a $\mathrm{PA}$-oracle $Z$ such that $\forall i\,(X_i$ is strongly $f_i$-random relative to $Z)$.

We also show that Theorems 1 and 2 fail badly with $\mathrm{KA}$ replaced by $\mathrm{KP}=$ prefix-free complexity.

The video for this talk should appear here if JavaScript is enabled.
If it doesn't, something may have gone wrong with our embedded player.
We'll get it fixed as soon as possible.
Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons