On the Autoreducibility of Random Sequences
MFCS '00 Proceedings of the 25th International Symposium on Mathematical Foundations of Computer Science
Limits on the computational power of random strings
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Limits on the computational power of random strings
Information and Computation
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In this paper languages "bounded reducible" to algorithmically random languages are studied; these are the languages whose characteristic sequences are algorithmically random (as defined by Martin-Löf [Inform. and Control, 9 (1966), pp. 602--619]); here RAND denotes the class of algorithmically random languages. The reducibilities $\le^{\cal R}$ are very general but are defined so that if $A\in\R(B)$, then there is a machine $M$ with the properties that $L(M,B)=A$ and every computation of $M$ relative to any oracle halts. Book, Lutz, and Wagner [Math. Systems Theory, 27 (1994), pp.~201--209] studied {\sf ALMOST}-$\R$, defined to be $\{A\mid$ for almost every $B,\ A\le_{\cal R} B\}$. They showed that {\sf ALMOST}-$\R=\R(RAND)\cap\rec$, where $\rec$ denotes the class of recursive languages, so that {\sf ALMOST}-$\R$ is the "recursive part" of $\R(RAND)$. In this paper this characterization is strengthened by showing that for every $B\in RAND$, {\sf ALMOST}-$\R=\R(B)\cap\rec$. A pair $(A,B)$ of languages is an independent pair of algorithmically random languages if $A\oplus B\in RAND$. In this paper it is shown that for every $\R$ and for every independent pair $(A,B)$, {\sf ALMOST}-$\R=\R(A)\cap\R(B)$.