On Languages Reducible to Algorithmically Random Languages

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-Lof [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}-$\R=\R(A)\cap\R(B)$.