Algorithmic randomness

Algorithmic randomness and lowness

We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0 0. This contrasts with a result of Ku cera and Terwijn 2] on sets that are low for the class of Martin-LL of random reals. The Cantor space 2 ! is the set of innnite binar...

Algorithmic Randomness and Complexity

Algorithmic Randomness and Computability

We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability. Mathematics Subject Classification Primary 68Q30, 68Q15, 03D15, 03D25, 03D28, 03D30.

Probabilistic algorithmic randomness

We introduce martingales defined by probabilistic strategies, in which random bits are used to decide whether and how much to bet. We show that different criteria for the success of computable probabilistic strategies can be used to characterize ML-randomness, computable randomness, and partial computable randomness. Our characterization of MLrandomness partially addresses a critique of Schnorr...

Thema Algorithmic Randomness

We consider algorithmic randomness in the Cantor space C of the infinite binary sequences. By an algorithmic randomness concept one specifies a set of elements of C, each of which is assigned the property of being random. Miscellaneous notions from computability theory are used in the definitions of randomness concepts that are essentially rooted in the following three intuitive randomness requ...