This is a dissertation in the field of Mathematics: Logic: Computability Theory: Algorithmic Randomness (Mathematics Subject Classification 03D80, 68Q30). Its focus is relative randomness as measured by rK-reducibility, a refinement of Turing reducibility defined as follows. An infinite binary sequence A is rK-reducible to an infinite binary sequence B, written A ≤rK B, if ∃d ∀n . K(A n|B n) < ...

We consider the situation for public-key encryption that the adversary knows the randomness which was used to compute the ciphertext. In some practical scenarios, there is a possibility that the randomness is revealed. For example, the randomness used to make a ciphertext may be stored in insecure memory, or the pseudorandom generator may be corrupted. We first formalize the security notion on ...

In the theory of algorithmic randomness, several notions of random sequence are defined via a game-theoretic approach, and the notions that received most attention are perhaps Martin-Löf randomness and computable randomness. The latter notion was introduced by Schnorr and is rather natural: an infinite binary sequence is computably random if no total computable strategy succeeds on it by bettin...

We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [14] (also Problem 5.5.19 in [33]). We fully characterize lowness for Demuth randomness using an appropriate notion of traceability. Central to this characterization is a partial relativization of Demuth randomness, which may be more natural than the fully relativized version. We also sho...

This article is a brief personal account of the past, present, and future of algorithmic randomness, emphasizing its role in inductive inference and artificial intelligence. It is written for a general audience interested in science and philosophy. Intuitively, randomness is a lack of order or predictability. If randomness is the opposite of determinism, then algorithmic randomness is the oppos...

Randomness is a fundamental concept, with implications from security of modern data systems, to fundamental laws of nature and even the philosophy of science. Randomness is called certified if it describes events that cannot be pre-determined by an external adversary. It is known that weak certified randomness can be amplified to nearly ideal randomness using quantum-mechanical systems. However...

Algorithmic randomness was originally defined for Cantor space with the fair-coin measure. Recent work has examined algorithmic randomness in new contexts, in particular closed subsets of 2ω ([2] and [8]). In this paper we use the probability theory of closed set-valued random variables (RACS) to extend the definition of Martin-Löf randomness to spaces of closed subsets of locally compact, Haus...

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...

We show that van Lambalgen’s Theorem fails with respect to recursive randomness and Schnorr randomness for some real in every high degree and provide a full characterization of the Turing degrees for which van Lambalgen’s Theorem can fail with respect to Kurtz randomness. However, we also show that there is a recursively random real that is not Martin-Löf random for which van Lambalgen’s Theore...