Mutual independence and algorithmic randomnessPustoslemšek, Andraž (Avtor) Simpson, Alexander Keith (Mentor) Turing machineinfinite binary sequenceprefixprefix-free Turing reducibilityoracleCantor spacecomputably open setpushforward measureMartin-Löf random sequenceproper sequencemutual independenceThe Swedish mathematician Per Martin-Löf defined in 1966 that a sequence is random if it passes all so-called Martin-Löf tests. We would like to seemingly strengthen this definition and say that a random sequence $X$ should not only pass all computable Martin-Löf tests, but also all $Y$-computable Martin-Löf tests, for any oracle $Y$ that is independent of $X$. To do this, we need a suitable notion of independence between infinite sequences. In this thesis, we define such a notion. Indeed, more generally, we define a relation of conditional mutual independence between sequences, and we show that this relation possesses many desirable properties. Furthermore, with this definition, Martin-Löf random sequences automatically satisfy the desired oracle-strengthened definition of randomness. In building towards these results, we shall recall some key statements from computability theory and prove important theorems from the field of algorithmic randomness.20252025-07-02 15:27:43Magistrsko delo/naloga170216UDK: 510.6VisID: 150686COBISS_ID: 240937987sl