Random numbers as probabilities of machine behaviour

TL;DR

This paper explores how probabilities of certain events in Turing machine computations relate to algorithmic randomness, especially in higher randomness classes, revealing complex interactions between arithmetical complexity and randomness strength.

Contribution

It characterizes higher randomness classes as probabilities of events in probabilistic oracle Turing machines and analyzes how these probabilities reflect arithmetical complexity and machine models.

Findings

Probabilities of events can reflect arithmetical complexity in many cases.

Not all probabilities of complex properties are maximally random.

Probabilities of properties like totality vary across different universal machines.

Abstract

A fruitful way of obtaining meaningful, possibly concrete, algorithmically random numbers is to consider a potential behaviour of a Turing machine and its probability with respect to a measure (or semi-measure) on the input space of binary codes. For example, Chaitin's Omega is a well known Martin-Loef random number that is obtained by considering the halting probability of a universal prefix-free machine. In the last decade, similar examples have been obtained for higher forms of randomness, i.e. randomness relative to strong oracles. In this work we obtain characterizations of the algorithmically random reals in higher randomness classes, as probabilities of certain events that can happen when an oracle universal machine runs probabilistically on a random oracle. Moreover we apply our analysis to different machine models, including oracle Turing machines, prefix-free machines, and models for infinite online computation. We find that in many cases the arithmetical complexity of a property is directly reflected in the strength of the algorithmic randomness of the probability with which it occurs, on any given universal machine. On the other hand, we point to many examples where this does not happen and the probability is a number whose algorithmic randomness is not the maximum possible (with respect to its arithmetical complexity). Finally we find that, unlike the halting probability of a universal machine, the probabilities of more complex properties like totality, cofinality, computability or completeness do not necessarily have the same Turing degree when they are defined with respect to different universal machines.