Christopher P. Porter

Strong reductions in effective randomness

We study generalizations of Demuths Theorem, which states that the image of a Martin-Lof random real under a tt-reduction is either computable or Turing equivalent to a Martin-Lof random real. We show that Demuths Theorem holds for Schnorr randomness and computable randomness (answering a question of Franklin), but that it cannot be strengthened by replacing the Turing equivalence in the statement of the theorem with wtt-equivalence. We also provide some additional results about the Turing and tt-degrees of reals that are random with respect to some computable measure.

Randomness and Semimeasures

A semimeasure is a generalization of a probability measure obtained by relaxing the additivity requirement to superadditivity. We introduce and study several randomness notions for left-c.e. semimeasures, a natural class of effectively approximable semimeasures induced by Turing functionals. Among the randomness notions we consider, the generalization of weak 2-randomness to left-c.e. semimeasures is the most compelling, as it best reflects Martin-Lof randomness with respect to a computable measure. Additionally, we analyze a question of Shen, a positive answer to which would also have yielded a reasonable randomness notion for left-c.e. semimeasures. Unfortunately, though, we find a negative answer, except for some special cases.

Random numbers as probabilities of machine behavior

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, Chaitins 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.

The Probability of a Computable Output from a Random Oracle

Consider a universal oracle Turing machine that prints a finite or an infinite binary sequence, based on the answers to the binary queries that it makes during the computation. We study the probability that this output is infinite and computable when the machine is given a random (in the probabilistic sense) stream of bits as the answers to its queries during an infinitary computation. Surprisingly, we find that these probabilities are the entire class of real numbers in (0,1) that can be written as the difference of two halting probabilities relative to the halting problem. In particular, there are universal Turing machines that produce a computable infinite output with probability exactly 1/2. Our results contrast a large array of facts (the most well-known being the randomness of Chaitin’s halting probability) that witness maximal initial segment complexity of probabilities associated with universal machines. Our proof uses recent advances in algorithmic randomness.

Algorithmically Random Functions and Effective Capacities

We continue the investigation of algorithmically random functions and closed sets, and in particular the connection with the notion of capacity. We study notions of random continuous functions given in terms of a family of computable measures called symmetric Bernoulli measures. We isolate one particular class of random functions that we refer to as random online functions \(F\), where the value of \(y(n)\) for \(y = F(x)\) may be computed from the values of \(x(0),\dots ,x(n)\). We show that random online functions are neither onto nor one-to-one. We give a necessary condition on the members of the ranges of random online functions in terms of initial segment complexity and the associated computable capacity. Lastly, we introduce the notion of Martin-Lof random online partial function on \(2^\omega \) and give a family of online partial random functions the ranges of which are precisely the random closed sets introduced in [2].

Trivial Measures are not so Trivial

Although algorithmic randomness with respect to various non-uniform computable measures is well-studied, little attention has been paid to algorithmic randomness with respect to computable trivial measures, where a measure μ on 2ω is trivial if the support of μ consists of a countable collection of sequences. In this article, it is shown that there is much more structure to trivial computable measures than has been previously suspected.

The Random Members of a

We examine several notions of randomness for elements in a given Π10

{\Pi }_{1}^{0}

{\Pi }_{1}^{0}

Class

class P

Randomness for computable measures and initial segment complexity

\mathcal {P}