When there is tax evasion, increased randomness about how much taxable income an auditor would assess generally leads to higher reported income and more revenue. When reducing randomness is costly, optimality requires some randomness in assessed taxable income. Even if reducing randomness is costless, taxpayers may prefer some randomness when the increased revenue can be rebated, so that the go...

This paper uses quick process machines to provide characterisations of computable randomness, Schnorr randomness and weak randomness. The quick process machine is a type of process machine first considered in work of Levin and Zvonkin. A new technique for building process machines and quick process machines is presented. This technique is similar to the KC theorem for prefix-free machines. Usin...

1 Computability and randomness 1 1.1 Studying randomness notions . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Martin-Löf’s randomness notion . . . . . . . . . . . . . . . . . 2 1.1.2 Notions weaker than ML-randomness . . . . . . . . . . . . . . 2 1.1.3 A notion stronger than ML-randomness . . . . . . . . . . . . . 3 1.2 Lowness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

We study randomness notions given by higher recursion theory, establishing the relationships Π1-randomness ⊂ Π1-Martin-Löf randomness ⊂ ∆1randomness = ∆1-Martin-Löf randomness. We characterize the set of reals that are low for ∆1 randomness as precisely those that are ∆1 -traceable. We prove that there is a perfect set of such reals.

We prove that there exists a noncomputable c.e. real which is low for weak 2-randomness, a definition of randomness due to Kurtz, and that all reals which are low for weak 2-randomness are low for Martin-Löf randomness.

Common misconceptions about randomness underlie the design and implementation of randomness sources in popular operating systems. We debunk these fallacies with a survey of the “realities of randomness” and derive a number of new architectural principles for OS randomness subsystems.