Natural Factors of the Medvedev Lattice Capturing IPC

Skvortsova showed that there is a factor of the Medvedev lattice which captures intuitionistic propositional logic (IPC). However, her factor is unnatural in the sense that it is constructed in an ad hoc manner. We present a more natural example of such a factor. We also show that for every non-trivial factor of the Medvedev lattice its theory is contained in Jankov's logic, the deductive closure of IPC plus the weak law of the excluded middle. This answers a question by Sorbi and Terwijn

Effective Genericity and Differentiability

We prove that a real x is 1-generic if and only if every differentiable computable function has continuous derivative at x. This provides a counterpart to recent results connecting effective notions of randomness with differentiability. We also consider multiply differentiable computable functions and polynomial time computable functions.Comment: Revision: added sections 6-8; minor correction

Monte Carlo Computability

We introduce Monte Carlo computability as a probabilistic concept of computability on infinite objects and prove that Monte Carlo computable functions are closed under composition. We then mutually separate the following classes of functions from each other: the class of multi-valued functions that are non-deterministically computable, that of Las Vegas computable functions, and that of Monte Carlo computable functions. We give natural examples of computational problems witnessing these separations. As a specific problem which is Monte Carlo computable but neither Las Vegas computable nor non-deterministically computable, we study the problem of sorting infinite sequences that was recently introduced by Neumann and Pauly. Their results allow us to draw conclusions about the relation between algebraic models and Monte Carlo computability

Model theory of measure spaces and probability logic

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