Uniform test of algorithmic randomness over a general space

The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These restrictions seem artificial. Some progress has been made to extend the theory to arbitrary Bernoulli distributions (by Martin-Loef), and to arbitrary distributions (by Levin). We recall the main ideas and problems of Levin's theory, and report further progress in the same framework. - We allow non-compact spaces (like the space of continuous functions, underlying the Brownian motion). - The uniform test (deficiency of randomness) d_P(x) (depending both on the outcome x and the measure P should be defined in a general and natural way. - We see which of the old results survive: existence of universal tests, conservation of randomness, expression of tests in terms of description complexity, existence of a universal measure, expression of mutual information as "deficiency of independence. - The negative of the new randomness test is shown to be a generalization of complexity in continuous spaces; we show that the addition theorem survives. The paper's main contribution is introducing an appropriate framework for studying these questions and related ones (like statistics for a general family of distributions).Comment: 40 pages. Journal reference and a slight correction in the proof of Theorem 7 adde

σ\sigma-locales in Formal Topology

A $\sigma$-frame is a poset with countable joins and finite meets in which binary meets distribute over countable joins. The aim of this paper is to show that $\sigma$-frames, actually $\sigma$-locales, can be seen as a branch of Formal Topology, that is, intuitionistic and predicative point-free topology. Every $\sigma$-frame $L$ is the lattice of Lindel\"of elements (those for which each of their covers admits a countable subcover) of a formal topology of a specific kind which, in its turn, is a presentation of the free frame over $L$. We then give a constructive characterization of the smallest (strongly) dense $\sigma$-sublocale of a given $\sigma$-locale, thus providing a ``$\sigma$-version'' of a Boolean locale. Our development depends on the axiom of countable choice.Comment: Paper presented at the conference Continuity, Computability, Constructivity - From Logic to Algorithms (CCC 2017), Nancy, France, June 26-30 201

Simulating Quantum Mechanics by Non-Contextual Hidden Variables

No physical measurement can be performed with infinite precision. This leaves a loophole in the standard no-go arguments against non-contextual hidden variables. All such arguments rely on choosing special sets of quantum-mechanical observables with measurement outcomes that cannot be simulated non-contextually. As a consequence, these arguments do not exclude the hypothesis that the class of physical measurements in fact corresponds to a dense subset of all theoretically possible measurements with outcomes and quantum probabilities that \emph{can} be recovered from a non-contextual hidden variable model. We show here by explicit construction that there are indeed such non-contextual hidden variable models, both for projection valued and positive operator valued measurements.Comment: 15 pages. Journal version. Only minor typo corrections from last versio

The internal description of a causal set: What the universe looks like from the inside

We describe an algebraic way to code the causal information of a discrete spacetime. The causal set C is transformed to a description in terms of the causal pasts of the events in C. This is done by an evolving set, a functor which to each event of C assigns its causal past. Evolving sets obey a Heyting algebra which is characterised by a non-standard notion of complement. Conclusions about the causal structure of the causal set can be drawn by calculating the complement of the evolving set. A causal quantum theory can be based on the quantum version of evolving sets, which we briefly discuss.Comment: Version to appear in Comm.Math.Phys. (minor modifications). 37 pages, several eps figure