Natural density and probability, constructively

We give here a constructive account of the frequentist approach to probability, by means of natural density. Using this notion of natural density, we introduce some probabilistic versions of the Limited Principle of Omniscience. Finally we give an attempt general definition of probability structure which is pointfree and takes into account abstractely the process of probability assignment

Verification of Information Flow Properties under Rational Observation

Information flow properties express the capability for an agent to infer information about secret behaviours of a partially observable system. In a language-theoretic setting, where the system behaviour is described by a language, we define the class of rational information flow properties (RIFP), where observers are modeled by finite transducers, acting on languages in a given family $\mathcal{L}$. This leads to a general decidability criterion for the verification problem of RIFPs on $\mathcal{L}$, implying PSPACE-completeness for this problem on regular languages. We show that most trace-based information flow properties studied up to now are RIFPs, including those related to selective declassification and conditional anonymity. As a consequence, we retrieve several existing decidability results that were obtained by ad-hoc proofs.Comment: 19 pages, 7 figures, version extended from AVOCS'201

Automatic sets of rational numbers

The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability.Comment: Previous version appeared in Proc. LATA 2012 conferenc

On Chaotic Dynamics in Rational Polygonal Billiards

We discuss the interplay between the piece-line regular and vertex-angle singular boundary effects, related to integrability and chaotic features in rational polygonal billiards. The approach to controversial issue of regular and irregular motion in polygons is taken within the alternative deterministic and stochastic frameworks. The analysis is developed in terms of the billiard-wall collision distribution and the particle survival probability, simulated in closed and weakly open polygons, respectively. In the multi-vertex polygons, the late-time wall-collision events result in the circular-like regular periodic trajectories (sliding orbits), which, in the open billiard case are likely transformed into the surviving collective excitations (vortices). Having no topological analogy with the regular orbits in the geometrically corresponding circular billiard, sliding orbits and vortices are well distinguished in the weakly open polygons via the universal and non-universal relaxation dynamics.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

A column of grains in the jamming limit: glassy dynamics in the compaction process

We investigate a stochastic model describing a column of grains in the jamming limit, in the presence of a low vibrational intensity. The key control parameter of the model, $\epsilon$, is a representation of granular shape, related to the reduced void space. Regularity and irregularity in grain shapes, respectively corresponding to rational and irrational values of $\epsilon$, are shown to be centrally important in determining the statics and dynamics of the compaction process.Comment: 29 pages, 14 figures, 1 table. Various minor changes and updates. To appear in EPJ

The invariant tori of knot type and the interlinked invariant tori in the Nos\'e-Hoover system

We revisit the famous Nos\'e-Hoover system in this paper and show the existence of some averagely conservative regions which are filled with an infinite sequence of nested tori. Depending on initial conditions, some invariant tori are of trefoil knot type, while the others are of trivial knot type. Moreover, we present a variety of interlinked invariant tori whose initial conditions are chosen from different averagely conservative regions and give all the interlinking numbers of those interlinked tori, showing that this quadratic system possesses so rich dynamic properties.Comment: 8 pages,8 figure