Random Closed Sets

We give resource bounded versions of the Completeness Theorem for propositional and predicate logic. For example, it is well known that every computable consistent propositional theory has a computable complete consistent extension. We show that, when length is measured relative to the binary representation of natural numbers and formulas, every polynomial time decidable propositional theory has an exponential time (EXPTIME) complete consistent extension whereas there is a nondeterministic polynomial time (NP) decidable theory which has no polynomial time complete consistent extension when length is measured relative to the binary representation of natural numbers and formulas. It is well known that a propositional theory is axiomatizable (respectively decidable) if and only if it may be represented as the set of infinite paths through a computable tree (respectively a computable tree with no dead ends). We show that any polynomial time decidable theory may be represented as the set of paths through a polynomial time decidable tree. On the other hand, the statement that every polynomial time decidable tree represents the set of complete consistent extensions of some theory which is polynomial time decidable, relative to the tally representation of natural numbers and formulas, is equivalent to P = NP. For predicate logic, we develop a complexity theoretic version of the Henkin construction to prove a complexity theoretic version of the Completeness Theorem. Our results imply that that any polynomial space decidable theory Δ possesses a polynomial space computable model which is exponential space decidable and thus Δ has an exponential space complete consistent extension. Similar results are obtained for other notions of complexity.

Analogous to Kolmogorov’s theorem for the existence of stochastic processes describing random functions, we consider theorems for the existence of stochastic processes describing random measures as limits of inverse measure systems. Specifically, given a coherent inverse system of random (bounded/signed/positive/probability) histograms on refining partitions, we study conditions for the existence and uniqueness of a corresponding random inverse limit, a Radon probability measure on the space of (bounded/signed/positive/probability) measures. Depending on the topology (vague/tight/weak/total-variational) and Kingman’s notion of complete randomness, the limiting random measure is in one of four phases, distinguished by their degrees of concentration (support/domination/discreteness). The results are applied in the well-known Dirichlet and Polya tree families of random probability measures and a new Gaussian family of signed inverse limit measures. In these three families, examples of all four phases occur, and we describe the corresponding conditions of defining parameters.

The optimal transport problem aims at finding an optimal way to transport a given probability measure into another. In contrast to the well-known Monge–Kantorovich problem, the ramified optimal transportation problem aims at modeling a tree-typed branching transport network by an optimal transport path between two given probability measures. An essential feature of such a transport path is to favor group transportation in a large amount. In previous works, the author has studied the existence of optimal transport paths between probability measures as well as their interior regularity, that is away from the supports of the measures. Such an optimal transport path may be understood as a 1-dimensional rectifiable current that has boundary the difference of the 2 measures and that minimizes a suitable cost functional. In this article, we study the regularity of such an optimal transport path nearby its boundary. Motivated from observing the vein structure of a tree leaf, we show that each superlevel set of an optimal transport path is locally supported on a bi-Lipschitz graph, which is a finite union of bi-Lipschitz curves.

Probabilistic model checkers like PRISM check the satisfiability of probabilistic CTL (pCTL) formulas against discrete-time Markov chains. We prove soundness and completeness of their underlying algorithm in Isabelle/HOL. We define Markov chains given by a transition matrix and formalize the corresponding probability measure on sets of paths. The formalization of pCTL formulas includes unbounded cumulated rewards.KeywordsMarkov ChainModel CheckProbability SpaceProof AssistantLinear Equation SystemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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-Löf 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.

On Oscillation-free ε-random Sequences

The necessity of enforcing conservation in computational elements or cells (element conservation) for discontinuous solutions is well understood and respected for solving conservation laws in computational fluid dynamics (CFD). In contrast, interface conservation, where the conservation across cell interfaces is enforced is long ignored, and yet is also ruled and required by the underlying physics just like element conservation. Violation of the interface conservation across discontinuities is the root cause why an exact discontinuous solution can never be achieved in shock capturing methods. The interface conservation is examined and explored in this paper. An error indicator based on the interface conservation is then designed for h-adaptive grid methods to effectively detect and identify discontinuities, as the interface conservation can never be attained across the discontinuities. A moving discontinuous Galerkin (MDG) finite element method with interface conservation enforcement (MDG-ICE) is also developed for solving compressible flow problems with discontinuities using an operator splitting strategy in the spirit of r-adaptation methods, based on the observation that the interface conservation can only be satisfied, only when mesh interfaces are aligned with discontinuities. A number of numerical experiments are conducted to assess effectiveness of the interface-based error indicator and performance of the MDG-ICE method. Numerical results for a variety of discontinuous flows obtained indicate that the error indicator based on the interface conservation can effectively detect all types of discontinuities and under-resolved flow regions and the MDG-ICE method is able to reposition grid, align mesh interfaces with all types of discontinuities, and achieve the designed order of both h- and p-convergence even for discontinuous solutions.

We consider denumerable stochastic processes with (or without) memory. Their evolution is encoded by a finite or infinite rooted tree. The main goal is to compare the entropy rates of a given base process and a second one, to be considered as a perturbation of the former. The processes are described by probability measures on the boundary of the given tree and by corresponding forward transition probabilities at the inner nodes. The comparison is in terms of Kullback–Leibler divergence. We elaborate and extend ideas and results of Bocherer and Amjad. Our extensions involve length functions on the edges of the tree as well as nodes with countably many successors. In particular, in Section V , we consider trees with infinite nonbacktracking paths and random perturbations of a given process.

For a Polish Sample Space with a Borel σ-field with a surjective measurable transformation, we define an equivalence relation on sample points according to their ergodic limiting averages. We show that this equivalence relation partitions the subset of sample points on measurable invariant subsets, where each limiting distribution is the unique ergodic probability measure defined on each set. The results obtained suggest some natural objects for the model of a probabilistic time-invariant phenomenon are uniquely ergodic probability spaces. As a consequence of the results gained in this paper, we propose a notion of randomness that is weaker than recent approaches to Schnorr randomness.

Computing Probability Bounds for Linear Time Formulas over Concurrent Probabilistic Systems

There is only one fully supported ergodic invariant probability measure for the adic transformation on the space of infinite paths in the graph that underlies the Eulerian numbers. This result may partially justify a frequent assumption about the equidistribution of random permutations.