Around Kolmogorov complexity: basic notions and results

Algorithmic information theory studies description complexity and randomness and is now a well known field of theoretical computer science and mathematical logic. There are several textbooks and monographs devoted to this theory where one can find the detailed exposition of many difficult results as well as historical references. However, it seems that a short survey of its basic notions and main results relating these notions to each other, is missing. This report attempts to fill this gap and covers the basic notions of algorithmic information theory: Kolmogorov complexity (plain, conditional, prefix), Solomonoff universal a priori probability, notions of randomness (Martin-L\"of randomness, Mises--Church randomness), effective Hausdorff dimension. We prove their basic properties (symmetry of information, connection between a priori probability and prefix complexity, criterion of randomness in terms of complexity, complexity characterization for effective dimension) and show some applications (incompressibility method in computational complexity theory, incompleteness theorems). It is based on the lecture notes of a course at Uppsala University given by the author

Subalgebras of FA-presentable algebras

Automatic presentations, also called FA-presentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of fundamental decision problems. This paper studies FA-presentable algebras. First, an example is given to show that the class of finitely generated FA-presentable algebras is not closed under forming finitely generated subalgebras, even within the class of algebras with only unary operations. However, it is proven that a finitely generated subalgebra of an FA-presentable algebra with a single unary operation is itself FA-presentable. Furthermore, it is proven that the class of unary FA-presentable algebras is closed under forming finitely generated subalgebras, and that the membership problem for such subalgebras is decidable.Comment: 19 pages, 6 figure

Mentoring Nurse Scientists to Meet Nursing Faculty Workforce Needs

Research indicates that mentoring has been highly effective in promoting faculty success. Strong mentors in the area of scholarship are extremely valuable for junior faculty, not only because of their research and academic expertise but also for their role modeling behaviors. This paper highlights key components of research mentoring used by a senior nursing faculty member. The senior faculty mentor and junior faculty mentee developed a common vision, relating to research interests in health promotion for vulnerable populations. Impact at the individual, school, university, and society level is discussed, and benefits of mentoring to meet nursing faculty workforce needs are emphasized

Polymers and Thermodynamics

Classic thermodynamic equilibrium considerations, supported by simple molecular models, may lead to useful predictions about phase relationships in partially miscible systems that contain polymers. How quantitative the prediction is depends on the amount of experimental information, as well as, on the complexity of the system. The solubility parameter and group contribution approaches present the first LeveL and allow a qualitative judgement whether a system is miscible or not. On this level, the entropy of mixing is not considered though it is higly important. A second, higher level of prediction is supplied by the Flory-Huggins-Staverman equation which permits estimations of the temperature and chain length dependence on the location of miscibility gaps. On this level the concentration ranges of partial miscibility are not well covered. Taking account of the ever present disparity in size and shape between molecules and repeat units improves the situation considerably and represents a third level of prediction. On this level the influence of pressure can reasonably accurately b~ dealt with. If predictions of a high precision are required, the present-day theory fails, even in the simple case of a linear, apolar homopolymer solution. Extensive measurements then remain needed to determine the many empirical and theoretical parameters. Predictions on such a high level have amore than academic value, since they may supply better mathematical frameworks to be applied in less demanding calculations

On Resource-bounded versions of the van Lambalgen theorem

The van Lambalgen theorem is a surprising result in algorithmic information theory concerning the symmetry of relative randomness. It establishes that for any pair of infinite sequences $A$ and $B$, $B$ is Martin-L\"of random and $A$ is Martin-L\"of random relative to $B$ if and only if the interleaved sequence $A \uplus B$ is Martin-L\"of random. This implies that $A$ is relative random to $B$ if and only if $B$ is random relative to $A$ \cite{vanLambalgen}, \cite{Nies09}, \cite{HirschfeldtBook}. This paper studies the validity of this phenomenon for different notions of time-bounded relative randomness. We prove the classical van Lambalgen theorem using martingales and Kolmogorov compressibility. We establish the failure of relative randomness in these settings, for both time-bounded martingales and time-bounded Kolmogorov complexity. We adapt our classical proofs when applicable to the time-bounded setting, and construct counterexamples when they fail. The mode of failure of the theorem may depend on the notion of time-bounded randomness

Phase Transition and Strong Predictability

The statistical mechanical interpretation of algorithmic information theory (AIT, for short) was introduced and developed in our former work [K. Tadaki, Local Proceedings of CiE 2008, pp.425-434, 2008], where we introduced the notion of thermodynamic quantities into AIT. These quantities are real functions of temperature T>0. The values of all the thermodynamic quantities diverge when T exceeds 1. This phenomenon corresponds to phase transition in statistical mechanics. In this paper we introduce the notion of strong predictability for an infinite binary sequence and then apply it to the partition function Z(T), which is one of the thermodynamic quantities in AIT. We then reveal a new computational aspect of the phase transition in AIT by showing the critical difference of the behavior of Z(T) between T=1 and T<1 in terms of the strong predictability for the base-two expansion of Z(T).Comment: 5 pages, LaTeX2e, no figure