An Algorithmic Approach to Information and Meaning

I will survey some matters of relevance to a philosophical discussion of information, taking into account developments in algorithmic information theory (AIT). I will propose that meaning is deep in the sense of Bennett's logical depth, and that algorithmic probability may provide the stability needed for a robust algorithmic definition of meaning, one that takes into consideration the interpretation and the recipient's own knowledge encoded in the story attached to a message.Comment: preprint reviewed version closer to the version accepted by the journa

Image Characterization and Classification by Physical Complexity

We present a method for estimating the complexity of an image based on Bennett's concept of logical depth. Bennett identified logical depth as the appropriate measure of organized complexity, and hence as being better suited to the evaluation of the complexity of objects in the physical world. Its use results in a different, and in some sense a finer characterization than is obtained through the application of the concept of Kolmogorov complexity alone. We use this measure to classify images by their information content. The method provides a means for classifying and evaluating the complexity of objects by way of their visual representations. To the authors' knowledge, the method and application inspired by the concept of logical depth presented herein are being proposed and implemented for the first time.Comment: 30 pages, 21 figure

Correlation of Automorphism Group Size and Topological Properties with Program-size Complexity Evaluations of Graphs and Complex Networks

We show that numerical approximations of Kolmogorov complexity (K) applied to graph adjacency matrices capture some group-theoretic and topological properties of graphs and empirical networks ranging from metabolic to social networks. That K and the size of the group of automorphisms of a graph are correlated opens up interesting connections to problems in computational geometry, and thus connects several measures and concepts from complexity science. We show that approximations of K characterise synthetic and natural networks by their generating mechanisms, assigning lower algorithmic randomness to complex network models (Watts-Strogatz and Barabasi-Albert networks) and high Kolmogorov complexity to (random) Erdos-Renyi graphs. We derive these results via two different Kolmogorov complexity approximation methods applied to the adjacency matrices of the graphs and networks. The methods used are the traditional lossless compression approach to Kolmogorov complexity, and a normalised version of a Block Decomposition Method (BDM) measure, based on algorithmic probability theory.Comment: 15 2-column pages, 20 figures. Forthcoming in Physica A: Statistical Mechanics and its Application

Global and local Complexity in weakly chaotic dynamical systems

In a topological dynamical system the complexity of an orbit is a measure of the amount of information (algorithmic information content) that is necessary to describe the orbit. This indicator is invariant up to topological conjugation. We consider this indicator of local complexity of the dynamics and provide different examples of its behavior, showing how it can be useful to characterize various kind of weakly chaotic dynamics. We also provide criteria to find systems with non trivial orbit complexity (systems where the description of the whole orbit requires an infinite amount of information). We consider also a global indicator of the complexity of the system. This global indicator generalizes the topological entropy, taking into account systems were the number of essentially different orbits increases less than exponentially. Then we prove that if the system is constructive (roughly speaking: if the map can be defined up to any given accuracy using a finite amount of information) the orbit complexity is everywhere less or equal than the generalized topological entropy. Conversely there are compact non constructive examples where the inequality is reversed, suggesting that this notion comes out naturally in this kind of complexity questions.Comment: 23 page