Coding-theorem Like Behaviour and Emergence of the Universal Distribution from Resource-bounded Algorithmic Probability

Previously referred to as `miraculous' in the scientific literature because of its powerful properties and its wide application as optimal solution to the problem of induction/inference, (approximations to) Algorithmic Probability (AP) and the associated Universal Distribution are (or should be) of the greatest importance in science. Here we investigate the emergence, the rates of emergence and convergence, and the Coding-theorem like behaviour of AP in Turing-subuniversal models of computation. We investigate empirical distributions of computing models in the Chomsky hierarchy. We introduce measures of algorithmic probability and algorithmic complexity based upon resource-bounded computation, in contrast to previously thoroughly investigated distributions produced from the output distribution of Turing machines. This approach allows for numerical approximations to algorithmic (Kolmogorov-Chaitin) complexity-based estimations at each of the levels of a computational hierarchy. We demonstrate that all these estimations are correlated in rank and that they converge both in rank and values as a function of computational power, despite fundamental differences between computational models. In the context of natural processes that operate below the Turing universal level because of finite resources and physical degradation, the investigation of natural biases stemming from algorithmic rules may shed light on the distribution of outcomes. We show that up to 60\% of the simplicity/complexity bias in distributions produced even by the weakest of the computational models can be accounted for by Algorithmic Probability in its approximation to the Universal Distribution.Comment: 27 pages main text, 39 pages including supplement. Online complexity calculator: http://complexitycalculator.com

A Survey of Cellular Automata: Types, Dynamics, Non-uniformity and Applications

Cellular automata (CAs) are dynamical systems which exhibit complex global behavior from simple local interaction and computation. Since the inception of cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention of several researchers over various backgrounds and fields for modelling different physical, natural as well as real-life phenomena. Classically, CAs are uniform. However, non-uniformity has also been introduced in update pattern, lattice structure, neighborhood dependency and local rule. In this survey, we tour to the various types of CAs introduced till date, the different characterization tools, the global behaviors of CAs, like universality, reversibility, dynamics etc. Special attention is given to non-uniformity in CAs and especially to non-uniform elementary CAs, which have been very useful in solving several real-life problems.Comment: 43 pages; Under review in Natural Computin

Stability and Complexity of Minimising Probabilistic Automata

We consider the state-minimisation problem for weighted and probabilistic automata. We provide a numerically stable polynomial-time minimisation algorithm for weighted automata, with guaranteed bounds on the numerical error when run with floating-point arithmetic. Our algorithm can also be used for "lossy" minimisation with bounded error. We show an application in image compression. In the second part of the paper we study the complexity of the minimisation problem for probabilistic automata. We prove that the problem is NP-hard and in PSPACE, improving a recent EXPTIME-result.Comment: This is the full version of an ICALP'14 pape

On the Algorithmic Nature of the World

We propose a test based on the theory of algorithmic complexity and an experimental evaluation of Levin's universal distribution to identify evidence in support of or in contravention of the claim that the world is algorithmic in nature. To this end we have undertaken a statistical comparison of the frequency distributions of data from physical sources on the one hand--repositories of information such as images, data stored in a hard drive, computer programs and DNA sequences--and the frequency distributions generated by purely algorithmic means on the other--by running abstract computing devices such as Turing machines, cellular automata and Post Tag systems. Statistical correlations were found and their significance measured.Comment: Book chapter in Gordana Dodig-Crnkovic and Mark Burgin (eds.) Information and Computation by World Scientific, 2010. (http://www.idt.mdh.se/ECAP-2005/INFOCOMPBOOK/). Paper website: http://www.mathrix.org/experimentalAIT

The Bottom-Up Position Tree Automaton, the Father Automaton and their Compact Versions

The conversion of a given regular tree expression into a tree automaton has been widely studied. However, classical interpretations are based upon a Top-Down interpretation of tree automata. In this paper, we propose new constructions based on the Gluskov's one and on the one of Ilie and Yu one using a Bottom-Up interpretation. One of the main goals of this technique is to consider as a next step the links with deterministic recognizers, consideration that cannot be performed with classical Top-Down approaches. Furthermore, we exhibit a method to factorize transitions of tree automata and show that this technique is particularly interesting for these constructions, by considering natural factorizations due to the structure of regular expression.Comment: extended version of a paper accepted at CIAA 201

Bridging the Gap between Enumerative and Symbolic Model Checkers

We present a method to perform symbolic state space generation for languages with existing enumerative state generators. The method is largely independent from the chosen modelling language. We validated this on three different types of languages and tools: state-based languages (PROMELA), action-based process algebras (muCRL, mCRL2), and discrete abstractions of ODEs (Maple).\ud Only little information about the combinatorial structure of the\ud underlying model checking problem need to be provided. The key enabling data structure is the "PINS" dependency matrix. Moreover, it can be provided gradually (more precise information yield better results).\ud \ud Second, in addition to symbolic reachability, the same PINS matrix contains enough information to enable new optimizations in state space generation (transition caching), again independent from the chosen modelling language. We have also based existing optimizations, like (recursive) state collapsing, on top of PINS and hint at how to support partial order reduction techniques.\ud \ud Third, PINS allows interfacing of existing state generators to, e.g., distributed reachability tools. Thus, besides the stated novelties, the method we propose also significantly reduces the complexity of building modular yet still efficient model checking tools.\ud \ud Our experiments show that we can match or even outperform existing tools by reusing their own state generators, which we have linked into an implementation of our ideas