Understanding maximal repetitions in strings

The cornerstone of any algorithm computing all repetitions in a string of length n in O(n) time is the fact that the number of runs (or maximal repetitions) is O(n). We give a simple proof of this result. As a consequence of our approach, the stronger result concerning the linearity of the sum of exponents of all runs follows easily

The Thermodynamics of Network Coding, and an Algorithmic Refinement of the Principle of Maximum Entropy

The principle of maximum entropy (Maxent) is often used to obtain prior probability distributions as a method to obtain a Gibbs measure under some restriction giving the probability that a system will be in a certain state compared to the rest of the elements in the distribution. Because classical entropy-based Maxent collapses cases confounding all distinct degrees of randomness and pseudo-randomness, here we take into consideration the generative mechanism of the systems considered in the ensemble to separate objects that may comply with the principle under some restriction and whose entropy is maximal but may be generated recursively from those that are actually algorithmically random offering a refinement to classical Maxent. We take advantage of a causal algorithmic calculus to derive a thermodynamic-like result based on how difficult it is to reprogram a computer code. Using the distinction between computable and algorithmic randomness we quantify the cost in information loss associated with reprogramming. To illustrate this we apply the algorithmic refinement to Maxent on graphs and introduce a Maximal Algorithmic Randomness Preferential Attachment (MARPA) Algorithm, a generalisation over previous approaches. We discuss practical implications of evaluation of network randomness. Our analysis provides insight in that the reprogrammability asymmetry appears to originate from a non-monotonic relationship to algorithmic probability. Our analysis motivates further analysis of the origin and consequences of the aforementioned asymmetries, reprogrammability, and computation.Comment: 30 page

Algorithmic Complexity for Short Binary Strings Applied to Psychology: A Primer

Since human randomness production has been studied and widely used to assess executive functions (especially inhibition), many measures have been suggested to assess the degree to which a sequence is random-like. However, each of them focuses on one feature of randomness, leading authors to have to use multiple measures. Here we describe and advocate for the use of the accepted universal measure for randomness based on algorithmic complexity, by means of a novel previously presented technique using the the definition of algorithmic probability. A re-analysis of the classical Radio Zenith data in the light of the proposed measure and methodology is provided as a study case of an application.Comment: To appear in Behavior Research Method

Protein Repeats from First Principles

Some natural proteins display recurrent structural patterns. Despite being highly similar at the tertiary structure level, repeating patterns within a single repeat protein can be extremely variable at the sequence level. We use a mathematical definition of a repetition and investigate the occurrences of these in sequences of different protein families. We found that long stretches of perfect repetitions are infrequent in individual natural proteins, even for those which are known to fold into structures of recurrent structural motifs. We found that natural repeat proteins are indeed repetitive in their families, exhibiting abundant stretches of 6 amino acids or longer that are perfect repetitions in the reference family. We provide a systematic quantification for this repetitiveness. We show that this form of repetitiveness is not exclusive of repeat proteins, but also occurs in globular domains. A by-product of this work is a fast quantification of the likelihood of a protein to belong to a family.Fil: Turjanski, Pablo Guillermo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Parra, Rodrigo Gonzalo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Química Biológica de la Facultad de Ciencias Exactas y Naturales. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Química Biológica de la Facultad de Ciencias Exactas y Naturales; ArgentinaFil: Espada, Rocío. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Química Biológica de la Facultad de Ciencias Exactas y Naturales. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Química Biológica de la Facultad de Ciencias Exactas y Naturales; ArgentinaFil: Becher, Veronica Andrea. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Ferreiro, Diego. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Química Biológica de la Facultad de Ciencias Exactas y Naturales. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Química Biológica de la Facultad de Ciencias Exactas y Naturales; Argentin