Algorithmically random series and Brownian motion

We consider some random series parametrised by Martin-Löf random sequences. The simplest case is that of Rademacher series, independent of a time parameter. This is then extended to the case of Fourier series on the circle with Rademacher coefficients. Finally, a specific Fourier series which has coefficients determined by a computable function is shown to converge to an algorithmically random Brownian motion.     

Rat pups and random robots generate similar self‐organized and intentional behavior

Biorobotic research continually demonstrates that behavior and cognition can be the emergent products of (1) embodied agents that are (2) dynamically embedded within an environment and (3) equipped with simple sensorimotor rules. Thigmotaxis is an orientation response to contact manifested in infant rats by wall following, corner burrowing, and group aggregation. Orientation responses have been long thought to be mediated only by sensory or central processes. Here we show that a random control architecture in a morphologically similar robot embedded in a scaled environment can reproduce thigmotaxic behaviors seen in infant rats. We conclude that (1) and (2) may play a larger role than previously thought in the generation of complex behaviors. © 2006 Wiley Periodicals, Inc. Complexity 12: 53–66, 2006     

C0复杂度的数学基础

对于许多同时具有强烈非线性和非平稳性的连续生物医学信号来说，计算其复杂度往往要求：1）在数据长度比较短的情况下也可以得出比较鲁棒的估计值；2）无需对原始信号作像二值化这样的过分的粗粒化，我们以前所提出的C0复杂度就是这样的一种度量，但是这种度量缺乏严格的数学基础，因而影响到它的应用，提出了一种改进形式，并严格证明了它的重要性质。从而表明这个量在一定条件下可以作为时间序列随机程度的指标，因而在随机性复杂度的意义下也可作为复杂性的一个定量指标，由于这个量有计算速度快的优点，因此特别适合于一些需要大量计算复杂度的场合，例如计算长时间过程中滑动窗口中复杂度的动态变化。     

Order-free Recursion on the Real Numbers

We investigate operations, computable in the sense of Recursive Analysis, which can be generated recursively from the arithmetic operations and the limit operation without using any tests on the real numbers. These functions, called order-free recursive, can be shown to include a large class of computable functions. As main tools we provide an effective version of the Stone-Weierstraß Approximation Theorem, as well as recursive partitions of unity.     

Correlated randomness: Some examples of exotic statistical physics

One challenge of biology, medicine, and economics is that the systems treated by these sciences have no perfect metronome in time and no perfect spatial architecture-crystalline or otherwise. Nonetheless, as if by magic, out of nothing but randomness one finds remarkably fine-tuned processes in time and remarkably fine-tuned structures in space. To understand this ‘miracle’, one might consider placing aside the human tendency to see the universe as a machine. Instead, one might address the challenge of uncovering how, through randomness (albeit, as we shall see, strongly correlated randomness), one can arrive at many spatial and temporal patterns in biology, medicine, and economics. Inspired by principles developed by statistical physics over the past 50 years-scale invariance and universality-we review some recent applications of correlated randomness to fields that might startle Boltzmann if he were alive today.     

Analyse de complexité pour un théorème de Hall sur les fractions continues

We give a polynomial time controlled version of a theorem of M. Hall: every real number can be written as the sum of two irrational numbers whose developments into a continued fraction contain only 1, 2, 3 or 4. Mathematics Subject Classification: 03D15, 03F60, 11A55, 68Q25.     

The Ackermann function. a theoretical,computational, and formula manipulative study

Ackermann's function is of highly recursive nature and of two arguments. It is here treated as a class of functions of one argument, where the other argument defines the member of the class. The first members are expressed with elementary functions, the higher members with a hierarchy of primitive recursive functions. The number of calls of the function needed in a straightforward recursive computation is given for the first members. The maximum depth in the recursion during the evaluation is investigated.Results from tests with the Ackermann function of recursive procedure implementations in ALGOL-60, ALGOL W, PL/I and SIMULA-67 on IBM 360/75 and CD 6600 are given.A SYMBAL formula manipulating program, that automatically solves recurrence relations for the first members of the function class and for the number of calls needed in their straightforward computation, is given.The Ackermann rating of programming languages is discussed.     

Quasi‐carousel tournaments

A tournament is called locally transitive if the outneighborhood and the inneighborhood of every vertex are transitive. Equivalently, a tournament is locally transitive if it avoids the tournaments W₄ and L₄, which are the only tournaments up to isomorphism on four vertices containing a unique 3‐cycle. On the other hand, a sequence of tournaments  with  is called almost balanced if all but  vertices of  have outdegree . In the same spirit of quasi‐random properties, we present several characterizations of tournament sequences that are both almost balanced and asymptotically locally transitive in the sense that the density of W₄ and L₄ in  goes to zero as n goes to infinity.     

A PGM framework for recursive modeling of players in simple sequential Bayesian games

We consider the situation where two agents try to solve each their own task in a common environment. In particular, we study simple sequential Bayesian games with unlimited time horizon where two players share a visible scene, but where the tasks (termed assignments) of the players are private information. We present an influence diagram framework for representing simple type of games, where each player holds private information. The framework is used to model the analysis depth and time horizon of the opponent and to determine an optimal policy under various assumptions on analysis depth of the opponent. Not surprisingly, the framework turns out to have severe complexity problems even in simple scenarios due to the size of the relevant past. We propose two approaches for approximation. One approach is to use Limited Memory Influence Diagrams (LIMIDs) in which we convert the influence diagram into a set of Bayesian networks and perform single policy update. The other approach is information enhancement, where it is assumed that the opponent in a few moves will know your assignment. Empirical results are presented using a simple board game.     

Subgame-perfection in free transition games

We prove the existence of a subgame-perfect ε-equilibrium, for every ε > 0, in a class of multi-player games with perfect information, which we call free transition games. The novelty is that a non-trivial class of perfect information games is solved for subgame-perfection, with multiple non-terminating actions, in which the payoff structure is generally not (upper or lower) semi-continuous. Due to the lack of semi-continuity, there is no general rule of comparison between the payoffs that a player can obtain by deviating a large but finite number of times or, respectively, infinitely many times. We introduce new techniques to overcome this difficulty.