We investigate notions of randomness in the space C½2 N of non-empty closed subsets of f0, 1g N . A probability measure is given and a version of the Martin-Lo¨f test for randomness is defined. Å 0 2 random closed sets exist but there are no random Å 0 1 closed sets. It is shown that any random closed set is perfect, has measure 0, and has box dimension log 2 ð4=3Þ. A random closed set has no n-c.e. elements. A closed subset of 2 N may be defined as the set of infinite paths through a tree and so the problem of compressibility of trees is explored. If T n ¼ T \ f0, 1g n , then for any random closed set ½T where T has no dead ends, KðT n Þ ! n À Oð1Þ but for any k, KðT n Þ 2 nÀk þ Oð1Þ, where K() is the prefix-free complexity of 2 f0, 1g à .
Abstract. We study weak 2 randomness, weak randomness relative to ∅ and Schnorr randomness relative to ∅ . One major theme is characterizing the oracles A such that ML[A] ⊆ C, where C is a randomness notion and ML[A] denotes the Martin-Löf random reals relative to A. We discuss the connections with LR-reducibility and also study the reducibility associated with weak 2-randomness.
Abstract-Schelling's model of segregation looks to explain the way in which particles or agents of two types may come to arrange themselves spatially into configurations consisting of large homogeneous clusters, i.e. connected regions consisting of only one type. As one of the earliest agent based models studied by economists and perhaps the most famous model of self-organising behaviour, it also has direct links to areas at the interface between computer science and statistical mechanics, such as the Ising model and the study of contagion and cascading phenomena in networks.While the model has been extensively studied it has largely resisted rigorous analysis, prior results from the literature generally pertaining to variants of the model which are tweaked so as to be amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory. In [2], Brandt, Immorlica, Kamath and Kleinberg provided the first rigorous analysis of the unperturbed model, for a specific set of input parameters. Here we provide a rigorous analysis of the model's behaviour much more generally and establish some surprising forms of threshold behaviour, notably the existence of situations where an increased level of intolerance for neighbouring agents of opposite type leads almost certainly to decreased segregation.
Thomas Schelling's spacial proximity model illustrated how racial segregation can emerge, unwanted, from the actions of citizens acting in accordance with their individual local preferences. One of the earliest agent-based models, it is closely related both to the spin-1 models of statistical physics, and to cascading phenomena on networks. Here a 1-dimensional unperturbed variant of the model is studied, which is open in the sense that agents may enter and exit the model. Following the authors' previous work [1] and that of Brandt, Immorlica, Kamath, and Kleinberg in [4], rigorous asymptotic results are established.This model's openness allows either race to take over almost everywhere. Tipping points are identified between the regions of takeover and staticity. In a significant generalization of the models considered in [1] and [4], the model's parameters comprise the initial proportions of the two races, along with independent values of the tolerance for each race.
Schelling's models of segregation, first described in 1969 [18] are among the best known models of self-organising behaviour. Their original purpose was to identify mechanisms of urban racial segregation. But his models form part of a family which arises in statistical mechanics, neural networks, social science, and beyond, where populations of agents interact on networks. Despite extensive study, unperturbed Schelling models have largely resisted rigorous analysis, prior results generally focusing on variants in which noise is introduced into the dynamics, the resulting system being amenable to standard techniques from statistical mechanics or stochastic evolutionary game theory [25]. A series of recent papers [6,3,4], has seen the first rigorous analyses of 1-dimensional unperturbed Schelling models, in an asymptotic framework largely unknown in statistical mechanics. Here we provide the first such analysis of 2-and 3-dimensional unperturbed models, establishing most of the phase diagram, and answering a challenge from [6].
We say that A ≤LRB if every B-random number is A-random. Intuitively this means that if oracle A can identify some patterns on some real γ, oracle B can also find patterns on γ. In other words, B is at least as good as A for this purpose. We study the structure of the LR degrees globally and locally (i.e., restricted to the computably enumerable degrees) and their relationship with the Turing degrees. Among other results we show that whenever ∝ is not GL2 the LR degree of ∝ bounds degrees (so that, in particular, there exist LR degrees with uncountably many predecessors) and we give sample results which demonstrate how various techniques from the theory of the c.e. degrees can be used to prove results about the c.e. LR degrees.
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