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].

In the distributed computing literature, consensus protocols have traditionally been studied in a setting where all participants are known to each other from the start of the protocol execution. In the parlance of the 'blockchain' literature, this is referred to as the permissioned setting. What differentiates the most prominent blockchain protocol Bitcoin [21] from these previously studied protocols is that it operates in a permissionless setting, i.e. it is a protocol for establishing consensus over an unknown network of participants that anybody can join, with as many identities as they like in any role. The arrival of this new form of protocol brings with it many questions. Beyond Bitcoin, what can we prove about permissionless protocols in a general sense? How does recent work on permissionless protocols in the blockchain literature relate to the well-developed history of research on permissioned protocols in distributed computing?To answer these questions, we describe a formal framework for the analysis of both permissioned and permissionless systems. Our framework allows for "apples-to-apples" comparisons between different categories of protocols and, in turn, the development of theory to formally discuss their relative merits. A major benefit of the framework is that it facilitates the application of a rich history of proofs and techniques in distributed computing to problems in blockchain and the study of permissionless systems. Within our framework, we then address the questions above. We consider the Byzantine Generals Problem [19,25] as a formalisation of the problem of reaching consensus, and address a programme of research that asks, "Under what adversarial conditions, and for what types of permissionless protocol, is consensus possible?" We prove a number of results for this programme, our main result being that deterministic consensus is not possible for decentralised permissionless protocols. To close, we give a list of seven open questions.

Abstract. We show that the index set complexity of the computably categorical structures is Π 1 1 -complete, demonstrating that computable categoricity has no simple syntactic characterization. As a consequence of our proof, we exhibit, for every computable ordinal α, a computable structure that is computably categorical but not relatively ∆ 0 α -categorical.

It is a classic result in algorithmic information theory that every infinite binary sequence is computable from an infinite binary sequence which is random in the sense of Martin-Löf. Proved independently by Kučera [Kuč85] and Gács [Gác86], this result answered a question by Charles Bennett and has seen numerous applications in the last 30 years. The optimal redundancy in such a coding process has, however, remained unknown. If the computation of the first n bits of a sequence requires n + g(n) bits of the random oracle, then g is the redundancy of the computation. Kučera implicitly achieved redundancy n log n while Gács used a more elaborate block-coding procedure which achieved redundancy √ n log n. Merkle and Mihailović [MM04] provided a different presentation of Gács' approach, without improving his redundancy bound. In this paper we devise a new coding method that achieves optimal logarithmic redundancy. For any computable non-decreasing function g such that i 2 −g(i) is bounded we show that there is a coding process that codes any given infinite binary sequence into a Martin-Löf random infinite binary sequence with redundancy g. This redundancy bound is exponentially smaller than the previous bound of √ n log n and is known to be the best possible by recent work [BLPT16], where it was shown that if i 2 −g(i) diverges then there exists an infinite binary sequence X which cannot be computed by any Martin-Löf random infinite binary sequence with redundancy g. It follows that redundancy ǫ · log n in computation from a random oracle is possible for every infinite binary sequence, if and only if ǫ > 1.

The halting probabilities of universal prefix-free machines are universal for the class of reals with computably enumerable left cut (also known as left-c.e. reals), and coincide with the Martin-Löf random elements of this class. We study the differences of Martin-Löf random left-c.e. reals and show that for each pair of such reals α, β there exists a unique number r > 0 such that qα − β is a Martin-Löf random left-c.e. real for each positive rational q > r and a Martin-Löf random right-c.e. real for each positive rational q < r. Based on this result we develop a theory of differences of halting probabilities, which answers a number of questions about Martin-Löf random left-c.e. reals, including one of the few remaining open problems from the list of open questions in algorithmic randomness [MN06].The halting probability of a prefix-free machine M restricted to a set X is the probability that the machine halts and outputs an element of X. These numbers Ω M (X) were studied in [BG05, BFGM06, BG07, BG09] as a way to obtain concrete highly random numbers. When X is a Π 0 1 set, the number Ω M (X) is the difference of two halting probabilities. Becher, Figueira, Grigorieff, and Miller asked whether Ω U (X) is Martin-Löf random when U is universal and X is a Π 0 1 set. This problem has resisted numerous attempts [BG05, BFGM06, FSW06]. We apply our theory of differences of halting probabilities to give a positive answer, and show that Ω U (X) is a Martin-Löf random left-c.e. real whenever X is a nonempty Π 0 1 set. Problem 1.1 (Question 8.10 in Miller and Nies [MN06]). If U is a universal machine and X ∅ is a Π 0 1 set, is the probability Ω U (X) always a Martin-Löf random number?This open problem was the starting point for our investigations, which led to the study of more general questions concerning the differences of Ω numbers, and revealed a missing theory which is complementary to the well developed theory of halting probabilities (see Downey and Hirschfeldt [DH10, Chapter 9] for an overview). Before we present our solution and, perhaps more interestingly, the intriguing theory of differences of halting probabilities that it inspired, we make our discussion precise by giving a formal definition of universality. Informally, a Turing machine is universal if it can simulate any other Turing machine.Definition 1.2 (Universal prefix-free machines). Given an effective list (M e ) of all prefix-free machines, a prefix-free machine U is universal if there exists a computable function e → σ e such that, for all τ ∈ 2 <ω , U(σ e * τ) ≃ M e (τ).As usual, the symbol ≃ denotes that either both sides of the relation are defined and equal, or both sides are undefined. In Kolmogorov complexity theory, universal machines are sometimes called universal by adjunction in order to distinguish them from a wider class of machines, the optimal prefix-free machines.

Abstract. The Kučera-Gács theorem [Kuč85, Gác86] is a landmark result in algorithmic randomness asserting that every real is computable from a Martin-Löf random real. If the computation of the first n bits of a sequence requires n + h(n) bits of the random oracle, then h is the redundancy of the computation. Kučera implicitly achieved redundancy n log n while Gács used a more elaborate coding procedure which achieves redundancy √ n log n. A similar bound is implicit in the later proof by Merkle and Mihailović [MM04]. In this paper we obtain optimal strict lower bounds on the redundancy in computations from Martin-Löf random oracles. We show that any nondecreasing computable function g such that n 2 −g(n) = ∞ is not a general upper bound on the redundancy in computations from Martin-Löf random oracles. In fact, there exists a real X such that the redundancy g of any computation of X from a Martin-Löf random oracle satisfies n 2 −g(n) < ∞. Moreover, the class of such reals is comeager and includes a ∆ 0 2 real as well as all weakly 2-generic reals. On the other hand, it has been recently shown in [BLP16] that any real is computable from a Martin-Löf random oracle with redundancy g, provided that g is a computable nondecreasing function such that n 2 −g(n) < ∞. Hence our lower bound is optimal, and excludes many slow growing functions such as log n from bounding the redundancy in computations from random oracles for a large class of reals. Our results are obtained as an application of a theory of effective betting strategies with restricted wagers which we develop.