Andy Lewis-Pye: the strange patterns of segregation

In 1969, the economist Thomas Schelling was seeking to understand some of the game theoretic considerations underlying the kind of racial segregation observed in large American cities at that time. Using only a chequerboard and some zinc and copper coins, he ran a simple experiment, known as the Schelling Segregation Model (chequerboard – SSM). With the coins originally randomly distributed on the board, he supposed that each coin might be satisfied in its present location so long as at least half of the coins in adjacent squares were of its own type (zinc or copper). Then he took each coin in turn, and moved those unsatisfied coins to the nearest position in which they would be satisfied, leaving satisfied coins where they were. This process of taking each coin in turn was repeated until no further moves were possible. What he observed might initially be considered rather surprising: while each individual would be happy in quite a mixed environment, the global structure which emerges is one in which large segregated regions appear, i.e. large clusters of individuals all of one type. For Schelling this provided evidence of a recurrent theme in his research: that the actions of individuals behaving according to their own locally-defined interests can lead to global results which are undesired by all

5 minutes with Maura Paterson

Maura Paterson (Birkbeck, University of London) visited our Department to present her seminar on “Applications of Disjoint Difference Families”. She also kindly took time out with Julia Böttcher (LSE) to answer a few questions on her research interests and how she takes a break from mathematics

5 minutes with Frank Wilczek

We were very pleased to have (Nobel Laureate) Frank Wilczek (Massachusetts Institute of Technology) visit the department and give a fantastic public lecture based on his new book to a packed Old Theatre. To view his presentation, “A Beautiful Question: finding nature’s deep design”, click here. Afterwards Andy Lewis-Pye (LSE) conducted a brief interview with Frank, to follow up on some of the points raised

5 minutes with Andre Nies

Andre Nies (University of Auckland) visited our Department to present his seminar on “Interactions of computability and randomness”. He also kindly took time out with Andy Lewis-Pye (LSE) to answer a few questions on his research, the future of algorithmic randomness and incorporating classical music into his presentations

5 minutes with Yannai A. Gonczarowski

We were pleased to have Yannai. A. Gonczarowski (Hebrew University of Jerusalem and Microsoft Research) visit our Department recently. During his time with us, he not only gave his excellent seminar on “Cascading to Equilibrium: Hydraulic Computation of Equilibria in Resource Selection Games“, but also generously gave his time to answer a short Q & A, covering everything from resource selection games, to creative thinking in mathematics, to finding the time to perform opera

Computing halting probabilities from other halting probabilities

The halting probability of a Turing machine is the probability that the machine will halt if it starts with a random stream written on its one-way input tape. When the machine is universal, this probability is referred to as Chaitin's omega number, and is the most well known example of a real which is random in the sense of Martin-L\"{o}f. Although omega numbers depend on the underlying universal Turing machine, they are robust in the sense that they all have the same Turing degree, namely the degree of the halting problem. In this paper we give precise bounds on the redundancy growth rate that is generally required for the computation of an omega number from another omega number. We show that for each ϵ>1, any pair of omega numbers compute each other with redundancy ϵlogn. On the other hand, this is not true for ϵ=1. In fact, we show that for each omega number there exists another omega number which is not computable from the first one with redundancy logn. This latter result improves an older result of Frank Stephan

Byzantine Generals in the Permissionless Setting

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 Bitcoin 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 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 eight open questions