The four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a non-trivial computer verification. Recently, a formal proof of the theorem was obtained with the equational logic program Coq. In this paper we use the computational method for evaluating (in a uniform way) the complexity of mathematical problems presented in [8, 6] to evaluate the complexity of the four colour theorem. Our method uses a Diophantine equational representation of the theorem. We show that the four colour theorem has roughly the same complexity as the Riemann hypothesis and almost four times the complexity of Fermat’s last theorem. 1

Interaction between sensornet nodes and the physical environment in which they are embedded implies real-time requirements. Application tasks are divided into smaller subtasks and distributed among the constituent nodes. These subtasks must be executed in the correct place, and in the correct order, for correct application behaviour. Sensornets generally have no global clock, and incur unacceptable cost if traditional synchronisation protocols are implemented. We present a lightweight primitive which generates a periodic sequence of synchronisation events which are coordinated across large sensornets structured into clusters or cells. Two biologically-inspired mechanisms are combined; desynchronisation within cells, and synchronisation between cells. This hierarchical coordination provides a global basis for local application-driven timing decisions at each node. 1

In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method, which is inspired by NKS1, is based on the possibility to completely describe complex mathematical problems, like the Riemann hypothesis, in terms of (very) simple programs. The method is illustrated on a variety of examples coming from different areas of mathematics and its power and limits are studied.

I, the undersigned, hereby declare that the work contained in this dissertation is my own original work and that I have not previously in its entirety or in part submitted it at any university for a degree.

From a philosophical viewpoint, mathematics has often and traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments — one of the corner stones of most modern natural science — have had no role to play in mathematics. However, during the last three decades, high speed computers and sophisticated software packages such as Maple and Mathematica have entered into the domain of pure mathematics, bringing with them a new experimental flavor. They have opened up a new approach in which computer-based tools are used to experiment with the mathematical objects in a dialogue with more traditional methods of formal rigorous proof. At present, a subdiscipline of experimental mathematics is forming with its own research problems, methodology, conferences, and journals. In this paper, I first outline the role of the computer in the mathematical experiment and briefly describe the impact of high speed computing on mathematical research within the emerging sub-discipline of experimental mathematics. I then consider in more detail the epistemological claims put forward within experimental mathematics and comment on some of the discussions that experimental mathematics has provoked within the mathematical community in recent years. In the second part of the paper, I suggest the notion of exploratory experimentation as a possible framework for understanding experimental mathematics. This is illustrated by discussing the so-called PSLQ algorithm.