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We intend holding a 3-day, 30-person, Interdisciplinary Research Workshop on Future Trends in Hypercomputation, in Sheffield in September 2006.Hypercomputation -- the study of systems that are provably capable of more than the standard Turing machine -- is a novel research area, with work conducted across a wide range of disparate displines, including computer science, mathematics, quantum theory, biology, neural networks, general relativity, the theory of intelligence and philosophy.Work in hypercomputation can be divided into 5 main themes:PHILOSOPHY Philosophy has been a major driving force behind modern hypercomputation theory; indeed, they coined the word. Of particular relevance is the question of computationalism -- is it reasonable to regard the brain as a computer, and human behaviour as the behaviour of a computer. Many philosophers believe not; they therefore argue that biological systems can behave hypercomputationally.PHYSICS Many physical systems can behave hypercomputationally. Hogarth showed in the early 90s that cosmological singularities can be exploited in this way, and more recently, Kieu has suggested that the quantum adiabatic theorem offers scope for hypercomputationalism. Hypercomputation is also possible under purely Newtonian circumstances; according to Xia, a body in an n-body system can be ejected to infinity in finite time -- the feasibility of such systems entails the implementation of hypercomputational `supertasks'.MATHS AND COMPUTER SCIENCE According to Wells, the noted mathematician Tarski posited the existence of a system that was 'decidable' but not 'recursive' -- in other words, a mathematical system whose behaviour violates the Church-Turing thesis. Similarly in Computer Science, models have frequently been proposed with behaviours that exceed those of the standard Turing machine; Moore, for example, has demonstrated a well-behaved model of real-time computation with manifestly super-Turing potential.RESEARCH QUESTIONS To date, the research community has been content to identify ad-hoc mathematical, physical or biological systems with hypercomputational power -- but if we are ever to create an industry based on hypercomputation, we will need to start codifying and designing hypercomputers. Doing so will present a number of problems that are unique to the field. For example, we clearly cannot describe a hypercomputational behaviour by giving an 'algorithm', because hypercomputation is non-algorithmic by definition -- so how do we do it? An important theme of the Workshop will be the identification of key research questions; the ensuing Research Network will take it as its mandate to attempt a solution to these, and other, questions.ARGUMENTS AGAINST Despite the growing interest in hypercomputation and hypercomputational systems, a small group of researchers feels passionately that hypercomputation is not a valid research area. We feel it is important to include these researchers within the debate, as they can be expected to pose difficult questions that will prompt important research themes. By including both sides of the argument, we can ensure that theoretical proposals for hypercomputation are consistently audited for physicality.
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