Hypercomputation is the computation of functions or numbers that cannot be computed in the sense of Turing's 1936 paper 'On Computable Numbers', i.e. cannot be computed with paper and pencil in a finite number of steps by a human clerk working effectively. It is often forgotten that, in 1936, a computer was not a machine at all, but a human being, a mathematical assistant who calculated by rote. This is the sense in which Turing used the term computer and its cognates, such as computable, in his 1936 paper; and the Turing machine (or as Turing called it, 'computing machine') is an idealization of the human computer.
A hypercomputer, on the other hand, is any information-processing machine, notional or real, that is able to achieve more than the traditional human clerk working by rote. Hypercomputers compute functions or numbers, or more generally solve problems or carry out tasks, that lie beyond the reach of the standard universal Turing machine of 1936. The additional computational power of a hypercomputer may arise because certain of the restrictions customarily imposed on the human computer are absent—for example, the restriction that data take the form of symbols on paper, or that all data be supplied in advance of the computation, or that the rules followed by the computer remain fixed for the duration of the computation. Or the additional power may arise because the machine has, among its repertoire of primitive capacities, one or more operations that no human rote-worker can carry out. All computation takes place relative to some set or other of primitive capacities, and the richer the capacities that are available, the greater the extent of the computable.
In this workshop we explore some simple logical models of hypercomputation. We also examine some apriori objections to the possibility of hypercomputation.
Will hypercomputation ever be observed? I argue that this is an empirical question whose answer is not yet known.