The Extended Turing Model as Contextual Tool

Computability concerns information with a causal – typically algorithmic – structure. As such, it provides a schematic analysis of many naturally occurring situations. We look at ways in which computabilitytheoretic structure emerges in natural contexts. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world. The adequacy of the classical Turing model of computation — as first presented in [18] — is in question in many contexts. There is widespread doubt concerning the reducibility to this model of a broad spectrum of real-world processes and natural phenomena, from basic quantum mechanics to aspects of evolutionary development, or human mental activity. In 1939 Turing [19] described an extended model providing mathematical form to the algorithmic content of structures which are presented in terms of real numbers. Most scientific laws with a computational content can be framed in terms of appropriate Turing reductions. This can be seen in implicit form in Newton’s Principia [14], published some 272 years before Turing’s paper. Newton’s work was formative in established a more intimate relationship between mathematics and science, and one which held the attention of Turing, in various guises, throughout his short life (see Hodges [10]). Just as the history of arithmetically-based algorithms, underlying many human activities, eventually gave rise to models of computation such as the Turing machine, so the oracle Turing machine schematically addresses the scientific focus on the extraction of predictions governing the form of computable relations over the reals. Whereas the inputting of data presents only time problems for the first model, the second model is designed to deal with possibly incomputable inputs, or at least inputs for which we do not have available an algorithmic presentation. One might reasonably assume that data originating from observation of the real world carries with it some level of computability, but we are yet to agree a mathematical model of physical computation which dispenses with the relativism of the oracle Turing machine. In fact, even as the derivation of recognisable incomputability in mathematics arises from quantification over algorithmic objects, so definability may play an essential role in fragmenting and structuring the computational ⋆ Preparation of this article supported by E.P.S.R.C. Research Grant No.