Non deterministic computation over the real numbers

We have defended in [2, 3] the idea that the physical Church thesis, if it holds, explains the efficiency of mathematics to describe the laws of nature. And also that it suggests that the laws of nature can be expressed, not only in a language of propositions, but also in a language of algorithms. The arguments of Gandy [4], that support the physical Church thesis, even suggest that these algorithms may operate on the elements of a countable set, such as the natural numbers, the rational numbers, the elements of an extension of finite degree of the field of rational numbers, the computable real numbers, ... Nevertheless, the desire to remain as close as possible to the traditional formulations of the laws of nature rather leads to try to describe these laws using algorithms operating on real numbers. This use of algorithms operating on real numbers may be just a first step, but it may also simplify the formulation of these laws, that might be much more complex, if formulated as algorithms operating on the elements of a countable set. An algorithmic formulation of the laws of nature should not predict more than a propositional formulation of the same laws. As some of the laws of physics are non deterministic, it seems that we must use non deterministic algorithms to express these laws in an algorithmic way. The notion of non deterministic algorithm and that of algorithm operating on real numbers have been defined and well studied. It seems however that the conjunction of these two notions, that is the notion of non deterministic algorithm operating on real numbers, remains to be defined. We discuss in this paper what such a definition may be. We assume that the notion of computable function operating on the natural numbers is known. The set of such functions can, for instance, be defined as the smallest set containing the projections, the null functions, the successor function and closed by composition, definitions by induction and minimization. Other definitions based on the notions of rewrite systems, terms of lambdacalculus, or Turing machines are possible. This notion extends to countable sets by numbering their elements.