## Universality on Real Computation

A paper of mine in French on this subject is already in arXiv: “Universality on Real Computation”. Chris Moore called my attention to his paper entitled “Recursion Theory on the Reals and Continuous-time Computation” ,  which arrives at similar results using a different approach. A paper I’m writing in English on this subject, and which I have been discussing with several people, is forthcoming.  A preliminary powerpoint presentation that I used when presenting on the topic to Gregory Chaitin is available here:Universality on Real Computation In computability theory, the theory of real computation deals with hypothetical abstract machines capable of infinite precision. They operate on the set of real numbers, most of which are non-computable by a Turing machine. These hypothetical computing machines can be viewed as idealised analog computers or differential machines. The question I am interested in concerns the consequences of computing on the set of real numbers for current foundational concepts in traditional computation,  for example universality. I have found that an infinite number -possibly denumerable- of  universal real computers of varying power allows what I have called “intrinsic universality” (there is another definition of intrinsic universality related to dynamical systems and universal computation), since real numbers are not bounded in complexity and there is therefore one for each class of complexity. However, taking the complete set of real numbers, what I claim is that a single and ultimate universal machine for real computation is not possible, since for each proposed universal real computer A –with pre-fixed real numbers R=r_1,r_2,…,r_n – it suffices to calculate the maximal Turing degree of the set of the real numbers involved, let’s say O_n, to conceive a new real computer B able to compute numbers of higher Turing degree, let’s say O_m with m>n, which the given universal real machine A won’t be able to emulate– which contradicts the definition of a universal abstract real computer. Therefore A won’t be able to emulate (in the way conceived for defining universality in the traditional Turing model)  any other real machine B which belongs to the same set –the real numbers– (which evidently can be seen as their class of languages) unless it allows non-recursive functions going through all the levels of the arithmetical and hyper-arithmetical hierarchy. In other words, real computation does not allow universality, which is the very foundation of computer science unless we take into consideration non-recursive functions of arbitrary power, which adds an aditional problem to the conception of a universal device and the convergence of real computing models, unlike the traditional model based on recursive functions theory and Turing machines. However, in real computation there is a place for relative universality, which is very rich in ideas and theories and gives rise to a hierarchical Church-Turing type thesis for each level of the arithmetical and hyper-arithmetical hierarchy. So talking about the universal abstract device E_n for example–because of  the arithmetical level E_n– for each n natural number would make sense, unlike a device E able to behave like any other E_n for any n natural number, which would need at least one function or the composite of several functions able to attain any arbitrary degree of computability. In other words, E wouldn’t be a field since it is not closed under the set of operators of the defined E.

Here is a related paper entitled on “The complexity of real recursive functions” by Manuel Lameiras Campagnolo. In France, “M. Olivier Bournez” has a particular interest in real computation, in particular analog and hybrid systems.Other important authors in the field include : Karp, da Costa, Blum, Cucker, Shub and Smale. Felix da Costa has let me know that  he is working on something related to my ideas on universality in real computation and relative universality.It is important to say that being interested in real computation is not the same thing as being an “hyper-computationalist”. In fact, some if not most  of the above authors think that real computation would at best be only as powerful as the digital models.  Their interest in the field is mainly operational, in the sense that these types of  machines perform operations commonly related to fields like real anaylisis.