On a theory of computation and complexity over the real numbers: 𝑁𝑃- completeness, recursive functions and universal machines

Abstract

We present a model for computation over the reals or an arbitrary (ordered) ring R. In this general setting, we obtain universal machines, partial recursive functions, as well as JVP-complete problems. While our theory reflects the classical over Z (e.g., the computable functions are the recursive functions) it also reflects the special mathematical character of the underlying ring R (e.g., complements of Julia sets provide natural examples of R. E. undecidable sets over the reals) and provides a natural setting for studying foundational issues concerning algorithms in numerical analysis.