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

  • Authors:

  • L. Blum;M. Shub;S. Smale

  • Affiliations:

  • Dept. of Math.&Comput. Sci., Mills Coll., Oakland, CA, USA;-;-

  • Venue:
  • SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
  • Year:
  • 1988

  • Proving programs robust

    Proceedings of the 19th ACM SIGSOFT symposium and the 13th European conference on Foundations of software engineering

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Abstract

A model for computation over an arbitrary (ordered) ring R is presented. In this general setting, universal machines, partial recursive functions, and NP-complete problems are obtained. While the theory reflects of 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 recursively enumerable undecidable sets over the reals) and provides a natural setting for studying foundational issues concerning algorithms in numerical analysis.