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On a theory of computation and complexity over the real numbers: NP- completeness, recursive functions and universal machines. (English) Zbl 0681.03020
A model for computation over the reals \({\mathbb{R}}\) (or an arbitrary ordered ring \({\mathcal R})\) is presented. A machine over \({\mathcal R}\) is a digraph with two kinds of nodes: computation (fan-out 1) nodes labelled by polynomial maps \({\mathfrak g}: {\mathcal R}^ n\to {\mathcal R}^ n\), and branching (fan-out 2) nodes labelled by tests “\({\mathfrak h}(x)\geq 0''\) where \({\mathfrak h}: {\mathcal R}^ n\to {\mathcal R}\) are polynomials. The concepts of universal machine, recursive function and NP-completeness are introduced for such a model of machines. The theory obtained reflects the classical one over \({\mathbb{Z}}\), and captures the special mathematical character of the underlying ring \({\mathcal R}\). Specifically, it is shown that complements of Julia sets (a concept from dynamic system theory) provide natural examples of undecidable sets over the reals. An analogue of Cook’s NP-completeness theorem over the reals is also proved, and the 4-feasibility problem (i.e. the problem of deciding whether or not a real degree 4 polynomial has a zero) is shown to be NP-complete over \({\mathbb{R}}\). The paper is almost self-contained and provides a natural setting for the theory of algorithms and complexity in numerical analysis.
Reviewer: S.Yukna

MSC:
03D15 Complexity of computation (including implicit computational complexity)
03D10 Turing machines and related notions
03D20 Recursive functions and relations, subrecursive hierarchies
03D80 Applications of computability and recursion theory
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
68Q25 Analysis of algorithms and problem complexity
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