 
Summary: Computing over the Reals: Where Turing Meets Newton lblum@cs.cmu.edu
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Computing over the Reals: Where Turing Meets Newton1
Alan Turing (19121954) Sir Isaac Newton (16421727)
Lenore Blum2
Computer Science Department
Carnegie Mellon University
Abstract
The classical (Turing) theory of computation has been extraordinarily successful in providing the
foundations and framework for theoretical computer science. Yet its dependence on 0's and 1's is
fundamentally inadequate for providing such a foundation for modern scientific computation
where most algorithms with origins in Newton, Euler, Gauss, et. al.  are real number
algorithms.
In 1989, Mike Shub, Steve Smale and I introduced a theory of computation and complexity over
an arbitrary ring or field R [BSS89]. If R is Z2 = ({0, 1}, +, ), the classical computer science
theory is recovered. If R is the field of real numbers , Newton's algorithm, the paradigm
algorithm of numerical analysis, fits naturally into our model of computation.
Complexity classes P, NP and the fundamental question "Does P = NP?" can be formulated
naturally over an arbitrary ring R. The answer to the fundamental question depends in general on
the complexity of deciding feasibility of polynomial systems over R. When R is Z2, this becomes
