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Iteration, Inequalities, and Differentiability in Analog Computers

Author

Listed:
  • Manuel Lameiras Campagnolo
  • Cristopher Moore
  • José Félix Costa

Abstract

Shannon's General Purpose Analog Computer (GPAC) is an elegant model of analog computation in continuous time. In this paper, we consider whether the set G of GPAC-computable functions is closed under iteration, that is, whether for any function f(x) 2 G there is a function F(x; t) 2 G such that F(x; t) = f t (x) for non-negative integers t. We show that G is not closed under iteration, but a simple extension of it is. In particular, if we relax the definition of the GPAC slightly to include unique solutions to boundary value problems, or equivalently if we allow functions x k f(x) that sense inequalities in a differentiable way, the resulting class, which we call G + fk, is closed under iteration. Furthermore, G + k includes all primitive recursive functions, and has the additional closure property that if T(x) is in G + k, then any function of x computable by a Turing machine in T(x) time is also.

Suggested Citation

  • Manuel Lameiras Campagnolo & Cristopher Moore & José Félix Costa, 1999. "Iteration, Inequalities, and Differentiability in Analog Computers," Working Papers 99-07-043, Santa Fe Institute.
    • Handle: RePEc:wop:safiwp:99-07-043
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    References listed on IDEAS

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    1. Vergis, Anastasios & Steiglitz, Kenneth & Dickinson, Bradley, 1986. "The complexity of analog computation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 28(2), pages 91-113.
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