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Iteration, Inequalities, and Differentiability in Analog Computers


  • Manuel Lameiras Campagnolo
  • Cristopher Moore
  • José Félix Costa


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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    6. Radner, Roy, 1980. "Collusive behavior in noncooperative epsilon-equilibria of oligopolies with long but finite lives," Journal of Economic Theory, Elsevier, vol. 22(2), pages 136-154, April.
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    Analog computation; recursion theory; iteration; differentially algebraic functions; primitive recursive functions;

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