## Iteration, Inequalities, and Differentiability in Analog Computers (1999)

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### BibTeX

@MISC{Campagnolo99iteration,inequalities,,

author = {Manuel Lameiras Campagnolo and Cristopher Moore and Jose Felix Costa},

title = {Iteration, Inequalities, and Differentiability in Analog Computers},

year = {1999}

}

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### 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 (x) that sense inequalities in a dierentiable way, the resulting class, which we call G + k , 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.