J.V. Tucker and J.I. Zucker (2007):
Computability of analog networks.
Theoretical Computer Science, 371, 115-146.
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Abstract. We define a general concept of a network of analogue processing units connected by analogue channels, processing data from a metric space A, and operating with respect to a global continuous clock T, modelled by the set of non-negative real numbers. The inputs and output of a network are continuous streams u: T→ A, and the input-output behaviour of a network with system parameters from Ar is modelled by a functional of the form
Φ: Ar × C[T,A]p → C[T,A]q (p,q>0, r≥0)
where C[T,A] is the set of all continuous streams equipped with the compact-open topology. We give an equational specification of the network, and a semantics which involves solving a fixed point equation over C[T,A], using a contraction principle based on the fact that C[T,A] can be approximated locally by metric spaces. We show that if the module functions are continuous then so is the network function Φ. We analyse in detail two case studies involving mechanical systems. Finally, we introduce a custom-made concrete computation theory over C[T,A] and show that if the module functions are concretely computable then so is Φ.