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Analog networks on function data streams. (English) Zbl 1418.68093

Summary: Most of the physical processes arising in nature are modeled by differential equations, either ordinary (example: the spring/mass/damper system) or partial (example: heat diffusion). From the point of view of analog computability, the existence of an effective way to obtain solutions (either exact or approximate) of these systems is essential.
A pioneering model of analog computation is the General Purpose Analog Computer (GPAC), introduced by C. E. Shannon [J. Math. Phys., Mass. Inst. Techn. 20, 337–354 (1941; Zbl 0061.29410)] as a model of the Differential Analyzer and improved by M. B. Pour-El [Trans. Am. Math. Soc. 199, 1–28 (1974; Zbl 0296.02022)], L. Lipshitz and L. A. Rubel [Proc. Am. Math. Soc. 99, 367–372 (1987; Zbl 0626.34012)], D. S. Graça and J. F. Costa [J. Complexity 19, No. 5, 644–664 (2003; Zbl 1059.68041)] and others. The GPAC is capable of manipulating real-valued data streams. Its power is known to be characterized by the class of differentially algebraic functions, which includes the solutions of initial value problems for ordinary differential equations.
We address one of the limitations of this model, which is its fundamental inability to reason about functions of more than one independent variable (the “time” variable), as noted by L. A. Rubel [Adv. Appl. Math. 14, No. 1, 39–50 (1993; Zbl 0805.68010)]. In particular, the Shannon GPAC cannot be used to specify solutions of partial differential equations. We extend the class of data types using networks with channels which carry information on a general complete metric space \(X\); here we take \(X = C ( \mathbb R )\) , the class of continuous functions of one real (spatial) variable.
We consider the original modules in Shannon’s construction (constants, adders, multipliers, integrators) and we add a differential module which has one input and one output. For input \(u\) , it outputs the spatial derivative \(v(t) = \partial_x u(t)\).
We then define an \(X\)-GPAC to be a network built with \(X\)-stream channels and the above-mentioned modules. This leads us to a framework in which the specifications of such analog systems are given by fixed points of certain operators on continuous data streams. Such a framework was considered by J. V. Tucker and the second author [Theor. Comput. Sci. 371, No. 1–2, 115–146 (2007; Zbl 1110.68046)]. We study the properties of these analog systems and their associated operators, and present a characterization of the \(X\)-GPAC-generable functions which generalizes Shannon’s results.

MSC:

68Q05 Models of computation (Turing machines, etc.) (MSC2010)
03D78 Computation over the reals, computable analysis
35E20 General theory of PDEs and systems of PDEs with constant coefficients
68Q10 Modes of computation (nondeterministic, parallel, interactive, probabilistic, etc.)

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