Abstract Computability and Its Relation to the General Purpose Analog Computer (Some Connections Between Logic, Differential Equations and Analog Computers)

Abstract

We present a mathematical definition of an analog generable function of a real variable. This definition is formulated in terms of a simultaneous set of nonlinear differential equations possessing a "domain of generation." (The latter concept is explained in the text.) Our definition includes functions generated by existing general-purpose analog computers. Using it we prove two theorems which provide a characterization of analog generable functions in terms of solutions of algebraic differential polynomials. The characterization has two consequences. First we show that there are entire functions which are computable (in the sense of recursive analysis) but which cannot be generated by any analog computer in any interval-e.g. l/r(x) and 2^-i (x°/n^')). Second we note that the class of analog generable functions is very large: it includes special functions which arise as solutions to algebraic differential polynomials. Although not all computable functions are analog generable, a kind of converse holds. For entire functions,/(x) = X"o b,x', the theorem takes the following form. If f(x) is analog generable on some closed, bounded interval then there is a finite number of bk such that, on every closed bounded interval, f(x) is computable relative to these bk. A somewhat similar theorem holds if/is not entire. Although the results are stated and proved for functions of a real variable, they hold with minor modifications for functions of a complex variable.

Information processing, computation, and cognition

“…Analog computation is often contrasted with digital computation, but analog computation is a vague and slippery concept. The clearest notion of analog computation is that of Pour-El [67]. Roughly, abstract analog computers are systems that manipulate continuous variables to solve certain systems of differential equations.…”

“…10 The claim that the brain is an analog computer is ambiguous between two interpretations. On the literal interpretation, "analog computer" is given Pour-El's precise meaning [67]. The theory that the brain is literally an analog computer in this literal sense was never very popular.…”

Algorithms of Reaction–Diffusion Computing

“…It may be possible to conceive that nature generates a computable function of a real variable directly and not necessarily by approximation as in the traditional approach [37].…”

“…We show how selected problems and tasks of computational geometry, robotics, and logics can be solved by encoding data in configuration of chemical medium's disturbances and programming wave dynamics and interaction. 5.2) we transform arcs to straight lines, and thus spanning tree is calculated.In the nineteenth century many of the fundamental theorems of function theory were discovered by Riemann by thinking of simple experiments concerning the flow of electricity in thin metallic sheets [16].At that time ideas on unconventional, or nature-inspired, computing were flourishing as ever, and Lord Kelvin made his famous differential analyzer, a typical example of a general-purpose analog computer generating functions of the time measure in volts [37]. Owing to surface tension the soap film between the pins, representing points, will try to minimize total surface area.…”

“…At that time ideas on unconventional, or nature-inspired, computing were flourishing as ever, and Lord Kelvin made his famous differential analyzer, a typical example of a general-purpose analog computer generating functions of the time measure in volts [37]. He wrote in 1876 FIGURE 5.2 Several steps of spanning tree constructions by soap film [41].…”