Digital computing can be functionally equivalent to analog computing, though the reverse is not true.

This is a two-part theoretical claim, judged on the computer-science and engineering understanding as of 2001: (a) digital computing can be functionally equivalent to analog computing, meaning it can emulate analog processes to any required precision; (b) analog computing cannot be functionally equivalent to digital computing. True: authoritative sources support both parts. Mostly true: both parts are broadly supported, with a minor caveat (for example a precision limit). Mixed: one part is supported and the other is contested or contradicted. Mostly false: (a) holds only in a much narrower sense than stated and (b) is unproven or contested. False: authoritative sources show both parts to be wrong.

Set before the research began.

1. E1 THE COMPLEXITY OF ANALOG COMPUTATION - Princeton University Princeton University (Vergis, Steiglitz, Dickinson; Mathematics & Computers in Simulation 1986) · 1986

   “we formulate a strong version of Church’s Thesis: that any analog computer can be simulated efficiently (in polynomial time) by a digital computer.”

2. E2 THE COMPLEXITY OF ANALOG COMPUTATION - Princeton University Princeton University (Vergis, Steiglitz, Dickinson; Mathematics & Computers in Simulation 1986) · 1986

   “We next prove Strong Church’s Thesis for a class of analog computers described by well-behaved ordinary differential equations, which we can take as representing part of classical mechanics.”

3. E3 Papers by H. T. Siegelmann and E.D. Sontag Sontag lab publication list (Rutgers) · 1 January 1995

   “The results suggest the following Church-like Thesis of Time-bounded Analog Computing: "Any reasonable analog computer will have no more power (up to polynomial time) than first-order recurrent networks."”

4. E4 Exponential Growth an Illusion?: Response to Ilkka Tuomi the Kurzweil Library · 13 September 2003 Own record

   “By using floating point numbers, digital computers can represent fractional values to any desired precision. In fact, digital computers are much more precise than analog computers in doing this.”

5. E5 The Complexity of Analog Computation Vergis, Steiglitz, Dickinson (Princeton / University of Minnesota) · 1986

   “we formulate a strong version of Church’s Thesis: that any analog computer can be simulated efficiently (in polynomial time) by a digital computer.”

6. E6 The Complexity of Analog Computation Vergis, Steiglitz, Dickinson (Princeton / University of Minnesota) · 1986

   “Theorem 3. The differential equation model described above can be simulated by a digital computer in a number of steps bounded by a polynomial in both R and 1/e”

7. E7 Exponential Growth an Illusion?: Response to Ilkka Tuomi KurzweilAI.net / the Kurzweil Library · 13 September 2003 Own record

   “By using floating point numbers, digital computers can represent fractional values to any desired precision. In fact, digital computers are much more precise than analog computers in doing this.”

8. E8 Some Arguments Favoring Non-Conventional Types of Computers (NASA TM X-1544) NASA / Electronics Research Center (Wayne A. Lea) · 1968

   “The assertion of their equivalence dictates that there are no problems which are solvable by any Turing machine that cannot be solved by such open-ended digital computers, and vice versa.”

9. E9 Essay Review Perspectives on Science and Christian Faith (American Scientific Affiliation), reviewed by Ian H. Hutchinson · December 2000

   “For example, he discusses specifically neural nets that can be approximated by digital computers, but which are inherently analog computers, though highly nonlinear.”

10. E10 Essay Review Perspectives on Science and Christian Faith (American Scientific Affiliation), reviewed by Ian H. Hutchinson · December 2000

    “Analog computers were quite widely used in the mid-twentieth century to solve complicated, ordinary differential equations.”

Independent sources support part (a) with caveats. E1/E5 formulate a Strong Church's Thesis that any analog computer can be simulated efficiently by a digital computer, and E2/E6 prove it for analog computers described by well-behaved differential equations, with polynomial cost in accuracy. E3 similarly proposes that reasonable analog computers have no more power than recurrent networks. None of the evidence addresses part (b), that analog computing cannot be functionally equivalent to digital computing; E8 concerns Turing machines versus digital computers, and E9/E10 do not speak to it. With part (b) neither supported nor contested in the evidence, the two-part criteria cannot be applied.