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The extended analog computer. (English)
Adv. Appl. Math. 14, No.1, 39-50 (1993).
People have been aware of the limitations of the “general purpose analog computer” (better called the “differential analyzer”) for some time. Just as examples, it cannot compute the Euler gamma function, or the Riemann zeta function, or solve the Dirichlet problem for Laplace’s equation in the disk, although it can compute most of the other special functions in, say, Whittaker and Watson. It works by solving (quasilinear first-order) systems of ordinary differential equations. In this paper, the “extended analog computer” is defined ‒ it solves systems of partial differential equations, and overcomes many of the limitations of the differential analyzer, including the three specific ones mentioned above. There is a lengthy introduction, followed by the formal definition of the extended analog computer. The concluding section shows that the extended analog computer really can compute a lot of functions. Indeed, one of the major open problems is to decide whether it can compute all real analytic functions of any finite number of real variables. To quote from p. 41, “Roughly speaking, the extended analog computer permits all the operations of ordinary analysis, except the unrestricted taking of limits.” The other major problem raised is that of implementation ‒ i.e., of constructing actual physical, biological, or chemical devices that function is reality like the black boxes of the conceptual extended analog computer. Jonathan Mills, of Indiana University’s Computer Science Department has recently informed the author that Kirchhoff, in 1845, solved Laplace’s equation with electrical currents in heavy sheets of copper and that he (Mills) is now doing similar things with pieces of pure silicon. Another problem discussed is that of Church’s Thesis in the context of the extended analog computer. Also, there is a brief discussion of the brain as an extended analog computer. There is a growing group of researchers exploring analog computation, as opposed to the presently dominant digital computation, and the next several decades may show some striking developments. There are five criticisms of the paper under review. First, (a minor one), it is loaded with abbreviations like EAC, GPAC, EWP, DA, ADE, that will turn some readers off. Next, the condition EWP (for extremely well- posed) is introduced to assure determinism and stability, but no real justification is given for choosing just this condition. (Incidentally, in the proof of Theorem 3.2, the author neglects to prove EWP, but this follows easily from Pour-El’s condition of “domain of generation” for the differential analyzer, after which it is modeled.) Also, on p. 44, it should be said that the “admissible test-errors” are to be real- analytic. Finally, there is a point about the boundary-value problems, that are the basis of the extended analog computer, that needs more thought. Namely, the boundaries themselves are generated by the computer, but are not subjected to any requirements of determinism or stability. Is this proper?
Reviewer: Lee A.Rubel (Urbana)