RESPONSE TO “PROLEGOMENA TO ANY FUTURE QUALITATIVE PHYSICS,” BY ELISHA SACKS AND JON DOYLE

Sacks’ and Doyle’s “Prolegomena” is entirely convincing. If anyone thinks that the current generation of qualitative physics systems can analyze differential equations as well as an expert equipped with years of training in theoretical and numerical analysis, with the huge library of mathematical techniques developed over the last three centuries, and with good judgment on such matters as when to use which techniques, how far to trust a numerical simulation, and so on, deriving from a combination of experience, formal mathematical knowledge, and intuition, then they should certainly read this paper. They should also have their heads examined. Not since I last read Weizenbaum have I seen A1 research put down by a comparison to such an impossibly high standard, nor such a thoroughly straw man put up. No one in their right minds believes or ever believed that SPQR “obviate[s] other mathematical and scientific reasoning.” Nor has anyone ever claimed that SPQR “represents a monumental advance in mathematics,” or that “most expert reasoning involves only a few elements of calculus and interval arithmetic.” The fact that “experts far outperform SPQR” is no more an embarrassment to the state of A1 research than the fact that Shakespeare outperforms current natural language generators. Sacks and Doyle criticize SPQR for “lack[ing] any examples where experts draw incorrect . . . conclusions . . . while SPQR does better.” But this is a ridiculous standard, if only because the SPQR community claims that experts do (internally and largely unconsciously) carry out qualitative reasoning before proceeding with more detailed calculations. What is more to the point is that SPQR can sometimes detect errors in the blind use of other techniques. For example, a numerical simulation of the equation X = -2tx x3,x(0) = 1 that uses the simple linear extrapolation x(t + At) = x(t) + i ( t ) A t will, for any fixed value of At , predict that x eventually becomes negative. By contrast, SPQR applied to the confluence ax = [ t ] [x ] [x] predicts correctly that x never becomes negative. Of course, one can use a more sophisticated numerical technique, but guaranteeing that an algorithm using floating-point arithmetic will avoid this error is no trivial matter. And the numerical methods will be substantially more expensive computationally, and the conclusion that x remains positive will apply only to the interval computed, not to all positive t . SPQR has two major objectives. The first objective is to give a partial analysis of dynamic systems where only partial specifications are available. Complete information may be unobtainable for any of a number of reasons: you may not be able to perceive or acquire or compute the information, or the system may be an imaginary one that you are currently in the process of designing, or you may be reasoning generically about a large class of systems. In any of these cases, the known constraints may be very weak. If so, the techniques advocated by Sacks and Doyle are likely to be inapplicable, and the results that they are interested in, such as asymptotic stability, are likely to be undetermined. Still, there may be useful information to be extracted. The second objective of SPQR is to develop reasoning systems that are decomposable into a series of simple, local inferences, of such forms as “If X goes up, then Y will go up and 2 will go down,” and “Since X is a lot bigger than Y, 2 will be a lot bigger than W.” Necessarily, such restricted forms of inference are weak. However, the fact that the