280 Common Sense Reasoning

Even with powerful numerical computers, exploring complex dynamical systems requires significant human effort and judgment to prepare simulations and to interpret numerical results. This paper describes one-aspect of a computer program, KAM, that can autonomously prepare numerical simulations, and can automatically generate high-level, qualitative interpretations of the quantitative results. Given a dynamical system, KAM searches in the phase space for regions where the system exhibits qualitatively distinct behaviors: periodic, almost-periodic, and chaotic motion. KAM uses its knowledge of dynamics to constrain its searches. This knowledge is encoded by a grammar of ’ dynamical behavior in which the primitives are geometric orbits, and in which the rules of combination are orbit adjacency constraints. A consistency analysis procedure analogous to Waltz’s constraint satisfaction algorithm is implemented. The utility of the approach is demonstrated by exploring the dynamics of nonlinear conservative systems with two degrees of freedom. From the study of stars and galaxies formation, to aircraft and wing design, to blood flow in the heart, and to microelectronics device modeling, scientists and engineers are confronted with nonlinear equations that are inaccessible to analytical methods. Although powerful numerical computers can painlessly generate quantitative particular solutions to the equations, understanding the qua3itative content of these equations requires substantial human effort and judgment to prepare numerical simulations, and interpret numerical results in qualitative terms. This paper demonstrates that by combining ideas from symbolic computation with methods incorporating deep knowledge of the underlying dynamics of the physical domain, one can build an effective program that autonomously explores the behaviors of a specified dynamical system, and generates high-level, qualitative interpretation of the quantitative data. ‘This rewrt describes research done at the ArtikiaI Intelligence Laboiatory of the Massachusetts Institute of Technology. Support for the Laboratory’s artificial intelligence research is provided in part by the Advanced Research Projects Agency of the Department of Defense under Office of Naval Research contract N00014-86-K-0180. Exploring a dynamical system involves two tasks: (1) generate all typical responses of the system by sampling a sufhcient number of starting conditions, and (2) describe how these typical responses change their characters as the system parameters are varied. Studying the swing of a simple pendulum provides a good illustration of these tasks. An experimenter observes how the pendulum starting at various initial states angular position and velocity of the blob swings. He may repeat the experiments by controlling the surrounding conditions such as air resistance and gravity. From the experimental results, he like the followingstable? Under wh oscillate? Will it short, one wants to know the typical responses of the aye tern without actually solving the equations governing the pendulum motion. Blind exhaustive testing of every point of the phase space the space of all possible starting conditions to find out the system responses is out of the question because the number of possible starts is overwhelming. Moreover, interesting qualitative changes in behavior often occur in so small a region in the phase space that unguided experiments will likely miss them. One way to meet these difficulties is to exploit knowledge of the natural constraints in the physical problem. Fluid flow provides a useful illustration. Fig. 1 depicts a flow pattern in some small region; it shows four flow lines. The flow pattern, as it stands, is not complete: some important feature is missing. Let us see why. Since the horizontal flow lines are going in opposite direction, there must be a line of fluid particles whose velocity vectors have zero horizontal components. A similar argument about the vertical flow lines shows there must be a line of fluid particles whose velocity vectors have zero vertical components. In general, these two lines of fluid particles will intersect transversely. The point of intersection, which has a zero velocity vector, is known as a stagnant point. 1 With the stagnant point added, the local flow pattern becomes consistent. Now, we want to turn this physical insight around, and use knowledge about possible local flow patterns to derive an understanding of the global behaviors of a physical syk tern. KAM 3 is an implemented program that embodies this type of knowledge. KAM describes dynamical behaviors ‘This argument can k made rigorous u&g the Index Theorem on singular points of vector fields [AmoId, 19731. ‘The initials stand for three mathematicians: Kobnogorov,