case, as we have indicated from the very beginning of this appendix, dynamical systems concepts will necessarily be at the very heart of any useful theory of VE.

It is important that VE researchers develop the kind of nonlinear intuition that the subject encourages and also build on existing methods for analysis of nonlinear systems. Both the concept of chaos —that apparent complexity and randomness can arise from deep simplicity —and the concept of emergence—that apparent order and simplicity can arise from deep complexity —are of great importance. On the other hand, they are empty without the more technical concepts such as phase space, bifurcation, strange attractors, Poincare maps, Lyapunov exponents, Hamiltonians, Euler-Lagrange equations, symplectic maps, integrability, self-organized criticality, ergodicity, and entropy. Unfortunately, there is no easy access to this deeper work.

To end this discussion, we might tentatively propose a notion of “soft complexity” analogous to, and including, “soft computing,” in the same way that we might propose a notion of “hard complexity” that is analogous to and includes “hard computing.” The flavor of the distinction would be as follows: Soft complexity equals emergence, fractals, artificial life, complex adaptive systems, edge of chaos, control of chaos, . . . plus soft computing, fuzzy logic, neural nets, and genetic algorithms. Hard complexity equals information theory, algorithmic complexity, computational complexity, dynamical systems, control theory, CASE/CAD, nonequilibrium physics, statistics, numerical analysis, and so on. This appendix has clearly advocated the relative importance of “hard” over “soft” complexity in VE. Some of the more extreme advocates for soft complexity claim it will revolutionize analysis and design of complex systems and obviate the need for the “structured and mathematical approach” advocated here. While we obviously disagree with this assessment, it is likely that soft complexity can help make concepts of hard complexity accessible, albeit in a limited way, to a nontechnical audience. It is also likely that the soft complexity concepts will be quite valuable in communication and, probably, for certain types of initial exploration of concepts. In any case, popular expositions of soft complexity will continue to emerge and will have effects on decisions about investment. Our hope is that papers such as the current appendix will help maintain perspectives. 9


For differing perspectives, see a selection of papers by users of fuzzy logic, including engineers, in Proceedings of the IEEE, March 1995. See also the collections of Zadeh's papers (Yager et al., 1987). And, in this volume, see Appendix G for examples of fuzzy logic research.

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