Abstract

e 17(a) as a discrete point set in the 3-simplex. Each point, somewhat enlarged there, corresponds to one of the states in Figure 15. Two more examples of ffl-machines are given in Figures 17(b) and 17(c). They suggest some of the richness of stochastic finite processes. In Figure 17(b), for example, the ffl-machine has a partial continuum of states; the causal states lie on a "fractal". Figure 17(c) shows an ffl-machine whose causal states limit on a full continuum of states. The different classes shown in Figure 17 are distinguished by a new complexity measure of the ffl-machines's state structure --- the ffl-machine dimension d fflM . d fflM is the information dimension of the state distribution on the simplex. In the case of the countable state ffl-machine of Figure 17(a) C ¯ is finite due to the strong localization of the state distri

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