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t is just the same thing as a periodic point whose period is t. The term is used because if the system is sampled every t units of time, then a periodic point with period t is fixed, or invariant.
5 Note that it is not impossible for there to be local dilation of phase space volumes, even in a highly dissipative system. But the overall trend must still be for volumes to contract; thus, local dilation in one neighbourhood requires quicker contraction in another neighbourhood.
6 Technically speaking, a fractal is not necessarily strange, and activity on a strange attractor is not necessarily chaotic; but these details do not concern us here.
7 Strictly speaking, we could also say that the mapping from reality to l space is fuzzy, because a system represented in l space has only a finite number of access states distinguishable in l space .
8 See a forthcoming paper on the interlocking ring architecture for more discussion on content addressable memory. Another forthcoming paper engages the frame problem from a neural network perspective.
9 See Mulhauser 1993c for an exploration of this idea in the form of a psychological Principle of Sufficient Reason.
10 The material in this section is drawn with minor revisions from my response to SmithÕs critique in Mulhauser 1993d.
11 Curiously, the paper which was the subject of SmithÕs commentary (Mulhauser 1993a) does not include any explicit argument of this kind, although I assume something similar to it in my mention of Horgan and TiensonÕs (1992) work. The purpose of my own paper was simply to provide a dynamical systems framework for analysing the interactions between various levels of description of the same intelligent system.
12 Smith has objected in private communication that he intended to mean Ôcontrol parameterÕ in a more general sense, which would include changes in afferent signals. But the standard use of Ôcontrol parameterÕ refer to a coefficient of one of the terms of the equations describing the dynamics of the system in question. It does not generally refer to anything like what we mean here by a change in afferent signals, which corresponds simply to a change in the systemÕs location in phase space but not to any overall change in dynamics. In any case, if we allow SmithÕs broader notion of Ôcontrol parameterÕ as a term which includes simple changes in the position of the system in phase space, then his assertion simply begs the question by assuming that psychological state changes in response to changes in afferent signals are deterministic with respect to psychological state level information.
13 Here I am referring to the state space describing the firing patterns of nodes in response to inputs after training; of course it is also useful to describe the evolution of a net through a state space of connection strengths during training.
14 Of course, we could make changes in such control parameters, and the shape of the possible phase trajectories would be altered accordingly; the point is that there are other ways the Ôpsychological stateÕ of the network could change.
15 Smith has pointed out in private communication that we shouldnÕt really be talking about anomalous monism here, since the original Davidsonian meaning of that term referred to physical monism without strict psycho-physical laws. Understanding psychological states as supervening upon volumes of points near particular attractors or within their boundaries of attraction is compatible with and even suggests that the correlation between y states and l states is law-like, the denial of anomalous monism.
16 This is why I noted previously that if we construe SmithÕs Ôcontrol parametersÕ very broadly, his assertion about deterministic changes in psychological state is question begging, and in fact it is incorrect.
17 This is not a straightforward implication, however, because it would still need to be shown that the aspects of analogue neural network behaviour which could not be satisfactorily simulated were indeed functionally relevant to instantiating human minds.
18 Much of the material in this section is adapted from Mulhauser 1993b. It is presently under review for the Springer-Verlag proceedings volume for that congress.
19 In private discussions, Smith has emphasised that when the set of critical points in a given system has measure zero, we are justified in throwing out that set because a randomly chosen point in the phase space of the system has probability zero of landing on a member of the set. But the phase space volume of relevant neighbourhoods around such points may not have measure zero.
20 Sommerer and Ott have coloured points within a very short distance of the invariant plane (|y| < 10-8, |vy| < 10-9, y·v < 0) as if they ultimately went to the plane (p. 140), but unlike systems without riddled attractors, we cannot be sure!
21 A Bernoulli-Turing Machine is simply a Universal Turing Machine equipped with a random number generator.
22 Smith gives this number as 1/2k (p. 67), but this must have been a typing error since there must be a whole number of sequences of n bits.
23 This is related to the idea that the graphs of an infinite number of polynomials may pass through any finite set of points.
24 It is unlikely such computers could be built in practice; see Mulhauser forthcoming.
25 Note also that it is very well to note that short programs for the Universal Quantum Computer can generate random strings rapidly, but if we are given a random string and asked to supply the Q-program to generate it, we are back where we were with BennettÕs measure.
26 We are not talking about the philosophersÕ possible worlds here, but rather about the different universes which can be interpreted as being home to the various possible states represented in a state vector which is a quantum linear superposition.
27 Someone might object that DES has actually increased the logical depth of the string which was randomised, since it is now a more computationally intensive task to extract the original string and compress it, rather than just trying to compress the DESed string. But of course doing this requires a specification of the entire DESed string as well as of the necessary decryption algorithm; thus, the shortest program which can generate the DESed string will be one which simply lists its elements, and the string will be both KCS random and logically shallow.
28 ItÕs interesting to notice that what we are doing when we compress a given string is really offering a different representation of the same data and including within the product string a definition of the new representational scheme in terms of the original one: an algorithm for one-way translating, or decompressing. For instance, we might search for occurrences of a particular string and represent that string in the compressed version by some special combination of bits which occurs infrequently or not at all in the original string. This is similar to what we do in ordinary English when, for instance, we say DNA instead of deoxyribonucleic acid.
29 The particulars of individual signal patterns will depend on the instruction set of the relevant chips, of course, but there is nothing in digital computer design which prevents us in principle from using the most random strings possible to represent whatever we want them to.