Crutchfield, J.P. & K. Young, 1990. Computation at the Onset of Chaos. In Zurek, W.H., ed. Complexity, Entropy and the Physics of Information (AddisonWesley, Reading, MA).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of dynamics does not correspond to that of computation, there are shift spaces whose languages are not recursively enumerable (r.e. even if the languages of functional shifts to give them are r.e. In recent years, relevance between dynamics and computation comes to a focus of study [1, 2, 3, 4]. The important idea in these studies is to regard time evolution of dynamics as a computational process. By corresponding unpredictability of dynamical systems to the halting problem, Moore insists the existence of dynamics more complex than chaos [1] The relation between complexity of dynamics ....

J. P. Crutchfield and K. Young, "Computation at the onset of chaos", Complexity, Entropy and the Physics of Information, Addison Wesley, 223-269, 1990.

.... (fi 2 ) H fi;n (S) n (w) and the fi order R enyi entropy rate h fi;n (S) H fi;n (S) 8) reduce to the block entropy Hn (S) and entropy rate hn (S) when fi = 1 [19] The formal parameter fi can be thought of as the inverse temperature in the statistical mechanics of spin systems [20]. In the infinite temperature regime, fi = 0, the R enyi entropy rate h 0;n (S) is just a logarithm of the number of allowed n blocks, divided by n. The limit h (0) S) lim n 1 h 0;n (S) gives the asymptotic exponential growth rate of the number of allowed n blocks, as the block length ....

J.P. Crutchfield and K. Young, "Computation at the onset of chaos," in Complexity, Entropy, and the physics of Information, SFI Studies in the Sciences of Complexity, vol 8, W.H. Zurek, Ed. 1990, pp. 223--269, Addison-Wesley.

....compare the RNN and extracted machine generated sequences through information theoretic measures. The experiments are performed on two chaotic sequences with different levels of computational structure expressed in the induced ffl machines. The ffl machines were introduced by Crutchfield et al. [11, 12] as computational models of dynamical systems. Size of the extracted machines from RNN that attain comparable performance to that of the original RNN serves as an indicator of the neural finite state complexity of the training sequence. We study the relationship between the computational ....

....n blocks in S is denoted by [S] n . Statistical n block structure in a sequence S is usually described through generalized entropy spectra. The spectra are constructed using a formal parameter fi that can be thought of as the inverse temperature in the statistical mechanics of spin systems [12]. The original distribution of n blocks, Pn (w) is transformed to the twisted distribution [43] also known as the escort distribution [4] Q fi;n (w) n (w) v2[S]n n (v) 1) The entropy rate h fi;n = w2[S]n Q fi;n (w) log Q fi;n (w) 2) of the twisted distribution Q ....

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J.P. Crutchfield and K. Young. Computation at the onset of chaos. In W.H. Zurek, editor, Complexity, Entropy, and the physics of Information, SFI Studies in the Sciences of Complexity, vol 8, pages 223--269. Addison-Wesley, 1990.

....a partial information concerning the sequence distribution P . A more fulfilling description is obtained through a spectrum of entropy measures. The spectrum is constructed using a formal parameter fi that can be thought of as the inverse temperature in the statistical mechanics of spin systems [3]. The original n block distribution Pn (w) is transformed to the twisted distribution [14] G fi;n (w) n P : 3) The most probable and the least probable n blocks of the original distribution Pn (w) become dominant in the positive zero and the negative zero temperature regimes, G1;n ....

....patterns [6] In this study a sequence S is considered complex if it appears to be random, i.e. the entropy rate h (eq. 2) is positive, and can be faithfully modeled only with nontrivial stochastic machines. Nontriviality of a machine M is measured by its topological complexity C 0 = log jQj [3]. 3 Neural models The RNN presented in figure 1 was shown to be able to learn mappings that can be described by finite state machines [11] A one of A encoding of symbols from the alphabet A is used with one input and one output neuron, I i and O i respectively, devoted to each symbol. ....

J.P. Crutchfield and K. Young. Computation at the onset of chaos. In W.H. Zurek, editor, Complexity, Entropy, and the physics of Information, SFI Studies in the Sciences of Complexity, vol 8, pages 223--269. Addison-Wesley, 1990.

.... H ;n (S) 1 n (w) and the order R enyi entropy rate h ;n (S) H ;n (S) 8) reduce to the block entropy H n (S) and entropy rate h n (S) when = 1 [18] The formal parameter can be thought of, for example, as the inverse temperature in the statistical mechanics of spin systems [19]. In the in nite temperature regime, 0, the R enyi entropy rate h 0;n (S) is just a logarithm of the number of allowed n blocks, divided by n. The limit h (0) S) lim n 1 h 0;n (S) gives the asymptotic exponential growth rate of the number of allowed n blocks, as the block length ....

J.P. Crutch eld, K. Young, Computation at the onset of chaos, in: W.H. Zurek (Ed.), Complexity, Entropy, and the physics of Information, SFI Studies in the Sciences of Complexity, vol 8, Addison-Wesley, 1990, pp. 223-269.

....and useful information reduction technique in symbolic dynamics. Under certain conditions, stochastic symbolic models of quantized chaotic time series represent, in a natural and compact way, the basic topological, metric and memory structure of the underlying real valued trajectories (see e.g. [4] [5] Analogous ideas in the context of stochastic real valued time series were recently put forward by Buhlmann [6] He introduces a new class of hybrid real valued symbolic models, the so called quantized variable length Markov chains (QVLMCs) that describes a class of real valued stochastic ....

J.P. Crutchfield and K. Young, "Computation at the onset of chaos," in Complexity, Entropy, and the physics of Information, SFI Studies in the Sciences of Complexity, vol 8, W.H. Zurek, Ed. 1990, pp. 223--269, Addison-Wesley.

....and computation in a more formal, and therefore more manageable, setting. Up to now, computational mechanics has been applied to formal dynamical systems (but see [3] leading to a determination of computational features of cellular automata [4] transitions in the period doubling route to chaos [5] and one dimensional spin systems [2] revealing some new properties of these systems not accounted for by classical measures such as entropies and algorithmic definitions of complexity [6] Computational mechanics has also been applied to globally coupled maps (GCM) which were taken as models of ....

....of its interesting dynamical behavior: Depending on , the asymptotic behavior of the logistic map may be a fixed point, a limit cycle or a chaotic attractor. Furthermore, it is a representative element of a very large class of dynamical systems: Those with a period dou2 bling route to chaos (see [5,6]) In GCMs we have N maps interacting with a sort of mean state of the system (mean field) h n = 1 N N X i=1 f (x i n ) 2) by means of the interaction parameter ffl, that is, we have i = 1 : N elements defined in the following way: x i n 1 = 1 Gamma ffl)f (x i n ) fflh n ....

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J.P. Crutchfield and K. Young, Computation at the onset of chaos, in: W. Zurek, ed., Complexity, Entropy and the Physics of Information (AddisonWesley, Reading, MA, 1990) 223--269.

....subjective information and the information measured by entropy or complexity. The approaches of the physicists to overcome this paradox seem to be not convincing, because most researchers in this community are constrained by their implicit goal to look for an observer independent solution [Crutchfield 1991]. Therefore we have to look for an observer dependent measure for the complexity of a perceived context. This problem is difficult, because we have to differentiate between the pre structured part of perception based on learned mental schemata (available information, see Figure 1) and the ....

CRUTCHFIELD, J. & YOUNG, K. (1991) Computation at the Onset of Chaos. In W. Zurek (ed.) Complexity, Entropy and the Physics of Information (pp. 223-269), Addison.

....subjective information and the information measured by entropy or complexity. The approaches of the physicists to overcome this paradox seem to be not convincing, because most researchers in this community are constrained by their implicit goal to look for an observer independent solution (cf. Crutchfield and Young 1991). input channel EC external activity output channel AC perception and attention action planning internal source of stimulation BC memory MC system context Figure 2 Two different sources for perception: the input channel of the context and the internal source of stimulation ....

Crutchfield, J. & Young, K. (1991) Computation at the Onset of Chaos, in Complexity, Entropy and the Physics of Information (ed. W. Zurek), Redwood: Addison-Wesley, pp. 223-269.

....biological, cultural and neurological systems are described as complex adaptive systems. The key feature being that they display self organised criticality a feature required for them to support the emergence of higher level adaptive systems and one not found in very simple nonadaptive systems [Crutchfield and Young, 1990; Mitchell et al. 1994] 9 . As engineers designing a robot control architecture, we need to strike a balance between simplicity and complexity. We desire simplicity but require enough complexity to support a complex adaptive system . The goal of this research was to investigate the nature of ....

Crutchfield, J. P. and Young, K., "Computation at the Onset of Chaos", in "Entropy, Complexity, and Physics of Information", W. Zurek, editor, SFI Studies in the Sciences of Complexity, VIII, Addison-Wesley, Reading, Massachusetts, pp 223-269, 1990.

....For B a set of symbols, I use B to denote the set of all finite strings of symbols drawn from B. By a language in B I mean a subset of B . 3 performing. Blair and Pollack combine this learning based approach with the kind of bottom up computational analysis developed by Crutchfield and Young [11], in which one tries to fit various finite state devices to an unknown machine in order to detect infinitestate computation. Moore examines his dynamical recognizers from the standpoint of their computing power and produces an elaboration of the Chomsky hierarchy involving many new language ....

....part because w 1 and w 2 are the two inverses of the much studied logistic map , f(x) rx(1 Gamma x) when r = 4. The logistic map has attracted attention because it is a relatively simple (one dimensional) function with chaotic trajectories (see [15] for an introduction) Crutchfield and Young [11], 16] analyzed f as a one string generator by considering f k (1=2) for k = 1; 2; 3; and outputting 1 if f k 1=2 and 0 if f k 1=2. They found that when r = 3:57 : the so called onset of chaos ) the set of initial 2 n character substrings of this single string for n 2 N ....

[Article contains additional citation context not shown here]

James P. Crutchfield and Karl Young. Computation at the onset of chaos. In Complexity, Entropy, and the Physics of Information, pages 223--70. Addison-Wesley, Redwood City, California, 1990.

....useful for instantiating powerful computing devices in metric space computers which exhibit graceful modification under small distortions. It is as though by embracing the caprice of a chaotic process, a computational system can stay in its good graces and make effective use of its complexity (cf. Crutchfield and Young, 1990; Crutchfield, 1994) Previous work has focused on how to instantiate complex symbolic computers in metric space computers like connectionist networks. The current work suggests that the most useful contribution of the metric space perspective is the revelation of geometric relationships among ....

....so far, the functions of the underlying GIFS s are contraction maps. One may well wonder if cascades only arise in standard iterated function systems (a la Barnsley, 1993[1988] in which all the functions are contraction maps. The following case is an interesting counterexample. 3.2. Example: Crutchfield and Young (1990) Consider the GIFS, S = 0, 1] w 1 , w 2 where the functions are given by w 1 = x 1 1 2 1 w 2 = x 1 1 2 1 Since w 1 maps the interval O = 0, 1) onto (1 2, 1) and w 2 maps this interval onto (0, 1 2) O is a pooling set of S. Moreover, the point 1 2 is (in) the crest of ....

[Article contains additional citation context not shown here]

CRUTCHFIELD, J. P. and K. YOUNG (1990) Computation at the onset of chaos. In Complexity, Entropy, and the Physics of Information, 223--70. Redwood City: AddisonWesley.

.... systems are regular; this corresponds to the existence of a nite Markov partition for their dynamics [20] At phase transitions such as the period doubling xed point, however, they can have a complicated scale invariant structure, and belong to an intermediate class called indexed context free [9]; and the iteration of smooth maps in the plane can correspond to universal Turing machines [43] Similarly, the image of a cellular automaton after a nite number of timesteps is regular [60] as is the set of xed points; but limit sets can be contextfree, context sensitive, or the complement ....

J.P. Crutcheld and K. Young, \Computation at the onset of chaos." In Complexity, Entropy, and the Physics of Information, W.H. Zurek, Ed. Addison-Wesley, 1990.

....for instantiating powerful computing devices in metric space computers which exhibit graceful modification under small distortions. It is as though by embracing the caprice of a chaotic process, a 2 computational system can stay in its good graces and make effective use of its complexity (cf. Crutchfield and Young, 1990; Crutchfield, 1994) Previous work has focused on how to instantiate complex symbolic computers in metric space computers. The current work suggests that the most useful contribution of the metric space perspective is the revelation of geometric relationships among familiar, effective symbolic ....

....is interesting in part because w 1 and w 2 are the two inverses of the much studied logistic map , f(x) rx(1 x) when r = 4. The logistic map has attracted attention because it is a relatively simple (one dimensional) function with chaotic trajectories (see Stogatz, 1994, for an introduction) Crutchfield and Young (1990, 1994) analyzed f as the generator for a string s = s 1 s 2 s 3 . where s i 0, 1 for all i by 12 considering f k (1 2) for k = 1, 2, 3, and letting s k = 1 if f k 1 2 and 0 otherwise. They found that when r = 3.57. the so called onset of chaos ) the set of initial 2 n ....

[Article contains additional citation context not shown here]

Crutchfield, J. P. and Young, K. (1990). Computation at the onset of chaos. In Complexity, Entropy, and the Physics of Information, pages 223--70. Addison-Wesley, Redwood City, California.

....bya #nite set Y i and the condition Y i 1 # Y i guarantees that the procedure will terminate to produce a discrete approximation A 0 for A 0 . In discrete form, the above procedure may be equated with the Hopcroft Minimization Algorithm #Hopcroft Ullman, 1979#, or the method of machines #Crutch#eld Young, 1990, Crutch#eld, 1994#, and was #rst used in the present context by Giles et al. #1992#. Using small values of r, their aim was to extract an FSA that, while it might not model the network s behavior exactly,would model it closely enough to faithfully classify the training data. We had in mind a ....

....the network. However, that approach is imprecise if the network has induced a non regular language and does not exactly model an FSA. Wehave provided a #ne grained analysis for a number of trained networks, both regular and non regular, using an approach similar to the method of machines which Crutch#eld Young #1990# used to analyse certain hand crafted dynamical systems. In particular, wewere able to measure empirically whether the induced language was regular or not. The fact that several of the networks induced non regular languages suggests a discrepancy between languages which are #simple for ....

Crutch#eld, J.P. & K. Young, 1990. Computation at the Onset of Chaos. In Zurek, W.H., ed. Complexity, Entropy and the Physics of Information #AddisonWesley, Reading, MA#.

....a partial information concerning the sequence distribution P . A more fulfilling description is obtained through a spectrum of entropy measures. The spectrum is constructed using a formal parameter fi that can be thought of as the inverse temperature in the statistical mechanics of spin systems [3]. The original n block distribution Pn (w) is transformed to the twisted distribution [14] G fi;n (w) P fi n (w) P w2A n P fi n (w) 3) The most probable and the least probable n blocks of the original distribution Pn (w) become dominant in the positive zero and the negative zero ....

....patterns [6] In this study a sequence S is considered complex if it appears to be random, i.e. the entropy rate h (eq. 2) is positive, and can be faithfully modeled only with nontrivial stochastic machines. Nontriviality of a machine M is measured by its topological complexity C 0 = log jQj [3]. 3 Neural models The RNN presented in figure 1 was shown to be able to learn mappings that can be described by finite state machines [11] A one of A encoding of symbols from the alphabet A is used with one input and one output neuron, I (t) i and O (t) i respectively, devoted to each ....

J.P. Crutchfield and K. Young. Computation at the onset of chaos. In W.H. Zurek, editor, Complexity, Entropy, and the physics of Information, SFI Studies in the Sciences of Complexity, vol 8, pages 223--269. Addison-Wesley, 1990.

....morph. Once these states are found, the temporal evolution of the process is given by a mapping from states to states, T : S S; that is, S t 1 = TS t . The available model building algorithms infer the states S via various approximations of the equivalence class conditions specified in (3) [7, 8, 9] These procedures are referred to generically as machine reconstruction . The result of machine reconstruction, then, is the discovery of the underlying process s hidden states. This should be contrasted with the ad hoc methods employed in hidden Markov modeling in which a set of states and a ....

....the amount of memory in the generating process. The entropy h , as a measure of the diversity of patterns, and the complexity C , as a measure of memory, have been taken as the two elementary information processing coordinates with which to analyze a range of nonlinear dynamical systems. [9] Although it falls somewhat outside of the present discussion, it is worthwhile noting that there is a general thermodynamics of ffl machines. This describes the process of optimal model estimation as one in which the observer comes to inferential equilibrium with the data stream. Additionally, ....

J. P. Crutchfield and K. Young, "Computation at the onset of chaos," in Entropy, Complexity, and the Physics of Information (W. Zurek, ed.), vol. VIII of SFI Studies in the Sciences of Complexity, (Reading, Massachusetts), p. 223, Addison-Wesley, 1990.

No context found.

Crutchfield, J.P. & K. Young, 1990. Computation at the Onset of Chaos. In Zurek, W.H., ed. Complexity, Entropy and the Physics of Information (AddisonWesley, Reading, MA).

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Crutchfield, J. P. & Young, K. (1990). Computation at the onset of chaos. In W. Zurek (Ed.), Complexity, entropy and the physics of information. Addison-Wesley, Reading, MA.

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Crutchfield, J. & Young, K. (1990), Computation at the onset of chaos, in W. Zurek, ed., `Complexity, Entropy and the Physics of Information', Addison-Wesley, Reading, MA. Das, S. & Das, R. (1991), `Induction of discrete-state machine by stabilizing a simple recurrent network using clustering', Computer Science and Informatics 21(2), 35--40.

No context found.

J. P. Crutchfield and K. Young. Computation at the onset of chaos. In W.H. Zurek, editor, Complexity, Entropy, and the Physics of Information. Addison-Wesley Pub. Co., Redwood City, CA., 1991.

No context found.

Crutchfield, J. P & Young, K. (1989). Computation at the Onset of Chaos. In W. Zurek, (Ed.), Complexity, Entropy and the Physics of INformation. Reading, MA: Addison-Wesley.

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James P. Crutchfield and Karl Young. Computation at the onset of chaos. In Complexity, Entropy, and the Physics of Information, pages 223--70. Addison-Wesley, Redwood City, California, 1990.

No context found.

J. Crutchfield and K. Young. Computation at the onset of chaos, pages 223--269. Addison-Wesley, 1990.

No context found.

J. P. Crutchfield and K. Young. Computation at the onset of chaos. In W. Zurek, editor, Entropy, Complexity, and the Physics of Information, volume VIII of SFI Studies in the Sciences of Complexity, page 223, Reading, Massachusetts, 1990. Addison-Wesley.

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