Cristopher Moore. Dynamical recognizers: Real-time language recognition by analog computers. Theoretical Computer Science, 201:99--136, May 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....The transition between regular and context free has been of particular interest as it demonstrates a certain in nite memory capacity. Rodrigez, Wiles and Elman [64] have explored how the dynamics of a predictor network allow it to recognize a context free language, a . Moore [56] showed how to construct Dynamical Recognizers [61] that can recognize arbitrary contextfree languages using the Cantor sets [82] In this work we look at a RAAM decoder model whose network is identical to the one used in chapter 4. Using understanding of the underlying dynamics of this network, ....

C. Moore. Dynamical recognizers: Real-time language recognition by analog computers. Theoretical Computer Science, July 1998.

....space so that the useful properties that we list here continue to hold with only slight modifications. For example we could use a onedimensional IFS with the maps i defined as i(x) x=A t i =A, t i = i Gamma 1. For a review of this kind of IFSs and an analysis of their computational power see [25]. Also, the chaos n block representation CBRn (S) includes the lower order representations CBRm (S) m n, as its (more or less crude) approximations: Let CBR m;n (S) denote the sequence CBRm (S) without the first n Gamma m points. Then, dS (CBRm;n (S) CBRn (S) 2. The chaos n block ....

C. Moore. Dynamical recognizers: real-time language recognition by analog computers. Technical Report Working Paper 96-05-023, to appear in Theoretical Computer Science, Santa Fe Institute, Santa Fe, NM, 1996.

.... variables and parameters How long do these computations take Are there parallels between the traditional, integer valued approach to computation and the new real valued approach These questions have attracted particular attention in the past decade (e.g. 1] 2] 3] 4] 5] 6] [7]) and the results have been invigorating. Indeed, the use of real numbers turns out to reveal interesting new avenues in a seemingly well explored field. In fact, there is another area in which the switch to real numbers alters the landscape significantly: the area of representation. It is ....

....term Dynamical Automaton is intentionally like the term Dynamical Recognizer which has been used in closely related contexts. The dynamical automata I describe here are similar to but not quite the same as the dynamical recognizers that Pollack [9] Blair and Pollack, 10] and Moore [6] [7] examine. All of these dynamical computing devices have in common that they perform computations in a real valued space in response to string input and involve iterative computations (a function is repeatedly applied to its own output) Pollack s, Moore s, and Blair s machine has the property that ....

Christopher Moore. Dynamical recognizers: Real-time language recognition by analog computers. Theoretical Computer Science., To appear.

....as observed for the two prediction tasks. To characterize each solution type and its potential, the general framework of dynamical systems is used. 2 Background Recurrent neural networks have been portrayed as offering novel computational features compared with automata and Turing Machines [9, 13]. Their representational power does not stem from an infinite tape (as for Turing Machines) but instead from infinite numerical precision of machine states. Siegelmann and Sontag [14] outline a specific neural network setup for a 2 stack machine equivalent to a 1 tape Turing Machine. Each stack ....

C. Moore. Dynamical recognizers: Real-time language recognition by analog computers. Theoretical Computer Science, 201:99--136, 1998.

....resulting from a certain kind of constraint on the processes involved. Other kinds of computation result from adopting different constraints. In particular, we can focus attention on some class of dynamical systems (Blum, Cucker, Shub, Smale, forthcoming; Blum, Shub, Smale, 1989; Moore, 1991; Moore, 1996). As long as there is some way to specify the questions and answers we can see dynamical processes as computing functions. For example, Hava Siegelmann has extensively studied the computational properties of one class of dynamical systems, recurrent neural networks (Siegelmann Sontag, 1994) ....

Moore, C. (1996) Dynamical Recognizers: Real-time Language Recognition by Analog Computers. No. 96-05-023, Santa Fe Institute.

....any string #=# 1 : # T 2A # , let x T = f #T # : #f # 1 #x 0 #. Then # is accepted if x T 2 X yes , rejected if x T 2 X no . This framework, introduced in #Pollack, 1991# has been extended in a number of directions both experimental #Giles et al. 1992, Watrous Kuhn, 1992# and theoretical #Moore, 1997, Casey, 1996#. A variety of methods have been developed for extracting a deterministic #nite automaton #DFA# from the dynamical recognizer once it is trained #Omlin Giles, 1996, Das Mozer, 1994, Manolios Fanelli, 1994, Zeng et al. 1994# although the recognizer will sometimes induce a ....

Moore, C., 1997. Dynamical Recognizers: Real-time Language Recognition by Analog Computers, Theoretical Computer Science #to appear#.

....into the representational differences between standard symbolic computers and connectionist networks. But the saturation analysis speaks in terms of typical properties of HRR vectors and does not give explicit insight into the geometric properties of particular encodings. A related line of work [196, 136] focuses on designing the geometry of specific trajectories of metric space computers, including many connectionist networks. One essential idea (proposed in embryonic form in Pollack [151, 152] Siegelmann and Sonntag [175] Wiles and Elman [218] Rodriguez, Wiles, and Elman, to appear [160] is ....

....many connectionist networks. One essential idea (proposed in embryonic form in Pollack [151, 152] Siegelmann and Sonntag [175] Wiles and Elman [218] Rodriguez, Wiles, and Elman, to appear [160] is to use fractals to organize recursive computations in a bounded metric space. Cris Moore [136] provides the first substantial development of this idea, relating it to the traditional practice of classifying machines based on their computational power. He shows, for example, that every context free language can be recognized by some dynamical recognizer that moves around on an elaborated, ....

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C. Moore. Dynamical recognizers: Real-time language recognition by analog computers. TR No. 96-05-023, Santa Fe Institute, 1996.

.... systems may provide a more complete model (Pollack, 1987; Elman, 1991; for a collection of articles on the issue, see Port and van Gelder, 1995) Results suggest that such systems may correspond to a di erent, though related, set of language classes, as compared to their symbolic counterparts (Moore, 1998). A better understanding of these relationships may provide important insights into the underlying mechanisms governing human languages. One class of dynamical system that provides a promising model of language processing is the class of recurrent neural networks (RNNs) Part of their appeal is ....

Moore, C. (1998). Dynamical recognizers: Real-time language recognition by analog computers. Theoretical Computer Science, 201(1-2):99-136.

....models and on interleaving them [25 27] The generalized shift map (GS) was discussed by Moore [28] He shows it to be computationally universal, and claims that this model is physically realizable. This is in contrast to his model [29] which he views as unphysical, and his dynamical recognizers [30]. An extension of the GS to include real constants was suggested in [22] This analog version of the GS has super Turing computational power as well. Cellular automata (CA) are a computational model which is an infinite lattice of discrete variables with a local homogeneous transition rule. CA s ....

C. Moore. Dynamical recognizers: Real-time language recognition by analog computers. Theoretical Computer Science.

....the regular tree language class which can be represented by a FA (and for which the FA can be effectively and efficiently trained to behave as recognizer) This conjecture is also supported by two computation models related to discrete time recurrent neural networks that recently emerged. Moore [52] presented classes of iterated functions systems that are (in the case of piecewise linear maps and beyond) strictly more powerful than deterministic real time pushdown automata. Tabor [73] specified classes of so called pushdown dynamical automata and metrics on its members that can be ....

.... BOL and OL (Relation 8) What is and how to describe the full computational power of the generic architecture that allows a discrete time analog computation in a potentially infinite state space The dynamical systems point of view (see e.g. Blair and Pollack [7] Casey [9] Kolen [43] Moore [52], Tabor [73] might be helpful in finding answers. The theoretical results presented here together with former empirical results on artificial data FA(r,2,dlog me, dlog ne,k,1, 1 ,oe t ) r = 2 r = 3 r = 4 r arbitrary O(nm k 2 k ) O(log m p nm k 2 k ) O iq nm k 2 k log m log n k ....

Christopher Moore. Dynamical Recognizers: Real-time Language Recognition by Analog Computers. Theoretical Computer Science, 1998. to appear.

....: oe T 2 A , let x T = f oe T ffi : ffi f oe 1 (x 0 ) Then Sigma is accepted if x T 2 X yes , rejected if x T 2 X no . This framework, introduced in (Pollack, 1991) has been extended in a number of directions both experimental (Giles et al. 1992, Watrous Kuhn, 1992) and theoretical (Moore, 1997, Casey, 1996) A variety of methods have been developed for extracting a deterministic finite automaton (DFA) from the dynamical recognizer once it is trained (Omlin Giles, 1996, Das Mozer, 1994, Manolios Fanelli, 1994, Zeng et al. 1994) although the recognizer will sometimes induce a ....

Moore, C., 1997. Dynamical Recognizers: Real-time Language Recognition by Analog Computers, Theoretical Computer Science (to appear).

....S over A, the relation between the nth order approximation Dn (CBRn (S) of the fractal dimension of CBRn (S) and the topological entropy rate estimate h 0;n (S) of S would read h 0;n (S) Dn (CBRn (S) log A. For a review of this kind of IFSs and an analysis of their computational power see [28]. 5 Recurrent neural network 5.1 Architecture The recurrent neural network (RNN) presented in figure 1 was shown to be able to learn mappings that can be described by finite state machines [30] We use a unary encoding of symbols from the alphabet A with one input and one output neuron devoted ....

C. Moore. Dynamical recognizers: real-time language recognition by analog computers. Technical Report Working Paper 96-05-023, to appear in Theoretical Computer Science, Santa Fe Institute, Santa Fe, NM, 1996.

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Cristopher Moore. Dynamical recognizers: Real-time language recognition by analog computers. Theoretical Computer Science, 201:99--136, May 1998.

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Moo98. C. Moore. Dynamical recognizers: real-time language recognition by analog computers. Theoretical Computer Science, 201:99-136, 1998.

.... in themselves, rather than strings of digits; and partly from a desire to use the tools of computation theory to better classify the variety of continuous dynamical systems we see in the world (or at least in classical idealizations of it) However, in most recent work on analog computation (e.g. [BSS89,Mee93,Sie98,Moo98] time is still discrete; just as in standard computation theory, the machines are updated with each tick of a clock. If we are to make the states of a computer continuous, it makes sense to consider making its progress in time continuous too. While a few e orts have been made in the direction of ....

C. Moore. Dynamical recognizers: real-time language recognition by analog computers. Theoretical Computer Science, 201:99-136, 1998.

.... values and replace unitarity of the transition matrices with stochasticity in which the elements of each row of the U a sum to 1, we get the stochastic automata of Rabin [24] see also the review in [20] If we generalize the U a to nonlinear maps in R n , we get real time dynamical recognizers [22]. If we generalize the U a to nonlinear Bayes optimal update maps of the n simplex, we get machine deterministic representations of recurrent hidden Markov models [8, 36] Note that the e ect of the matrix product Uw = Uw1 Uw2 is to sum over all possible paths that the machine can take. ....

C. Moore, \Dynamical recognizers: real-time language recognition by analog computers." Theo. Comp. Sci. 201 (1998) 99-136.

.... in themselves, rather than as strings of digits; and partly from a desire to use the tools of computation theory to better classify the variety of continuous dynamical systems we see in the world (or at least in its classical idealization) However, in most recent work on analog computation (e.g. [BSS89,Mee93,Sie98,Moo98]) time is still discrete. Just as in standard computation theory, the machines are updated with each tick of a clock. If we are to make the states of a computer continuous, it makes sense to consider making its progress in time continuous too. While a few e orts have been made in the direction of ....

C. Moore. Dynamical recognizers: real-time language recognition by analog computers. Theoretical Computer Science, 201:99-136, 1998.

.... words: Analog computation, recursion theory, iteration, di erentially algebraic functions, primitive recursive functions 1 Introduction There has been a recent resurgence of interest in analog computation, the theory of computers whose states are continuous rather than discrete (see for instance [BSS89,Meer93,SS94,Moo98]) However, in most of these models, time is still discrete; just as in classical computation, the machines are updated with each tick of a clock. If we are to make the states of a computer continuous, it makes sense to consider making its progress in time continuous too. While a few e orts have ....

C. Moore. Dynamical recognizers: real-time language recognition by analog computers. Theoretical Computer Science 201:99-136, 1998.

.... in themselves, rather than as strings of digits; and partly from a desire to use the tools of computation theory to better classify the variety of continuous dynamical systems we see in the world (or at least in its classical idealization) However, in most recent work on analog computation (e.g. [BSS89,Mee93,Sie98,Moo98]) time is still discrete. Just as in standard computation theory, the machines are updated with each tick of a clock. If we are to make the states of a computer continuous, it makes sense to consider making its progress in time continuous too. While a few e#orts have been made in the direction of ....

C. Moore. Dynamical recognizers: real-time language recognition by analog computers. Theoretical Computer Science, 201:99--136, 1998.

.... in themselves, rather than as strings of digits; and partly from a desire to use the tools of computation theory to better classify the variety of continuous dynamical systems we see in the world (or at least in its classical idealization) However, in most recent work on analog computation (e.g. [BSS89,Mee93,Sie98,Moo98] time is still discrete. Just as in standard computation theory, the machines are updated with each tick of a clock. If we are to make the states of a computer continuous, it makes sense to consider making its progress in time continuous too. While a few e orts have been made in the direction of ....

C. Moore. Dynamical recognizers: real-time language recognition by analog computers. Theoretical Computer Science, 201:99-136, 1998.

....to other models of analog computation; in particular, it can be seen as a real time, constant space, off line version of Blum, Shub and Smale s real valued machines. Many additional results, showing inclusions with classical language classes and complexity and decidability results, are given in [15]. 2 Definitions Define a dynamical recognizer as follows (using slightly different notation from [17] Let A be the set of finite words in an alphabet A, with ffl the empty word. The length of a word w is jwj, and a k is a repeated k times. The concatenation of two words is u Delta v or ....

....recognized by elementary functions, meaning compositions of algebraic, trigonometric, and exponential functions. In this paper, we will further assume that the coefficients (and component boundaries in the piecewise linear case) of all the f a and h are rational. We study real coefficients in [15]. As a partial motivation for this work, we note that recurrent neural networks are being studied as models of language recognition [17] for regular [7] context free [5] and context sensitive [20] languages, as well as fragments of natural language [6] where grammars are represented dynamically ....

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C. Moore, "Dynamical Recognizers: Real-time Language Recognition by Analog Computers." Santa Fe Institute Working Paper 96-05-023, submitted to Theoretical Computer Science.

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C. Moore. Dynamical recognizers: Real-time language recognition by analog computers. Theoretical Computer Science, 201:99--136, 1998.

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C. Moore. Dynamical recognizers: Real-time language recognition by analog computation. Theoretical Computer Science, 201(1):99--136, 1998.

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Christopher Moore. Dynamical recognizers: Real-time language recognition by analog computers. TR No. 96-05-023, Santa Fe Institute, 1996.

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Christopher Moore. Dynamical recognizers: Real-time language recognition by analog computers. Theoretical Computer Science., To appear.

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