Learning, representation and synthesis of discrete dynamical systems in continuous recurren


Learning, Representation, and Synthesis of Discrete Dynamical Systems in Continuous Recurrent Neural Networks
b

C. Lee Giles Christian W. Omlin NEC Research Institute, 4 Independence Way, Princeton, NJ 08540 Institute for Advanced Computer Studies, University of Maryland, College Park, MD 20742 E-mail: fgiles,omlincg@research.nj.nec.com
a;b a a

Abstract| This paper gives an overview on learning and representation of discrete-time, discrete-space dynamical systems in discretetime, continuous-space recurrent neural networks. We limit our discussion to dynamical systems (recurrent neural networks) which can be represented as nite-state machines (e.g. discrete event systems 53]). In particular, we discuss how a symbolic representation of the learned states and dynamics can be extracted from trained neural networks, and how (partially) known deterministic nitestate automata (DFAs) can be encoded in recurrent networks. While the DFAs that can be learned exactly with recurrent neural networks are generally small (on the order of 20 states), there exist subclasses of DFAs with on the order of 1000 states that can be learned by small recurrent networks. However, recent work in natural language processing implies that recurrent networks can possibly learn larger state systems 35].
Answering the questions \What are the rules that govern the dynamics of a complex system?" and \How can a complex system with desired dynamics be designed?" is often very hard for systems with many degrees of freedom. However, the consensus is that methods for answering such questions become necessary for intelligent control applications 40]. In such applications, control strategies are not hard-wired at the time of system design; instead, the complexity or performance criteria dictate that control strategies be exible, and that control systems learn from past experience and adapt to the current
Proceedings of the IEEE Workshop on Architectures for Semiotic Modeling and Situation Analysis in Large Complex Systems, Monterey, CA, August 27-29, 1995. Copyright IEEE Press.

situation. Symbolic dynamics has been shown to be an appropriate tool for analyzing the behavior of many di erent complex system, e.g. hybrid systems 51], optical systems 5], nonlinear Hamiltonian systems 64], general dynamical systems 10], and recently discrete-time, continuous-space recurrent neural networks 8, 15, 22, 43, 52, 62, 63]. In particular, extraction of symbolic knowledge from recurrent networks trained on message sequences was used to learn the structure of the computer interconnection network 25]. In this paper, we focus on the analysis and synthesis of discrete-time, discrete-space systems with discrete-time, continuous-space recurrent neural networks and primarily on our own work. Such networks are able to model a variety of dynamical processes, including systems that can be represented as deterministic nite-state automata (DFAs). Discrete event systems 27] are one such class of dynamical systems. We will show how recurrent neural networks with a large number of degrees of freedom can be trained to correctly classify strings of some regular language which have been encoded as temporal sequences. Thus, design is performed through training. Contrary to popular belief, it is possible to extract a symbolic model of the learned knowledge by a network in the form of DFAs. We brie y summarize the DFA extraction algorithm and also address the quality of the extracted rules, i.e. how well the extracted automaton represents the (unknown) regular source grammar that generated the training data. Recent theoretical results assert that neural networks are appropriate tools for re ning initial do-

I. Introduction

main knowledge 1, 19]. Given partial prior knowledge about some application, performance can be signi cantly improved when that prior knowledge is used e ectively, e.g. by initializing a network with the knowledge prior to training. We summarize theoretical results which show that it is possible to encode discrete-time, discrete-space dynamical systems in discrete-time, continuous-space recurrent networks such that the behavior of the system and its neural network model is identical, even for very large systems and even in the presence of noise. Finally, we show that the recurrent neural network approach to modeling dynamical systems can scale well for two subclasses of DFAs. Motivation for this work comes from the increased use of neural networks implemented as VLSI 39, 58]. DFAs as neural networks and vice versa could have signi cant implications for design and e ective implementation in this area. Since we will be training on strings of regular languages, we give a brief description of regular grammars; see 28] for more details. Regular languages represent the smallest and simplest class of formal languages in the Chomsky hierarchy and are generated by regular grammars. A regular grammar G is a quadruple G =< S; N; T; P > where S is the start symbol, N and T are respectively nonterminal and terminal symbols and P are productions of the form A ! a or A ! aB where A; B N and a T . The regular language generated by G is denoted L(G). A deterministic nite-state automaton (DFA) M is the recognizer of each regular language L; L(G) = L(M ). Formally, a DFA M is a 5-tuple M =< ; Q; R; F; > where = fa1 ; : : : ; aK g is the alphabet of the language L, Q = fs1; : : : ; sN g is a set of states, R Q is the start state, F Q is a set of accepting states and : Q ! Q de ne state transitions in M . A string x is accepted by the DFA M and hence is a member of the regular language L(M ) if an accepting state is reached after the entire string x has been read by M . Alternatively, a DFA M can be interpreted as grammar which generates the regular language L(M ).
s

unit delay

W ijk t+1 Si t+1 S 0

I

t k

Figure 1: Second-order fully-connected recurrent neural network. course in high level VLSI design 4].
A. Network Architecture Recurrent neural networks have been shown to be at least computationally equivalent to Turing machines 60, 59] and represent canonical forms of automata 43]. Their computational power and training ability make them useful tools for modeling nonlinear dynamical systems 13, 41]. DFAs can be represented in many discrete-time, recurrent network architectures 14, 16, 22, 29, 52]. (To review the large variety of recurrent neural network architectures, please see the papers in 21]. We choose for ease of representation networks with second-order weights Wijk shown in gure 1. The continuous network dynamics are described by the following equations: t Sit+1 = g(ai (t)) = 1 + e1 a (t) ; ai (t) = bi + Wijk Sjt Ik ; ? j;k where bi is the bias associated with hidden recurrent state neurons Si ; Ik denotes input neurons; g is the nonlinearity; and ai is the activation of the ith neuron. An aspect of the second order recurrent neural t network is that the product Sjt Ik in the recurrent network directly corresponds to the state transition (qj ; ak ) = qi in the DFA. After a string has been processed, the output of a designated neuron S0 decides whether the network accepts or rejects a string. The network accepts a given string if the value of the t output neuron S0 at the end of the string is greater than some preset value such as 0.5; otherwise, the network rejects the string. For the remainder of this
i

III. Recurrent Neural Networks

II. Regular Languages

X

Regular languages and DFAs have proven useful in the analysis and design of discrete event systems for control and manufacturing processes 42, 53] and of

paper, we assume a one-hot encoding for input symt bols ak , i.e. Ik 2 f0; 1g. B. Training Algorithm For any iterative training algorithm based on optimization, we must specify the training data, the form in which it will be presented, the error function that is to be minimized, the weight update algorithm and the training criterion, i.e. when to stop training. The training data consists of strings that are members of some regular language (positive example strings) and strings which are not member of that language (negative example strings) along with a boolean label indicating membership in the unknown regular language. Example strings are presented as inputs to the network one symbol at a time. We use a one-hot encoding for input symbols, i.e. each symbol of the alphabet is assigned its own input neuron. Training is performed by updating the network weights at the end of each sample string presentation. We use a gradient descent optimization algorithm for nding a minimum on the network error surface which is de ned by a quadratic cost function 55]. As with any optimization method based on gradient-descent, the training algorithm is prone to nding local minima for which the network does not correctly classify the training data. However, other methods such as simulated annealing are computationally prohibitive 31]. We have found that an incremental learning strategy whereby we start training on the shortest strings rst and gradually train on more strings until the network correctly classi es the entire training set facilitates convergence. Theoretical results indicate that training with this incremental learning strategy is more likely to succeed compared to training on the entire data set from the start 6]. Symbolic rules about the learned grammar can be extracted in the form of DFAs 8, 12, 22, 46, 62, 63]. The extraction algorithms are generally based on the hypothesis that the outputs of the recurrent state neurons tend to cluster when the network is welltrained and that these clusters correspond to the states of the learned DFA. Thus, rule extraction is reduced to nding clusters in the output space of state neurons and transitions between clusters. many clustering methods exist 22, 62, 63, 66]. The clustering method we use is based on partitioning; partitions and transitions between partitions correspond to DFA states and state transitions, respectively.
IV. Extraction of Symbolic Knowledge

b

0

=?H/2
S t+1 0

b =?H/2 b =?H/2 j i t+1
Si
t+1 Sj

It k

q

a k j

q

i

(a)

W =?H 0ik or W =+H 0ik
St 0
t Si

W = + H Wjjk = ? H ijk

t Sj

It k

a (b)

k

Figure 2: (a) A known DFA transition is programmed into a network. (b) Recurrent network unfolded over two time steps t and t + 1. The insertion algorithm consists of two parts: Programming the network state transition and programming the output of the response neuron. Neurons Si and Sj correspond to DFA states qi and qj , respectively; Ik denotes the input neuron for symbol ak . Programming the weights Wijk ; Wjjk and biases bi and bj as shown in the gure ensures a nearly orthonormal internal representation of DFA states qi and qj (active/inactive neurons are illustrated by shaded/white circles). The value of the weight W0jk connected to the response neuron S0 depends on whether DFA state qi is an accepting or rejecting state. H denotes the rule strength. The operation Si Ik is shown as . Our algorithm employs a dynamical state space exploration along with pruning heuristics to make the extraction computationally feasible 22]. The cluster analysis is followed by a standard DFA minimization procedure 28]. Di erent minimized DFA's may be extracted for di erent partitionings. We have empirical evidence suggesting that, among the many possible DFA's, the DFA extracted with the coarsest partitioning which is consistent with the training data best models the knowledge learned by the neural network. Let Mq denote the minimized DFA extracted with partitioning parameter q. We extract DFA's M2; M3 ; M4 ; : : :; as the best representation of the network's knowledge, we choose the rst DFA Mq which is consistent with the training data 50]. It is important to note that the above knowledge extraction algorithm may not capture the true longterm behavior of a trained network for an arbitrary number of time steps 44], i.e. several DFA states that are extracted with the above heuristic may de-

generate to one state due to the presence of attracting xed points. It has also been argued from a dynamical systems point of view that the DFA extraction approach may be problematic 34]. However, our empirical results show the approach works well for extracting a model of the unknown regular source grammar. The mathematical foundation of DFA extraction based on partitioning of a network's state space has been subsequently established in 7, 44]. The approach for learning and extracting the grammatical rules from trained networks has been extended to context-free languages 28] by augmenting the basic recurrent network architecture with an external continuous stack 11, 24, 61, 65]. The augmented architecture implements a neural network pushdown automaton. There is an increased interest in implementing neural network architectures in analog VLSI 3, 26, 30, 32, 38, 39, 54, 56]. Analog implementations o er the advantage of speed and low power consumption. Several methods for mapping DFAs into recurrent neural network architectures have been proposed 2, 16, 18, 36, 45]. Our algorithm takes advantage of the second-order structure of the recurrent network model which allows for a natural mapping of DFA state transitions into network state changes. A. DFA Encoding Algorithm Our encoding algorithm illustrated in gure 2 achieves a nearly orthonormal internal representation of the desired DFA dynamics; it constructs a network with n + 1 recurrent state neurons (including the output neuron) and m input neurons from a DFA with n states and m input symbols. There is a one-to-one correspondence between state neurons Si and DFA states qi . For each DFA state transition (qj ; ak ) = qi , we set Wijk to a large positive value +H . The self connection Wjjk is set to ?H (i.e. neuron Sj changes its state from high to low) except for state transitions (qj ; ak ) = qj (self-loops) where Wjjk is set to +H (i.e. state of neuron Sj remains high). Furthermore, if state qi is an accepting state, then we program the weight W0jk to +H ; otherwise, we set W0jk to ?H . We set the bias terms bi of all state neurons Si to ?H=2. For each DFA state transition, at most three weights of the network have to be programmed. The initial state S0 of the network 0 is S0 = (S0 ; 1; 0; 0; : : :; 0). The value of the response 0 neuron S0 is 0 if the DFA's initial state q0 is a rejecting state and 1 otherwise. All weights that are
V. Synthesis of Large Systems

not set to ?H , ?H=2 or +H are set to zero. B. Learning with Prior Knowledge Empirical studies have shown that partial prior knowledge of a DFA (states and transitions) can signi cantly improve the training time 23]. Recurrent networks can even perform rule revision, i.e. re ne incomplete initial rules 17, 36, 37, 57] and correct incorrect prior knowledge through learning from data 48]. C. Stability of Designed Networks The following theorem asserts that time-discrete, continuous recurrent neural networks can represent DFAs 47]: Theorem C. Let L(MDFA) denote the regular lan.1 guage accepted by some DFA with n states over an alphabet = fa1 ; : : : ; aK g. Then, a second-order recurrent neural network (RNN) with n + 1 sigmoidal state neurons and K input neurons (\one-hot encoding") can be constructed such that L(MDFA) = L(MRNN ) for a nite value of the rule strength H . The signi cance of that theorem is that it is possible for a nonlinear discrete dynamical system to be modeled exactly by a nonlinear continuous dynamical system, even in the presence of noise 45, 49]. Thus, discrete nonlinear systems can be mapped onto analog VLSI designs. D. Scaling to Large Networks It would seem only natural that the value of the rule strength H scale with increasing network size n. While there exist degenerate cases where we can observe such a scaling behavior, the rule strength H is largely independent of the size of `average' DFAs 49]. To date, it has been very hard to train recurrent networks on strings generated by DFAs with more than about 20 states. The complexity and poor convergence properties of training algorithms are one reason why learning regular languages with recurrent neural networks does not scale well with DFA size. While partial prior knowledge about the DFA can signi cantly improve the convergence time, it does not overcome the scaling problem, i.e. learning of large DFA with on the order of hundreds of states. However, it has recently been reported that certain subclasses of DFAs with on the order of hundreds and even thousands of states can be learned rather
VI. Large Systems

nant function. Extraction of symbolic knowledge from a trained network is useful since it makes the knowledge which is stored in a network's weights accessible to inspection, analysis, re nement and manipulation. Prior knowledge encoded in the network can be checked and can signi cantly reduce the learning time when used e ectively. The stable insertion of automata (and their rules) into recurrent neural networks is important for robust computing and could possibly in neural network VLSI design. Adaptive fuzzy systems have proven to be very powerful tools for solving control problems. Their parameters often have clear physical meanings which facilitates the choice of their initial values. Although there has been much interest and success in combining fuzzy systems with neural networks, the e orts have focused on static models, i.e. approximating unknown dynamical systems with static input output mappings. However, there exist applications which require state information to be available across inde nite periods of time. Thus, an interesting and open issue is how fuzzy rules and automata can be embedded in recurrent neural networks. 1] Y. Abu-Mostafa, \Learning from hints in neural networks," Journal of Complexity, vol. 6, p. 192, 1990. 2] R. Alquezar and A. Sanfeliu, \An algebraic framework to represent nite state automata in single-layer recurrent neural networks," Neural Computation, 1995. In press. 3] J. Alspector and R. Allen, \A neuromorphic VLSI learning system," in Advanced Research in VLSI: Proceedings of the 1987 Stanford Conference (P. Losleben, ed.), (Cambridge), pp. 313{ 349, MIT Press, 1987. 4] P. Ashar, S. Devadas, and A. Newton, Sequential Logic Synthesis. Norwell, MA: Kluwer Academic Publishers, 1992. 5] J. Bajer and P. Lisonek, \Symbolic computation approach to nonlinear dynamics," Journal of Modern Optics, vol. 38, no. 4, p. 719, 1991. 6] Y. Bengio, P. Simard, and P. Frasconi, \Learning long-term dependencies with gradient descent is di cult," IEEE Transactions on Neural Networks, vol. 5, no. 2, pp. 157{166, 1994.
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