Fool's Gold: Extracting Finite State Machines from Recurrent Network Dynamics

Several recurrent networks have been proposed as representations for the task of formal language learning. After training a recurrent network recognize a formal language or predict the next symbol of a sequence, the next logical step is to understand the information processing carried out by the network. Some researchers have begun to extracting finite state machines from the internal state trajectories of their recurrent networks. This paper describes how sensitivity to initial conditions and discrete measurements can trick these extraction methods to return illusory finite state descriptions. INTRODUCTION Formal language learning (Gold, 1969) has been a topic of concern for cognitive science and artificial intelligence. It is the task of inducing a computational description of a formal language from a sequence of positive and negative examples of strings in the target language. Neural information processing approaches to this problem involve the use of recurrent networks that embody the internal state mechanisms underlying automata models (Cleeremans et al., 1989; Elman, 1990; Pollack, 1991; Giles et al, 1992; Watrous & Kuhn, 1992). Unlike traditional automata-based approaches, learning systems relying on recurrent networks have an additional burden: we are still unsure as to what these networks are doing.Some researchers have assumed that the networks are learning to simulate finite state Fool’s Gold: Extracting Finite State Machines From Recurrent Network Dynamics machines (FSMs) in their state dynamics and have begun to extract FSMs from the networks' state transition dynamics (Cleeremans et al., 1989; Giles et al., 1992; Watrous & Kuhn, 1992). These extraction methods employ various clustering techniques to partition the internal state space of the recurrent network into a finite number of regions corresponding to the states of a finite state automaton. This assumption of finite state behavior is dangerous on two accounts. First, these extraction techniques are based on a discretization of the state space which ignores the basic definition of information processing state. Second, discretization can give rise to incomplete computational explanations of systems operating over a continuous state space. SENSITIVITY TO INITIAL CONDITIONS In this section, I will demonstrate how sensitivity to initial conditions can confuse an FSM extraction system. The basis of this claim rests upon the definition of information processing state. Information processing (IP) state is the foundation underlying automata theory. Two IP states are the same if and only if they generate the same output responses for all possible future inputs (Hopcroft & Ullman, 1979). This definition is the fulcrum for many proofs and techniques, including finite state machine minimization. Any FSM extraction technique should embrace this definition, in fact it grounds the standard FSM minimization methods and the physical system modelling of Crutchfield and Young (Crutchfield & Young, 1989). Some dynamical systems exhibit exponential divergence for nearby state vectors, yet remain confined within an attractor. This is known as sensitivity to initial conditions. If this divergent behavior is quantized, it appears as nondeterministic symbol sequences (Crutchfield & Young, 1989) even though the underlying dynamical system is completely deterministic (Figure 1). Consider a recurrent network with one output and three recurrent state units. The output unit performs a threshold at zero activation for state unit one. That is, when the activation of the first state unit of the current state is less than zero then the output is A. Otherwise, the output is B. Equation 1 presents a mathematical description. is the current state of the system is the current output.