Equivalence in Knowledge Representation : Automata , Recurrent Neural Networks , andDynamical Fuzzy

Neuro-fuzzy systems-the combination of artiicial neural networks with fuzzy logic-are becoming increasingly popular. However, neuro-fuzzy systems need to be extended for applications which require context (e.g., speech, handwriting, control). Some of these applications can be modeled in the form of nite-state automata. Previously, it was proved that deterministic nite-state automata (DFAs) can be stably synthesized or mapped into second-order recurrent neural networks with sigmoidal discriminant functions and sparse interconnection topology by programming the networks' weights to +H or ?H. Based on those results, this paper proposes a synthesis method for mapping fuzzy nite-state automata (FFAs) into recurrent neural networks which is suitable for implementation in VLSI, i.e. the encoding of FFAs is a generalization of the encoding of DFAs. The synthesis method requires FFAs to undergo a transformation prior to being mapped into recurrent networks. Their neurons have a slightly enriched functionality in order to accommodate a fuzzy representation of FFA states, i.e. any state can be occupied with a fuzzy membership that takes on values in the range 0; 1] and several fuzzy states can be occupied at any given time 1. The enriched neuron functionality allows fuzzy parameters of FFAs to be directly represented as parameters of the neural network. In this paper we prove the stability of fuzzy nite-state dynamics of constructed neural networks for nite values of network weight H and through simulations give empirical validation of the proofs. 1 This is in contrast to stochastic nite-state automata where there exists no ambiguity about which is an automaton's current state. The automaton can only be in exactly one state at any given time and the choice of a successor state is determined by some probability distribution. For a discussion of the relation between probability and fuzziness, see for instance 46].