Diversity-Based Inference of Finite Automata

We present new procedures for inferring the structure of a finite-state automaton 1994 from its input/output behavior, using access to the automaton to perform experiments. Our procedures use a new representation for finite automata, based on the notion of equivalence between tests. We call the number of such equivalence classes the diversity of the automaton; the diversity may be as small as the logarithm of the number of states of the automaton. For the special class of permutation automata, we describe an inference procedure that runs in time polynomial in the diversity and log(1/δ), where δ is a given upper bound on the probability that our procedure returns an incorrect result. (Since our procedure uses randomization to perform experiments, there is a certain controllable chance that it will return an erroneous result.) We also discuss techniques for handling more general automata. We present evidence for the practical efficiency of our approach. For example, our procedure is able to infer the structure of an automaton based on Rubik's Cube (which has approximately 10¹⁹ states) in about 2 minutes on a DEC MicroVax. This automaton is many orders of magnitude larger than possible with previous techniques, which would require time proportional at least to the number of global states. (Note that in this example, only a small fraction (10^-14) of the global states were even visited.) Finally, we present a new procedure for inferring automata of a special type in which the global state is composed of a vector of binary local state variables, all of which are observable (or visible) to the experimenter. Our inference procedure runs provably in time polynomial in the size of this vector (which happens to be the diversity of the automaton), even though the global state space may be exponentially larger. The procedure plans and executes experiments on the unknown automaton; we show that the number of input symbols given to the automaton during this process is (to within a constant factor) the best possible.