Probabilistic Generalization of Simple Grammars and Its Application to Reinforcement Learning

Abstract

Recently, some non-regular subclasses of context-free grammars have been found to be efficiently learnable from positive data. In order to use these efficient algorithms to infer probabilistic languages, one must take into account not only equivalences between languages but also probabilistic generalities of grammars. The probabilistic generality of a grammar G is the class of the probabilistic languages generated by probabilistic grammars constructed on G. We introduce a subclass of simple grammars (SGs), referred as to unifiable simple grammars (USGs), which is a superclass of an efficiently learnable class, right-unique simple grammars (RSGs). We show that the class of RSGs is unifiable within the class of USGs, whereas SGs and RSGs are not unifiable within the class of SGs and RSGs, respectively. We also introduce simple context-free decision processes, which are a natural extension of finite Markov decision processes and intuitively may be thought of a Markov decision process with stacks. We propose a reinforcement learning method on simple context-free decision processes, as an application of the learning and unification algorithm for RSGs from positive data.