Absence of phase transition in random language model

Abstract

The random language model, proposed as a simple model of human languages, is defined by the averaged model of a probabilistic context-free grammar. This grammar expresses the process of sentence generation as a tree graph with nodes having symbols as variables. Previous studies proposed that a phase transition, which can be considered to represent the emergence of order in language, occurs in the random language model. We discuss theoretically that the analysis of the ``order parameter'' introduced in previous studies can be reduced to solving the maximum eigenvector of the transition probability matrix determined by a grammar. This helps analyze the distribution of a quantity determining the behavior of the order parameter and reveals that no phase transition occurs. Our results suggest the need to study a more complex model such as a probabilistic context-sensitive grammar, in order for phase transitions to occur.