Mitchener proposes to study stochastic processes and dynamical systems that model language learning and change, supported by statistical analysis of manuscript and corpus data. The proposed models are designed to account for variability in speech in ways that most existing models do not. Some of the proposed models are deterministic, and represent the distribution of speech patterns in a population as probability density functions governed by Banach-space-valued differential equations. Other proposed models are stochastic, based on discrete Markov chains and stochastic differential equations. The Markov chain models are the most detailed, including simulated learning algorithms based on Bayesian statistics. The PI plans to develop perfect sampling algorithms to determine the mixing behavior of these Markov chains, which should give some insight into linguistic questions of how typical the set of existing human languages is with respect to the set of all possible human languages. The stochastic differential equation models represent the simplification of the Markov chain simulations to an infinite population and continuous time. For all of these models, the PI plans to run computer simulations and prove theorems about their behavior using tools from dynamical systems theory, functional analysis, and probability. The research will first address the simple case of unstructured, well mixed populations, then add social structure, and then delay dynamics as a model of literacy. The PI plans to find hypotheses under which the delay dynamics give rise to well posed problems. The PI is collaborating with linguist Misha Becker at the University of North Carolina at Chapel Hill to develop realistic models of human language learning and to test these models against linguistic data.
Mitchener proposes to study mathematical models of language learning and change. Most existing models assume an idealized form of speech. For example, each person might have just one way of stating a given meaning. In contrast, true speech is variable. The same person can convey the same meaning by using a variety of word orders, for example: John gave Fred a book. John gave a book to Fred. Fred was given a book by John. As an improvement upon models that assume idealized speech, the proposed models account for language variability in two ways. Some of them represent each speaker's utterances with a list of random choices among the alternatives. The others are not explicitly random, but instead keep track of how often a speaker prefers the possible options for expressing a meaning. In a model with idealized speech, learning takes place when a speaker chooses one alternative from a list. Under a variable speech model, learning is much more complicated because each speaker must learn which of many
