A Dynamical Systems Model for Language Change
Partha Niyogi
Electronic mail address: niyogi@research.bell-labs.com.
Robert C. Berwick
Electronic mail address: berwick@ai.mit.edu.
Artificial Intelligence Laboratory,
Center for Biological and Computational Learning,
Massachusetts Institute of Technology,
Cambridge, MA 02139
Abstract
This paper formalizes linguists' intuitions about language change, proposing a dynamical systems model for language change derived from a model for language acquisition. Linguists must explain not only how languages are learned but also how and why they have evolved along certain trajectories and not others. While the language learning problem has focused on the behavior of individuals and how they acquire a particular grammar from a class of grammars 
, this paper considers a population of such learners and investigates the emergent, global population characteristics of linguistic communities over several generations. It is argued that language change follows logically from specific assumptions about grammatical theories and learning paradigms. Roughly, as the end product of two types of learning misconvergence over several generations, individual language learner behavior leads to emergent, population language community characteristics.
In particular, it is shown that any triple 
of grammatical theory, learning algorithm, and initial sentence distributions can be transformed into a dynamical system whose evolution depicts the evolving linguistic composition of a population. It is explicitly shown how this transformation can be carried out for memoryless learning algorithms and parameterized grammatical theories. As the simplest case, the example of two grammars (languages) differing by exactly one binary parameter is formalized, and it is shown that even this situation leads directly to a quadratic (nonlinear) dynamical system, including regions with chaotic behavior. The computational model is applied to some actual data, namely the observed historical loss of "verb second" from old French to modern French. Thus, the formal model allows one to pose new questions about language phenomena that one otherwise could not ask, such as the following.
1. Do languages (grammars) correctly follow observed historical trajectories? This is an evolutionary criteria for the adequacy of grammatical theories.
2. What are the logically possible dynamical change envelopes given a posited grammatical theory? These are rates and shapes of linguistic change, including the possibilities for the past and the future.
3. What can be the effect of quantified variation in initial conditions? For example, population differences resulting from socio-political facts.
4. Other intrinsically interesting mathematical questions regarding linguistic dynamical systems.
