A mathematical model of human languages: The interaction of game dynamics and learning processes.

摘要

Human language is a remarkable communication system, apparently unique among animals. All humans have a built-in learning mechanism known as universal grammar or UG. Languages change in regular yet unpredictable ways due to many factors, including properties of UG and contact with other languages. This dissertation extends the standard replicator equation used in evolutionary biology to include a learning process. The resulting language dynamical equation models language change at the population level. In a further extension, members of the population may have different UGs. It models evolution of the language faculty itself.; We begin by examining the language dynamical equation in the case where the parameters are fully symmetric. When learning is very error prone, the population always settles at an equilibrium where all grammars are present. For more accurate learning, coherent equilibria appear, where one grammar dominates the population. We identify all bifurcations that take place as learning accuracy increases. This alternation between incoherence and coherence provides a mechanism for understanding how language contact can trigger change.; We then relax the symmetry assumptions, and demonstrate that the language dynamical equation can exhibit oscillations and chaos. Such behavior is consistent with the regular, spontaneous, and unpredictable changes observed in actual languages, and with the sensitivity exhibited by changes triggered by language contact.; From there, we move to the extended model with multiple UGs. The first stage of analysis focuses on UGs that admit only a single grammar. These are stable, immune to invasion by other UGs with imperfect learning. They can invade a population that uses a similar grammar with a multi-grammar UG. This analysis suggests that in the distant past, human UG may have admitted more languages than it currently does, and that over time variants with more built-in information have taken over.; Finally, we address a low-dimensional case of competition between two UGs, and find conditions where they are stable against one another, and where they can coexist. These results imply that evolution of UG must have been incremental, and that similar variants may coexist.; This research was conducted under the supervision of Dr. Martin A. Nowak (Program in Theoretical Biology at the Institute for Advanced Study, and Program in Applied and Computational Mathematics at Princeton University).