Rules are an efficient feature of natural languages which allow speakers to use a finite set of instructions to generate a virtually infinite set of utterances. Yet, for many regular rules, there are irregular exceptions. There has been lively debate in cognitive science about how individual learners acquire rules and exceptions; for example, how they learn the past tense of preach is preached, but for teach it is taught. However, for most population or language-level models of language structure, particularly from the perspective of language evolution, the goal has generally been to examine how languages evolve stable structure, and neglects the fact that in many cases, languages exhibit exceptions to structural rules. We examine the dynamics of regularity and irregularity across a population of interacting agents to investigate how, for example, the irregular teach coexists beside the regular preach in a dynamic language system. Models show that in the absence of individual biases towards either regularity or irregularity, the outcome of a system is determined entirely by the initial condition. On the other hand, in the presence of individual biases, rule systems exhibit frequency dependent patterns in regularity reminiscent of patterns found in natural language. We implement individual biases towards regularity in two ways: through child agents who have a preference to generalise using the regular form, and through a memory constraint wherein an agent can only remember an irregular form for a finite time period. We provide theoretical arguments for the prediction of a critical frequency below which irregularity cannot persist in terms of the duration of the finite time period which constrains agent memory. Further, within our framework we also find stable irregularity, arguably a feature of most natural languages not accounted for in many other cultural models of language structure.
