The Fundamental Theorem of Language Change (Yang, 2000) implies the impossibility of stable variation in the Variational Learning framework, but only in the special case where two, and not more, grammatical variants compete. Introducing the notion of an advantage matrix, I generalize Variational Learning to situations where the learner receives input generated by more than two grammars, and show that diachronically stable variation is an intrinsic feature of several types of such multiple-grammar systems. This invites experimentalists to take the possibility of stable variation seriously and identifies one possible place where to look for it: situations of complex language contact.

中文翻译：

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多维竞争中的稳定变化

语言变化的基本定理 (Yang, 2000) 意味着在变分学习框架中不可能有稳定的变化，但仅限于两个而不是更多的语法变体竞争的特殊情况。引入优势矩阵的概念，我将变分学习推广到学习者接收由两种以上语法生成的输入的情况，并表明历时稳定变化是几种类型的多语法系统的内在特征。这邀请实验者认真对待稳定变异的可能性，并确定一个可能的地方去寻找它：复杂语言接触的情况。