Human communication is unparalleled in the animal kingdom. The key distinctive feature of our language is productivity: we are able to express an infinite number of ideas using a limited set of words. Traditionally, it has been argued or assumed that productivity emerged as a consequence of very specific, innate grammatical systems. Here we formally develop an alternative hypothesis: productivity may have rather solely arisen as a consequence of increasing the number of signals (e.g. sentences) in a communication system, under the additional assumption that the processing mechanisms are algorithmically unconstrained. Using tools from algorithmic information theory, we examine the consequences of two intuitive constraints on the probability that a language will be infinitely productive. We prove that under maximum entropy assumptions, increasing the complexity of a language will not strongly pressure it to be finite or infinite. In contrast, increasing the number of signals in a language increases the probability of languages that have—in fact—infinite cardinality. Thus, across evolutionary time, the productivity of human language could have arisen solely from algorithmic randomness combined with a communicative pressure for a large number of signals.

中文翻译：

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交流压力下的偶然性可以产生无限生产力的语言

人类的交流在动物界是无与伦比的。我们语言的关键显着特征是生产力：我们能够使用有限的词组表达无限数量的想法。传统上，有人争论或假设生产力的出现是非常具体的、先天的语法系统的结果。在这里，我们正式提出了一个替代假设：在处理机制在算法上不受约束的额外假设下，生产力可能只是作为增加通信系统中信号（例如句子）数量的结果而产生的。使用算法信息论中的工具，我们研究了两种直观约束对语言无限生产力概率的影响。我们证明在最大熵假设下，增加语言的复杂性不会强烈地迫使它成为有限或无限。相比之下，增加语言中的信号数量会增加语言具有（实际上是）无限基数的可能性。因此，在整个进化过程中，人类语言的生产力可能完全来自算法随机性以及大量信号的交流压力。