Revisiting the Optimality of Word Lengths

Tiago Pimentel, Clara Meister, Ethan Wilcox, Kyle Mahowald, Ryan Cotterell

Submission Type: Regular Long Paper

Submission Track: Linguistic Theories, Cognitive Modeling, and Psycholinguistics

Keywords: word length, uniform information density, zipf, law of abbreviation

TL;DR: We formalize the lexicalization problem: how to select wordforms in a lexicon to minimize a specific communicative cost function. We then use it to derive Zipf's law of abbreviation and optimal word lengths under the channel capacity hypothesis.

Abstract: Zipf (1935) posited that wordforms are optimized to minimize utterances' communicative costs. Under the assumption that cost is given by an utterance's length, he supported this claim by showing that words' lengths are inversely correlated with their frequencies. Communicative cost, however, can be operationalized in different ways. Piantadosi et al. (2011) claim that cost should be measured as the distance between an utterance's information rate and channel capacity, which we dub the channel capacity hypothesis (CCH) here. Following this logic, they then proposed that a word's length should be proportional to the expected value of its surprisal (negative log-probability in context). In this work, we show that Piantadosi et al.'s derivation does not minimize CCH's cost, but rather a lower bound, which we term CCH-lower. We propose a novel derivation, suggesting an improved way to minimize CCH's cost. Under this method, we find that a language's word lengths should instead be proportional to the surprisal's expectation plus its variance-to-mean ratio. Experimentally, we compare these three communicative cost functions: Zipf's, CCH-lower , and CCH. Across 13 languages and several experimental settings, we find that length is better predicted by frequency than either of the other hypotheses. In fact, when surprisal's expectation, or expectation plus variance-to-mean ratio, is estimated using better language models, it leads to worse word length predictions. We take these results as evidence that Zipf's longstanding hypothesis holds.

Submission Number: 3938