Elements of uniformly bounded word-length in groups

AbstractA function on a discrete group is weakly combable if its discrete derivative with respect to a combing can be calculated by a finite-state automaton. A weakly combable function is bicombable if it is Lipschitz in both the left- and right-invariant word metrics. Examples of bicombable functions on word-hyperbolic groups include:(1)homomorphisms to ℤ;(2)word length with respect to a finite generating set;(3)most known explicit constructions of quasimorphisms (e.g. the Epstein–Fujiwara counting quasimorphisms).We show that bicombable functions on word-hyperbolic groups satisfy acentral limit theorem: if$\overline {\phi }_n$is the value of ϕ on a random element of word lengthn(in a certain sense), there areEandσfor which there is convergence in the sense of distribution$n^{-1/2}(\overline {\phi }_n - nE) \to N(0,\sigma )$, whereN(0,σ) denotes the normal distribution with standard deviationσ. As a corollary, we show that ifS1andS2are any two finite generating sets forG, there is an algebraic numberλ1,2depending onS1andS2such that almost every word of lengthnin theS1metric has word lengthn⋅λ1,2in theS2metric, with error of size$O(\sqrt {n})$.

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The Challenges of Large‐Scale, Web‐Based Language Datasets: Word Length and Predictability Revisited

Cognitive Science ◽

10.1111/cogs.12983 ◽

2021 ◽

Vol 45 (6) ◽

Author(s):

Stephan C. Meylan ◽

Thomas L. Griffiths

Keyword(s):

Word Length ◽

Large Scale ◽

Web Based

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EXPRESS: Improving reading rate prediction with word length information: Evidence from Dutch

Quarterly Journal of Experimental Psychology ◽

10.1177/17470218211017100 ◽

2021 ◽

pp. 174702182110171

Author(s):

Marc Brysbaert ◽

Longjiao Sui ◽

Wouter Duyck ◽

Nicolas Dirix

Keyword(s):

Word Length ◽

Word Processing ◽

Reading Rate ◽

English Word ◽

Length Effect ◽

Non Fiction ◽

Word Length Effect ◽

Best Fitting ◽

Dutch Language ◽

Processing Effort

Previous research in English has suggested that reading rate predictions can be improved considerably by taking average word length into account. In the present study, we investigated whether the same regularity holds for Dutch. The Dutch language is very similar to English, but words are on average half a letter longer: 5.1 letters per word (in non-fiction) instead of 4.6. We collected reading rates of 62 participants reading 12 texts with varying word lengths, and examined which change in the English equation accounts for the Dutch findings. We observed that predictions were close to the best fitting curve as soon as the average English word length was replaced by the average Dutch word length. The equation predicts that Dutch texts with an average word length of 5.1 letters will be read at a rate of 238 word per minute (wpm). Texts with an average word length of 4.5 letter will be read at 270 wpm, and texts with an average word length of 6.0 letters will be read at a rate of 202 wpm. The findings are in line with the assumption that the longer words in Dutch do not slow down silent reading relative to English and that the word length effect observed in each language is due to word processing effort and not to low-level, visual factors.

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