# A Cultural Evolutionary Model for the Law of Abbreviation

Olivier Morin, Alexey Koshevoy

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## To cite this version:

Olivier Morin, Alexey Koshevoy. A Cultural Evolutionary Model for the Law of Abbreviation. Topics in cognitive science, 2024, 10.1111/tops.12782. hal-04861584

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A cultural evolutionary model for the law of abbreviation

Olivier Morin¹ & Alexey Koshevoy²,¹

1. Institut Jean Nicod, Département d’études cognitives, ENS, EHESS, CNRS, PSL University,
UMR 8129, 29, rue d’Ulm 75230 Paris. olivier.morin@ens.psl.eu

2. Centre de Recherche en Psychologie et Neurosciences, Aix-Marseille Université, UMR 7077,
3, place Victor Hugo 13003 Marseille, France. alexey.koshevoy@univ-amu.fr

Word count (including references): 7200 words + 5 figures

Word count (excluding references): 6300 words + 5 figures

**Keywords:** Zipf; Law of abbreviation; Law of brevity; Communicative efficiency; Cultural evolution; Informativeness; Word length; Functionalism.

Paper accepted in TopiCS in Cognitive Science — this is the last submitted version.

DOI: https://doi.org/10.1111/tops.12782
A cultural evolutionary model for the law of abbreviation

**Abstract.** Efficiency principles are increasingly called upon to study features of human language and communication. Zipf's law of abbreviation is widely seen as a classic instance of a linguistic pattern brought about by language users' search for efficient communication. The "law"—a recurrent correlation between the frequency of words and their brevity—is a near-universal principle of communication, having been found in all of the hundreds of human languages where it has been tested, and a few non-human communication systems as well. The standard explanation for the law of abbreviation derives it from pressures for efficiency: speakers minimize their cumulative effort by using shorter words for frequent occurrences. This explanation, we argue here, fails to explain why long words exist at all. It also fails to explain why the law of abbreviation, despite being robust, is systematically weakened by many short and rare words. We propose an alternative account of the law of abbreviation, based on a simple cultural evolutionary model. Our model does not require any pressure for efficiency. Instead, it derives the law of abbreviation from a general pressure for brevity applying to all words regardless of their frequency. This model makes two accurate predictions that the standard model misses: the correlation between frequency and brevity is consistently weak, and it is characterized by heteroskedasticity, with many short and rare words. We argue on this basis that efficiency considerations are neither necessary nor sufficient to explain the law.

# 1. Introduction
Thinking of our mental life as resting on a range of cognitive technologies can help us better
understand some aspects of language, writing, numerical notations, and other tools of the mind.
Our cognitive toolkit includes cultural inventions like writing, mnemonic routines (Riggsby,
2024; Yates, 1974), diagram-making and picture-based reasoning (Fusaroli & Tylén, 2014;
Giardino & Greenberg, 2015), navigation tools (Hutchins, 1995), and other innovations that
emerged over historical time, in specific cultures. These techniques serve specific purposes: their
form can be, to some extent, predicted from their function.

A promising research program consists in studying forms of communication in this light, as if
they were cognitive technologies. Spoken or signed languages, writing systems, the alarm calls
or mating songs of many non-human animals, pictographs, emojis, etc. can all be seen as tools to
transmit information (Everett, 2012). Taking this angle has several benefits. It encourages a
unification of the study of communication in all its forms, going beyond human language. It
opens up the possibility of comparing forms of communication for efficiency—asking, for
instance, whether the notion that all languages are equally complex and equally expressive has a
factual basis (Lupyan & Dale, 2010; Sampson et al., 2009; Shcherbakova et al., 2023). Finally
and most importantly, it invites us to adopt a *functionalist* perspective—to explain the form of
communication systems by their function.

Functionalism, in this sense, is the focus of this paper—a specific brand of functionalism, one
that is concerned with communicative efficiency. (Linguistics has produced several other
functionalist schools of thought, mostly concerned with explaining syntactic structures on the
basis of communicative efficiency—Croft, 1995; Van Valin Jr, 2003—but we are not directly
concerned with them here.) Efficiency principles require that the costs of communication be as
low as possible, relative to its benefits (Gibson et al., 2019; Martinet, 1971; Zipf, 1949). A
prominent example is Zipf's principle of economy (Zipf, 1949: 21), which posits that spoken
languages are shaped by a constraint that consists in balancing the benefits of communication
with its energetic costs. On the benefits side, Zipf counted the information transmitted by a given
message; on the costs side, the articulatory effort expended by the speaker of the message
(today's researchers often add the cognitive cost associated with processing, on the speaker's
side). Based on the principle that spoken languages occupy a point of equilibrium between these
two constraints, Zipf was able to deduce a number of regularities, such as the law of
abbreviation, according to which frequent words tend to be short, or the law of meaning,
according to which short words have more meanings (Zipf, 1945). This principle and certain
laws derived from it have been generalized to non-human communication (Heesen et al., 2019)
and to non-verbal symbols such as letter shapes (Koshevoy et al., 2023).

We often took this functionalist perspective in our own work (Koshevoy et al., 2023; Pavlek et
al., 2020): we know how useful it can be. In this paper, however, we will play devil's advocate
and stress important lacunas of theories grounded in efficiency principles. The functionalist
perspective, we will argue, lacks mechanistic foundations. Functionalism offers post-hoc
rationalizations of well-known phenomena, but its predictive power is weak: it cannot tell us
why functional pressures prevail, or fail to prevail, over other pressures that run counter to them.
It progresses by accumulating positive evidence (cases compatible with the theory) instead of successfully confronting its predictions with those of alternative theories. The reason for this is a dearth of plausible null models that could serve as touchstones against which the functionalist account might be tested.

Instead of a systematic overview, we will illustrate this general point with an examination of Zipf's law of abbreviation, a highly robust correlation between word brevity and word frequency. The next section (section 2) introduces the law, and presents its standard functionalist explanation, arguing that this explanation to be incomplete and empirically inadequate. Section 2 explains how the standard account recognises, but fails to tease apart, three possible mechanisms that could lead speakers to align words' lengths with their frequency. Section 4 presents a cultural evolutionary model for the law of abbreviation. This model eschews functionalist assumptions and, contrary to the standard account, it explains why the frequency-brevity correlation is weakened by an abundance of rare, short words. In the conclusion (Section 5) we reflect on the role that such null models as the one we present could play. Rather than refuting the functionalist perspective, they provide a demanding baseline against which improved versions of the functionalist account can be tested.

## 2. Functionalism cannot explain the law of abbreviation
The Law of Abbreviation (also known as “Zipf’s law of abbreviation”, or the “law of brevity”, and not to be confused with “Zipf’s Law”, which names the fact that word frequencies are approximately power-law distributed, but does not concern word lengths) is a correlation between the brevity of words in a language and their frequency (see **Fig. 1** as an example of this relation, in the Brown corpus of American English). Although the word “law” suggests a perfect correlation (or at least a rather tight one), the correlation is in fact never statistically strong (for instance, it is equal to -0.17 in the case of the 40,234 English words in the Brown corpus). The correlation is, however, remarkably constant. Across 1,498 languages, Bentz & Ferrer i Cancho (2016) found a negative correlation between word length and word frequency in all the languages studied, with no exception (see also Petrini et al., 2023, Piantadosi et al., 2011).

<figure><img src="image_3.png" /></figure>
**Fig. 1.** Zipf's law of abbreviation in the Brown University Standard Corpus of Present-Day American English. Notice the abundance of short rare words (bottom left).

The most common explanations of the law of abbreviation in different modalities, as derived from Zipf's own work, are functionalist (Gibson et al., 2019; Kanwal et al., 2017; Koshevoy et al., 2023; Martinet, 1971; Piantadosi et al., 2011; Zipf, 1949). Such explanations start from the premise that communication is efficient when communicators manage a trade-off between the benefits of conveying information to others, and the effort devoted to communication, trying to minimize the costs of communication without diminishing its benefits. From this perspective, the time and effort devoted to pronouncing a word is a cost that increases with word length, while the successful transmission of information to one's interlocutor is a benefit that can be achieved with words of various lengths. The law of abbreviation is supposed to maximize efficiency because, as Zipf remarked, the overall cost of pronouncing words is lower when shorter words are more frequent. The same principle lies at the heart of Huffman coding in information theory: longer symbols, which require more bits of information to encode, should be reserved for less frequent uses, in order to minimize the overall weight of encoded sequences (Huffman, 1952).

Long before Huffman and Zipf theorized it, Morse had intuited this principle when designing his Morse code: he spontaneously associated the shortest signals with the most frequent letters
(whose frequency he knew by the size of the corresponding cells in the printers' type box)
(Pfrehm, 2018)<fnref n="1" />.

The standard account is intuitive, and indeed it rests on two incontrovertible assumptions. The Huffman coding principle is mathematically true: it tells us that any optimally efficient (binary) code (of reasonable size) would necessarily respect the law of abbreviation. Another true assumption: all other things being equal, the greater the correlation between length and brevity, the lower the cost of producing words. Thus, the law of abbreviation is undoubtedly a feature that makes communication more efficient; what is more, any optimal communication system would have to conform to the law.

When confronted with real languages, however, the functionalist account of the law of abbreviation presents, in our view, two main flaws.

First, efficiency wrongly predicts that words should be shorter than they really are. If languages are designed in such a way as to minimize the cost of producing long words, there should be as few long words as possible. This is true of the ideal language posited by Huffman coding. In that language, words are made of binary strings. When creating new words, all possible binary strings of length n are used before moving on to strings of length n+1, and so on. Creating words longer

<footnote n="1">¹ The scope and limitations of the analogy with Morse code are worth stressing. Samuel Morse's coding scheme transcribed the symbols of English writing (mostly alphabet letters) into telegraphic signals. He attempted to match the length of signals to letter frequency, but his system was not strictly speaking optimal. Morse's initiative resembles the law of abbreviation, since he implemented a correlation between symbol frequency and symbol length. However, Morse did not concern himself with word length.</footnote>
than a certain length is only efficient when all the possible words of that length already exist.

Natural languages, however, do not work in this way: they typically leave unused a large number of possible words that would be shorter than the average word, even when phonotactic rules are met (Dautriche et al., 2017). Consider these English monosyllables: *blop*, *grix*, *snurk*, *quomp*, *zilth*. They all respect the phonological constraints of English; none of them are English words.

One way to solve this difficulty is to say that, in natural languages (as opposed to the artificial languages that Huffman coding principles are based on), length can be efficient, because it makes words easier to discriminate, and less easily corrupted by noise. This is how Zipf implicitly approached this issue: communication was not about minimising effort, but about achieving a trade-off between communicative costs and communicative benefits. It is probably true that long words are more distinctive, and desirable for this reason (e.g. Ernestus et al., 2002). However, admitting this point leads to difficulties when it comes to solving functionalism's second major problem, to which we now turn.

Second, *efficiency does not explain why the law of abbreviation is weak, due to an abundance of words that are both short and rare*. Straightforwardly, the functionalist logic predicts that short words be used more frequently than longer words. But real languages systematically contain many short and rare words. Statistically speaking, this results, first, in relatively low values for the brevity-frequency correlation, typically ranging around 0.3 (Fig. 2), second, in systematic heteroskedasticity: there is much more variance in length for low-frequency words than for high-frequency ones. Almost all very frequent words are short, but infrequent words may be short as
well as long. This cannot just be statistical noise (i.e. due to imperfect measurements or random departures from the correlation), for two reasons. First, the frequency-length correlation is weak more or less independently of the way length and frequency are measured; in particular, it remains weak when length measurements are based on phonological length instead of letter counts — a more direct and accurate measure (Petrini et al., 2023). Second, statistical noise would degrade the correlation between length and frequency homogeneously, but this is not what we see. The law of abbreviation always breaks down in the same way: languages contain too many short, rare words. They do not contain too many long, frequent words.

<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Histogram of the negative correlation between word length and word frequency in 1,498 languages, using data from Bentz & Ferrer-i-Cancho (2016).</figcaption></figure>
This second problem is not impossible to solve. One may argue that languages maximise word brevity, and thus use all (or nearly all) the possible short words they could possibly invent, and only then move on to using longer words. Since there are many possible short words, and only a few highly frequent words, this results in a small number of short, frequent words, followed by a large set of infrequent words that may be indifferently long or short. This is consistent with Huffman coding. But this solution only works if words are designed solely to maximise brevity. It is not compatible with the claim that having long words can be efficient. If length is, in some cases at least, a desirable property in a word, one has to explain why so many words are so short, even though they are infrequent, meaning that there is little pressure to shorten them. If length is a desirable property, why do speakers not lengthen rare words? This would make the law of abbreviation strong; yet the law of abbreviation is weak.

The inadequacies of functionalism as applied to the law of abbreviation illustrate, in our view, the more general issues that we mentioned earlier.

*Post-hoc rationalization.* Functionalism rationalizes the robust correlation between word brevity and word frequency that Zipf discovered, by showing that it makes speech more efficient. But this rationalization is not a complete explanation: even assuming that a pressure to make communication efficient is really at work, there are other mechanisms acting against the law of abbreviation (Pimentel et al., 2021), and the functionalist account does not explain why they should fail to prevail.
*A focus on confirmation, not refutation.* Since the functionalist account was put forward, the field has progressed largely by accumulating positive evidence—phenomena consistent with the functionalist view. Much attention is paid to the fact that speakers tend to shorten words in contexts where they are predictable (Jaeger & Buz, 2018; Kanwal et al., 2017; Mahowald et al., 2013), or that informativeness is a better predictor of word length than frequency (Piantadosi et al., 2011; but see Ferrer-i-Cancho & Martín, 2011, Levshina, 2022;); unpredicted phenomena, like the abundance of short, rare words, receive less scrutiny.

*A dearth of plausible null models.* Anomalies like short, rare words receive little attention because we do not know what to do with them: there are few plausible null models against which the functionalist account might be assessed. Random models have been proposed for the law of abbreviation (Caplan et al., 2020); but they rely on cognitively implausible assumptions (words are produced purely at random, as if by monkeys at a typewriter) and do not attempt to model the characteristic features of the law of abbreviation in natural languages.

Our goal in this paper is to propose an adequate null model for the law of abbreviation. This model should explain why the frequency-brevity correlation exists, but also how it can be as weak as we observe it to be. It should predict that the law is weak due to an abundance of short, rare words (but not frequent long words); and it should do all this without calling upon functionalist assumptions. To do this, we start by reconsidering the role that word frequency plays in the law of abbreviation.
### 3. Three roles for word frequency in producing the law of abbreviation

A relation between word frequency and word length may come about in at least three different ways.

(1) **The "thinning coins" mechanism.** If all words are somewhat likely to be shortened when they are uttered, frequent words, being uttered more, will on average be shortened more. We call this the "thinning coins" mechanism after Schuchardt (Schuchardt, 1885), who likened words to coins that get thin from passing from hand to hand. This mechanism simply posits a general tendency for speakers to shorten words, whatever words they utter. Frequent words, being more often uttered, mechanically get shortened more. Under this mechanism, speakers indifferently shorten any word they utter. In other words, they are equally likely to shorten a frequent or an infrequent word, conditional on uttering it. However, speakers happen to utter some words more frequently than others, and so they shorten them more.

(2) **The “selective shortening of frequent words” mechanism:** Speakers are more likely to shorten frequent words than infrequent ones. In this mechanism, speakers take into account (consciously or not) the fact that the cumulative cost of uttering frequent words is (all else being equal) higher than the cost of infrequent words. The total cost of a frequent word weighs more heavily in the speakers' energy budget. Speakers may also (depending on the accounts) represent
the fact that frequent words are easier for listeners to predict, and thus convey less information.

Given this, an efficient speaker will prioritise shortening frequent words. In this scenario, frequent words get shorter because speakers specifically target frequent words for shortening, thus reducing their cumulative effort. This in contrast with (1), the thinning coins mechanism, where speakers are as likely to shorten frequent or infrequent words, conditional on uttering them.

**(3) The “preferential re-use of frequent words” mechanism.** Speakers preferentially select short words when given a choice between synonyms, or when searching for a word to express a new meaning. Contrary to mechanisms (1) and (2), this mechanism does not require speakers to reduce long words into shorter ones. It simply posits that, among existing words, short ones are preferentially selected by speakers for new uses.

All three mechanisms are compatible with one another; all three can plausibly contribute to the law of abbreviation. Accounts of the law, since Zipf himself, usually avail themselves of all three types of explanation, with emphasis laid on the second and third mechanisms. Those three mechanisms are nonetheless quite distinct. Each could exist without the other two, and each makes distinct assumptions about the evolution of the law of abbreviation and about speakers' decisions.

The first mechanism is recognised by some early accounts of the law of abbreviation (Schuchardt, 1885; Zipf, 1949) but as far as we know it has never been used to explain the law of
abbreviation on its own. The “thinning coins” mechanism does not assume that speakers aim to
make their lexicon efficient in any way. No one, as far as we know, assumes that shortening all
the words in a lexicon might make it more efficient. Scholars who explain the law of
abbreviation on the basis of efficiency generally believe the lexicon to be efficiently structured,
and there are very few (if any) real-world lexicons where all words are equally short. Having
long words in addition to short ones in a lexicon is arguably useful, because long words are less
vulnerable to noise and easier to distinguish from one another (Ernestus et al., 2002; Jaeger &
Buz, 2018; Mahowald et al., 2013). Obviously, a lexicon made only of one-phoneme words
would raise impossible challenges for efficient communication, having too few distinct
wordforms.

Thus, the “thinning coins” mechanism cannot be seen as a pressure specifically acting to make
communication efficient. Indeed, it arguably has little to do with communication *per se*. A robust
result of transmission chain experiments is progressive loss of the content travelling down the
chain: stories get shorter (Bartlett, 1932), shapes get simpler (Tamariz & Kirby, 2015). This is
true even when the content being transmitted plays no communicative role, having no meaning,
like the meaningless doodles used in Tamariz and Kirby (2015). Simplification, there, merely
results from memory limitations: simpler (or shorter) content is more likely to be memorised
(e.g. Baddeley et al., 1975), and thus travels further down the transmission chain.

The second mechanism (“selective shortening of frequent words”), is more central to standard
explanations of the law of abbreviation (e.g. Zipf 1949, Mahowald et al. 2013), and also plays a
significant role in linguistics at large, as a prominent factor in account for phonological change
(Bybee, 2010; Jaeger & Buz, 2018; Schuchardt, 1885). There is abundant evidence that speakers
preferentially shorten high-frequency words (Bybee, 2007; Pierrehumbert, 2001) as well as
contextually predictable words (Jaeger & Buz 2018, Mahowald et al. 2013) (contextual
predictability is strongly correlated with frequency, and frequency is identical to one-gram
contextual predictability).

One easily sees how the selective shortening of frequent words gives rise to a brevity-frequency
correlation: the two are almost the same. This makes the second mechanism particularly
intuitive: the law of abbreviation exists because speakers directly shape language in accordance
with it. These theories give diverging accounts of the reasons why speakers act in this way, and
on whether these reasons are consciously represented or not; a constant theme, however, is that
speakers attempt to communicate efficiently. What efficiency means may differ from one account
to the other: it could be mere energetic efficiency (a reduction of the cumulative costs of
speaking), as in Zipf's original account, or pragmatic concerns rooted in Grice's maxim of
manner (Grice, 1989; see Levinson, 2000 for a Gricean account of the law of abbreviation seen
as a consequence of manner implicatures).

The last mechanism, selective re-use of short words does not involve reduction, contrary to the
other two. The law of abbreviation may emerge because speakers favor short words when
choosing which words to recycle for novel uses. In this hypothesis, when a new meaning
emerges, speakers are much more likely to re-use an existing word than to create a neologism;
and the word they re-use tends to be short. Why re-use short words specifically? Here again,
because it makes the lexicon more efficient, as it ensures that the most frequently used words are
also the least costly. Consistent with this, polysemous words indeed tend to be shorter
(Piantadosi et al. 2012, see also Hermalin & Regier 2019). As Zipf (e.g. 1949: 62) pointed out,
the selective re-use of short words may result in a positive feedback loop, whereby frequent
words are shortened, short words get even more frequent, get shortened even more, etc. We dwell
on this possibility at greater length in the conclusion.

There has been, as far as we know, no attempt to tease apart the contribution these three
mechanisms make to the law of abbreviation. Our model makes a first step in this direction by
showing that the law of abbreviation can be derived from the “thinning coins” mechanism alone,
without calling upon the other two. In doing so, we also shed light on one anomaly that the
standard account does not readily explain: the abundance of short, rare words.

4. Culturally evolving the law of abbreviation

Our simple model (see https://github.com/alexeykosh/2023-dripf-model) simulates an evolving
population of agents who learn elements of a lexicon from one another; this lexicon is composed
of words of a certain length, associated with meanings of a certain frequency (Fig. 3). A number
Na of agents interact at each generation, communicating with each other. Each agent has a
repertoire of N<sub>m</sub> words, associated with N<sub>m</sub> meanings. Each word's length is initially determined
by sampling from a uniform distribution (ranging between 0 and 20).

A number N<sub>i</sub> of interactions take place at each generation. This number is a multiple (which we
set at 5) of the number of meanings in the lexicon, N<sub>m</sub>. Having the number of interactions depend
on the number of meanings allows us to avoid cases where a great number of meanings are never
subject to any interaction between agents, which would happen if we could make the number of
meanings arbitrarily high and the number of interactions arbitrarily low.

At each interaction, two agents are drawn at random and each can learn from the other a word
corresponding to a meaning. In real conversations, some meanings are mentioned much more
frequently than others (compare “day” to “xylophone”). To reflect this, our model assigns a
frequency to each meaning, each meaning’s frequency being drawn from a power law
distribution (with exponent 1.5, which is similar to what is attested for natural languages). This
choice of distribution, which schematically reflects empirical data (Piantadosi, 2014), is not
crucial: we replicate our main finding with a flat distribution of meaning frequencies instead of a
power-law distribution.

Each interaction focuses on one meaning, picked by weighting the probability of drawing each
meaning according to its frequency, with more frequent meanings having a greater chance of
being picked. This reflects the fact that frequent meanings are, by definition, more likely to be
mentioned in conversations.
When an interaction takes place, a meaning having been picked, one of the agents adopts the word that the other agent associates with that meaning. Adoption can take two forms: random adoption or selection. When adoption takes place at random (which occurs with probability 1-S), one of the two agents is selected at random to adopt the other agent's word. In case selection occurs instead, which it does with probability S, we compare the lengths of the words that the two agents associate with the meaning in question; the agent with the longest word adopts the word of the other agent.

This selection of shorter words implements the “thinning coins” mechanism described earlier: agents are equally likely to shorten any word, frequent or not, conditional on encountering it; but more frequent words are unlikely to be encountered. This is different from the “selective shortening of frequent words”, since agents do not preferentially shorten frequent words. As we discussed earlier, the “thinning coins” mechanism is distinct from any pressure to make frequent words easier to produce (for speakers) or process (for listeners). Word shortening, in our model, is not guided by a trade-off between informativeness and economy. We never assume that word shortening is caused by a pressure for efficiency (as distinct from, say, memory constraints); and shortening can make words less informative, for instance by making short words insufficiently distinctive from one another.

Once the number $N_i$ of interactions has taken place, the population of agents is partially renewed: a fraction $\mu$ of the population is replaced by entirely different agents who associate different,
randomly drawn words with the N<sub>m</sub> meanings. When turn-over occurs, the new agents' lexicon is completely reset, with no overlap between this lexicon and that of their predecessors. This is meant to reflect the fact that languages change through time, with the arrival of new individuals who have not yet learned the group's lexicon.

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** A synthetic description of our model.</figcaption></figure>

Our model reproduces the law of abbreviation for a broad range of parameter values (Fig. 4): we observe a negative correlation between the frequency of a meaning and the average length of the words associated with that meaning across agents. The correlations are generally weak;
importantly, they are weak for the right reason: we reproduce the abundance of short, rare words that weakens the correlation (Fig. 5).

<figure><img src="image_6.png" /></figure>

**Figure 4. The main results of our model.** Panels (A) and (B) show the values we get for the law of abbreviation, i.e., the correlation between word brevity and the frequency of associated meanings (Spearman's rho), for the full range of values taken by the selection parameter (S) and the turnover rate (μ). In (A), each cell represents the average correlation value obtained at the end of 100 simulations for the given values of S and μ. In (B), the blue curve represents the average value of the abbreviation law, as a function of the value of the selection rate S, over the set of all simulations we ran for this value of S. The orange curve represents the mean value of the abbreviation law, as a function of the value of turnover rate μ, over the set of all simulations we ran for this value of μ. Panel (C) shows the values we get for the law of abbreviation, for a
range of values assigned to the number of agents in the population, and the number of meanings
in their language (each cell averaging the result of 100 simulations). **Panel (D)** shows how the
law of abbreviation increases as a function of the number of interactions inside a given
generation; each point represents an average of 100 simulations (with 95% confidence interval).
In all simulations, for all four panels and unless specified otherwise, number of agents = 1,000,
number of meanings = 100, number of interactions = number of meanings times 5, number of
generations = 20, selection rate S = 0.5, turnover rate μ = 0.5.

<figure><img src="image_7.png" /></figure>
**Fig. 5.** An example of a weak correlation between the frequency of the meaning associated with each word (y axis) and the mean brevity of that word, computed across agents (x axis). The correlation is weak (Spearman's rho = 0.1) due to the abundance of short, rare words. There is massive heteroskedasticity, with considerably more variance in the low-frequency range: many short and rare words. The same configuration emerges for a broad range of parameters, i.e. whenever the model yields a negative frequency-length correlation.

The only configurations where we fail to get realistic values for the law of abbreviation are when there is no selection for brevity (S is equal to zero); when the turnover rate is maximally high (so the population is renewed every generation and the law of abbreviation has no time to emerge); and when the number of meanings is very low relative to the number of agents (e.g. a thousand times lower). We ran our model for 100 generations when generating our results, but using a much larger number of generations would not change our results drastically; in particular, it would not make the law of abbreviation markedly stronger, since the correlation between brevity and frequency plateaus after a small number of generations.

How does the law of abbreviation emerge without a pressure for efficient communication? Even though selection for brevity is the same for all words irrespective of their length, more frequent words are more likely to be the object of an interaction. Thus, words associated with frequent meanings are more likely to undergo selection for brevity, resulting in lower mean length and reduced variance in the high-frequency range. Interestingly, this effect does not depend much on
the strength of selection (the value of S), as long as S is higher than zero. In other words, *the relative weakness of the law of abbreviation does not mean selection for brevity is itself weak*: even when agents always adopt the shorter of two words during their encounters, the fact that infrequent words are rarely encountered ensures that many rare words are never selected for brevity. Even high rates of selection for brevity can result in realistically low values for the frequency-brevity correlation; conversely, even relatively low rates of selection for brevity are sufficient for the correlation to emerge.

Simplistic as it may be, our model shows how the law of abbreviation can be derived without assuming that speakers have any tendency to shorten frequent words specifically.

**5. Conclusion**

Our model shows that the law of abbreviation can emerge even when agents do not specifically seek to reduce the length of words associated with frequent meanings. This model helps explain the systematic weakness of the law of abbreviation, in the sense that it generates plausibly low values for the correlation between brevity and frequency, for a wide range of parameter values, reproducing the large number of rare, short words typical of the law in real languages. It also suggests that a strong general tendency to shorten words is perfectly compatible with relatively weak values for the law of abbreviation.
This simple model has some clear limitations, worth pointing out. Real language change is more
gradual than we make it out to be: in our model, new words are introduced through population
turn-over, that is, through new agents that appear in the population with a fully reinitialised
lexicon, comprising only novel words. A more realistic version of the model would make lexical
change less abrupt, with the probable result of slowing down evolutionary change. Another
simplification concerns the mapping of words onto meanings. In our model, each word
corresponds to a single meaning. Real lexicons obviously differ, with potential consequences for
the law of abbreviation, as polysemy is not evenly distributed but favors words that are frequent
(Zipf 1949) and short (Piantadosi et al. 2012). Taking this mechanism into account would likely
increase the strength of the law of abbreviation in our model (as we argue below).

For these reasons, our model is not meant to provide an accurate picture of the law of
abbreviation or its causes. We only intend it to illustrate a very general mechanism that
contributes to the correlation between word brevity and word frequency: the fact that frequently
encountered words are much more likely to be shortened than infrequently encountered ones. We
cannot conclude from this that the mechanism modeled here is the only one driving the law of
abbreviation.

The exact values that we obtain for the frequency-brevity correlation can be quite variable and
may be above or below the range of real values in human languages, depending on the way
parameters are set. The model obviously needs to be constrained using more realistic estimates
for its parameters; when we do this, we may end up with a range of values for the law of
abbreviation that is lower than the real range. That would suggest that our mechanism is by itself
insufficient to explain the full strength of the law of abbreviation, and needs complementing.
This is exactly the function of a null model: to be discarded if a less parsimonious hypothesis
makes better predictions.

Thus, we do not rule out the two common explanations of the law of abbreviation reviewed in
section 3: the “selective shortening of frequent words” mechanism and the “selective re-use of
short words” mechanism. The evidence that speakers do in fact selectively shorten frequent
words is overwhelming (especially if we include evidence bearing on contextual predictability
instead to raw frequency) (Bybee, 2007; Jaeger & Buz, 2018), and evidence for the preferential
re-use of short words, though more indirect, cannot be dismissed out of hand (Zipf 1949,
Piantadosi et al. 2012). Our model suggests, however, that a substantial part of the law of
abbreviation can be explained using only the “thinning coins” mechanism alone. That
mechanism has several advantages compared to the other two.

The “thinning coins” mechanism is a parsimonious hypothesis, as it merely requires a general
tendency to shorten words as they get transmitted, a claim consistent with research on memory
constraints and with transmission chain experiments. It is parsimonious, too, in eschewing any
functionalist assumption regarding word lengths. Our model does not assume that word lengths
are optimized to balance the costs of uttering words with the amount of information that one can
transmit in doing so. On the contrary, words in our model have an arbitrary length when
invented, and speakers shorten all words regardless of their frequency. Thus, our model is
compatible with the possibility that word lengths are arbitrary.

In contrast, the functionalist account is taking up a bigger burden of explanation regarding word
lengths, since it is facing a dilemma. If functionalism assumes that long words are always
detrimental to efficient communication, it needs to explain why many languages use long words
instead of using all the possible short words they have access to. Why do long words exist at all?
If, on the other hand, functionalism assumes that long words can make communication more
efficient (for instance because long words are more distinctive or less vulnerable to noise—
Ernestus et al. 2002), it needs to explain why many rare words are short, even though it would be
easy to lengthen these short, rare words, thus making the lexicon more efficient, and the law of
abbreviation stronger.

One of the main advantages of our “thinning coins” model is its capacity to generate realistically
low values for the law of abbreviation. More complex models could include additional
mechanisms, starting with the “selective shortening of frequent words” and “selective re-use of
short words” mechanisms, and see whether these additions strengthen the correlation between
length and brevity. We expect (speculatively) that the full causal scenario proposed by Zipf
would likely result in excessively high values. As Zipf pointed out, when put together, the
selective re-use of short words and the selective shortening of frequent words create a positive
feedback loop: short words get more frequent, frequent words get shortened, etc. This should
result in the evolution of highly frequent and extremely short words, in contrast with large numbers of rare, long words. The actual weakness of the law of abbreviation belies this.

Efficiency matters to communication, at the speakers’ individual level; but does speaker efficiency explain the law of abbreviation? It may have nothing to do with it. Efficiency principles by themselves do not explain the weakness of the law of abbreviation and the importance of short, rare words; and the law of abbreviation can emerge when speakers indifferently shorten any word they encounter instead of fitting length to frequency. Our take on the law of abbreviation does not invalidate the claim that language is an efficient cognitive technology; just that we cannot take it for granted that all the features that make human languages efficient evolved for this reason.

Open code: https://github.com/alexeykosh/2023-dripf-model

**Acknowledgements.** We thank Stephen Chrisomalis, Helena Miton, and Andrea Bender for organizing this special issue of TopiCS. Our work is supported by the ANR grant ComCogMean, ANR-23-CE28-0016 as well as the EUR FrontCog grant ANR-17-EURE-0017 and ANR-10-IDEX-0001-02.

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