Irregularity is a pervasive phenomenon in the inflectional morphology of the world’s languages and raises a number of questions about language design, learnability, and change. Nevertheless, irregularity remains an understudied phenomenon and many basic questions remain unanswered (Kiefer, 2000; Stolz et al., 2012). Do all languages exhibit irregularity? What is the relationship between irregularity and frequency? Is irregularity best thought of as a property of individual forms, or a property of more abstract objects like morphological paradigms? In this paper, we examine these questions, focusing in particular on the relationship between irregularity and frequency.
One of the fundamental challenges in studying irregularity is defining the phenomenon in a way that is applicable across languages. We begin the paper by addressing this question. First, we formalize the problem of inflectional morphology and present a novel, information-theoretic measure of the degree of irregularity of an inflected form. This definition builds on recent work that defines (ir)regularity in terms of the probabilistic predictability of a form given the rest of the language (Cotterell et al., 2018a; Ackerman and Malouf, 2013). Making use of a state-of-the-art model of morphological inflection, we estimate our measure of irregularity across a large number of word forms from 28 languages drawn from the UniMorph database (Kirov et al., 2018)
. Based on these estimates we perform three studies. First, we validate our estimates by examining the predictions on English past tense forms—showing that the model’s predicts accord with human judgements of irregularity. We also examine the overall rate of accuracy of our model. Second, we examine the degree of irregularity across languages, showing that the model predicts wide variance in the average amount of irregularity between the languages in our sample. Finally, we provide empirical evidence for a correlation between irregularity and frequency across languages. While this relationship has been observed for individual languages(e.g., English: Marcus et al., 1992; Bybee, 1985), this is the first confirmation of the effect across this many languages. This result is especially relevant given recent discussions calling the relationship into question (e.g., Fratini et al., 2014; Yang, 2016). We find, furthermore, that the correlation between irregularity and frequency is much more robust when irregularity is considered as a property of whole lexemes (or stems/paradigms) rather than as a property of individual word forms. We discuss the implications of these findings.
2 Formalizing Inflectional Morphology
In this work, each word type is represented as a triple consisting of the following components:
A lexeme111This terminology is characteristic of word-and-paradigm approaches to morphology. In item-and-arrangement approaches, this might be called the stem (Hockett, 1954). : An arbitrary integer or string that indexes an abstract word (e.g., go, which provides an index to forms of the verb go such as goes and went).
A slot : An arbitrary integer, string, or more structured object that indicates how the word is inflected (e.g., for the form went).
A surface form : A string over a fixed phonological or orthographic alphabet (e.g., went).
A paradigm (boldface ) is a lexeme-specific map from slots to surface forms for lexeme .222See (Baerman et al., 2015, Part II) for a tour of alternative views of inflectional paradigms. Typically, slots are indexed by structured entities—known as morpho-syntactic feature vectors or morpho-syntactic tags—represented by a set of key-value pairs: . For example, the English verb form runs
, which has the feature vector. In what follows, the keys and the corresponding values are taken from the universal inventory, defined by the UniMorph annotation scheme and denoted (Kirov et al., 2018). We use dot notation to refer to specific forms or sets of forms in a paradigm indexed by some slot went.
Given the pieces just sketched, a complete model of inflectional morphology will specify a joint distribution over surface forms, lexemes, and slots, that is, or one of its associated conditional distributions, such as —the distribution over lexemes and features, given a surface form; or
—the conditional probability of a surface form given a lexeme and inflectional features. In this paper, we will focus on the latter, defining a probabilistic model to approximate this distribution and using that to estimate degrees of irregularity.
3 Operationalizing Irregularity
The informal distinction between regular and irregular forms is an important one for many theories of grammar (e.g., Siegel, 1974), language processing (e.g., Hay, 2003), and language acquisition (e.g., Pinker, 1999; Marcus et al., 1992; McClelland and Patterson, 2002a, b; Pinker and Ullman, 2002b, a; Rumelhart and McClelland, 1986; Prasada and Pinker, 1993; Pinker and Prince, 1988). However, there have been few proposals for how the notion can be characterized precisely or measured quantitatively.
Clearly, the regularity of a form (or rule) can only be defined with respect to the language as a whole—what makes something irregular is that it does not behave in the way that would be expected given other forms in the language. But what is meant by expected? Here, we follow recent work by defining the notion of expectedness in terms of a probabilistic model of inflection which approximates (Cotterell et al., 2018a; Ackerman and Malouf, 2013). However, there remains a wrinkle. A form like went is highly expected as the past tense of go for an adult speaker of English, but is also irregular. How do we capture this?
We take the correct notion of expectedness to be the expectedness of the word form treated as if it were the first instance of that lexeme which had been observed. Thus, we base our measures of regularity on the conditional probability of a word type given the rest of the forms in the language with the target lexeme removed.
Of course, since the target language is generally infinite, we will need to make use of some model-based estimate of this probability . In essence, our definition of irregularity is based on wug-testing (Berko, 1958) such a probabilistic model to see how robustly it generalizes to the target form . In practice, we will estimate this quantity by performing a holdout evaluation of the target form under our model.
More irregular forms will tend to have a lower wug-test probability
than most regular forms. However, the absolute value of such a probability is not directly interpretable. To turn these probabilities into interpretable values which directly measure irregularity, we take the negative log odds of the probability of the correct word form.
We refer to this quantity as the degree of irregularity of a form. If probability of the correct form is exactly , then eq. 2 will be . However, if , then eq. 2 will be negative. Otherwise, the quantity is positive. In other words, the metric is more strongly positive when a form is less predictable given other forms in the language and more strongly negative when a form is more strongly predictable. The midpoint at occurs when there is an equal amount of probability mass on the correct form and all other forms.
Note that this definition of neatly addresses several challenges in studying the notion of (ir)regularity. First, it doesn’t require us to define a binary notion of regular versus irregular or even to explicitly define any such notion at all—a model may treat regularity as an implicit rather than explicit feature of a form or paradigm. Second, and relatedly, we do not require data annotated with the regularity of forms to train or test our model. Third, this definition inherently captures the idea of degree of regularity, for instance, capturing the distinction between wholly suppletive forms such as went and semi-productive inflectional classes such as ring/rang, sing/sang, etc. Fourth and finally, regularity is known to be correlated with other features of morphological structure, such as productivity. Our definition sidesteps the tricky issue of disentangling these different properties of inflection.
Note that our definition of conditions on —the language without the target lexeme—rather than on —the language without the target word. Thus, we are measuring the probability that the model will generalize to the correct form without any evidence of a lexeme at all. Thus, we rule out predictability that comes from similar forms within a paradigm . For example, in our approach a model cannot make use of the irregularity of the past tense form ring to guess that the past participle form was more likely to be rung. We discuss the implications of this assumption in more detail below § 5.4.
4 Modeling Morphological Inflection
Our goal is to estimate from data. We do this by using a structured probabilistic model of string transduction which we call . In the following sections, we describe this model, how we handle syncretism in the model, our training (holdout and test) scheme, and our estimates of the degree of irregularity .
4.1 A Lemma-Based Model
In linguistic morphology, a major division is between item-and-arrangement or morpheme-based models and word-and-paradigm or word-based models (Hockett, 1954). Following (Cotterell et al., 2017b), we adopt a word-based approach. To do this, we designate a unique surface form for each paradigm known as the lemma. The lemma is associated with a slot which we notate : . The lemma can be thought of as a dictionary or citation form of a word and is traditionally chosen by lexicographers of a language. For example, in many Western European languages the lemma of verb forms is the infinitive. Figure 1 shows several of the forms of the Spanish verb poner
(“to put”) organized around the lemma form. In what follows, we use the lemma to identify lexemes, and wherever a probability distribution would condition on the abstract lexemewe instead condition on the lemma .
Our probabilistic model of string transduction is a monotonic model with hard attention described in Wu and Cotterell (2019) and can be viewed as a graphical model over strings like the one shown in Figure 1. It is expressed as follows.
The definition of the model includes a sum over all monotonic (non-crossing) alignments between the lemma and the output surface form . The inner term of this sum is estimated using a sequence to sequence model. The sum itself is computable in polynomial time using a variant of the forward algorithm (Rabiner, 1989). The model achieves state-of-the-art performance on the SIGMORPHON 2017 shared task on morphological reinflection (Cotterell et al., 2017a)
. We follow the hyperparameter used byWu and Cotterell (2019).
4.2 Handling Syncretism
Many inflectional systems display syncretism—the morphological phenomenon whereby two slots with distinct morpho-syntactic tags may have an identical surface form. In contrast to many models of inflectional morphology, we collapse syncretic forms of a word into a single paradigm slot, thereby assuming that every every surface form in a paradigm is distinct. An example of such a collapsed paradigm in German is given in Table 1. Our formalization includes a slot that merges the genitive, accusative and dative singular into a single slot due to the word Herr.
To accomplish this we assume that each lexeme is associated with a set of syncretism classes denoted by . is a map from a slot to a citation form slot which indexed the canonical suface citation form for that combination of features. is used to collapse paradigm cells with identical surface forms. For instance, all forms of the lexeme go are realized as went in the English past tense, regardless of person and number features; thus, for example, . We say that two lexemes and are syncretically equivalent if for all . We assume the mappings are known and given in advance in what follows.
We will use this syncretism-collapsed representation for all simulations below. In particular, this assumption will allow us to simply count the surface forms of each word in Wikipedia without dealing with the tricky issue of assigning individual words to the correct combination of morphosyntactic features (see, Cotterell et al., 2018b, for detailed discussion).
4.3 Handling Derived Forms
As discussed above, we hold out whole lexemes, including all of their
inflected forms during training. However, derivational morphology
presents a potential challenge for this approach. Consider the
irregular verb do/did/done. This verb appears in
a number of derived prefixed forms such as redo and
undo. These forms all inflect identically to the base form
do—for example, redo/redid/redone.333An anonymous reviewer points out that in some languages, such as Dutch, forms derived from irregular verbs become regular (e.g., zeggen/zei but toezeggen/toezegde ). In those languages, it should be unnecessary to apply our heuristic approach.
). In those languages, it should be unnecessary to apply our heuristic approach.If we train our probability model on such derived forms, it is likely to estimate too high a wug-test probability for all forms which are built from the shared stem.
To obviate this problem, we remove all derived forms from the data we consider. To do so we develop a heuristic approach to isolate all words that may have been derived from another. Note that a key desideratum of heuristic is that it should be high precision with respect to finding derivational transformation—we would rather overexclude forms as potentially derivative of another, rather than leave a derived form in the data.
We consider a lexeme to be derived from a lexeme if and only if there is a string such that or where denotes string concatenation of strings and . For example, do and redo satisfy this condition, while sing and ring do not. We perform a search for candidate for all pairs of lexemes in each language and remove all that meet this criterion.
4.4 Measuring Irregularity
With the above definitions in place, we can define an approximation to our degree of irregularity .
In our analyses below, we will also wish to measure the irregularity of lexemes as a whole. To do this, we take the average irregularity score over the entire paradigm .
5 Studies of Irregularity
The empirical portion of our work consists of three studies. We first validate and examine the accuracy of the model (§ 5.2.1). Second, we examine the distribution of irregularity across the languages in our sample (§ 5.3). Finally, we examine the correlation between irregularity and frequency (§ 5.4). Before presenting these studies we first give an overview of the data and simulations common to all of them.
All word forms, paradigms, and morphosyntactic features are taken from the UniMorph project Kirov et al. (2018). Specifically, we examine the following 28 languages: Albanian, Arabic, Armenian, Basque, Bulgarian, Czech, Danish, Dutch, English, Estonian, French, German, Hebrew, Hindi, Irish, Italian, Latvian, Persian, Polish, Portuguese, Romanian, Russian, Spanish, Swedish, Turkish, Ukrainian, Urdu, and Welsh. The languages come from 4 stocks (Indo-European, Afro-Asiastic, Finno-Urgic and Turkic) with Basque, a language isolate, included as well. Although this sample represents a reasonable degree of typological diversity, the Indo-European family is overrepresented in the UniMorph dataset, as is the case for most current multilingual corpora. However, within the Indo-European family, we consider a diverse set of subfamilies: Albanian, Armenian, Slavic, Germanic, Romance, Indo-Aryan, Baltic, and Celtic. For each subfamily, we sample subset of languages randomly.
All of our form-level frequencies were computed from Wikipedia.444Wikipedia data retrieved on Feb 1, 2019. Lexeme counts are achieved by summing over all entries in the paradigm associated with a lexeme. In all simulations, we predict the orthographic form of the target word from the orthographic form of the lemma as a proxy for phonological transcriptions which do not exist for our all languages in UniMorph.
Lexeme-Based Cross Validation.
In the studies that follow, we train a separate instance of our model on the forms in each language using the following procedure. We first remove morphologically-complex forms that are derived from other lemmas in the corpus using the heuristic technique described in § 4.3. We then randomly assign the remaining lexemes of each language to one of ten splits. Note that each split will contain all of the forms associated with each lexeme and a lexeme will never be divided across splits. We then perform 10-fold cross-validation, training the model on 8 splits, tuning on one of the remaining two splits, and testing on the final remaining split. Note that this approach to cross-validation allows us to approximate without the costly procedure of retraining for every held-out lexeme. However, also note that this approach has a potential confound. Lexemes can often be grouped into inflectional classes in which all lexemes mark different slots in the same way. For example, verbs such as sing/sang/sung and ring/rang/rung form an inflectional class in English. Inflectional classes vary in their size and regularity (Stump, 2001). If all or most lexemes in the same irregular inflectional class end up together in the test split under our approach, we may systematically overestimate their irregularity.
5.2 Validation and Accuracy
5.2.1 Validation on English Verbs
|Albright and Hayes (2003)||O’Donnell (2015)|
The first question we wish to ask is whether the irregularity predictions made by our model are consistent with human intuitions. To answer this question, we examine the predictions of our model on the English past tense—a morphological system which has been intensely studied for decades (see Pinker, 1999, for overview) and for which there is general agreement about which forms are regular or irregular. We make use of the databases of Albright and Hayes (2003) which consists of 4039 English verb forms and the dataset of O’Donnell (2015) which consists of 15202 verb forms, both hand-annotated for irregularity by experts.
We present our results in Table 2. We find that our measure of irregularity strongly correlates with human intuitions on English verbs. We take this as tentative validation of our metric. Future work will investigate the linguistic plausibility of our metric on a greater diversity of languages.
5.2.2 Wug-Test Accuracy
|Language||Family||Avg. Accuracy||Lexemes||Forms||Avg. Forms/Lexeme|
Our lexeme-based cross-validation setup differs substantially from the form-based setup typically used to evaluate models of inflectional morphology (see, e.g., Cotterell et al., 2017a). In the typical evaluation setup, individual surface word forms are heldout, rather than all of the forms associated with entire lexemes. This means, amongst other things, that words from irregular lexemes will often be split between test and train, giving models an opportunity to learn partially productive and semi-regular patterns of inflection. Our approach however makes this impossible by strictly assigning all forms from each lexeme to either train or test.
It is important to ask, therefore, how well does our model predict the forms of heldout lexemes given this stricture? The results are displayed in Table 3. This table displays the average accuracy for each language in our sample as well as the number of lexemes for that language, the total number of forms, and the average number of forms per lexeme. The majority of languages show very high generalization accuracy to our lexeme-based wug-tests: 21 out of 28 have an average accuracy of 75% or higher. Three languages stand out in terms of their low accuracy and are highlighted in Table 3: Basque, Urdu, and Welsh. These languages, Basque especially, are characterized by smaller numbers of lexemes and larger numbers of forms per lexeme.
In the § 5.4, we discuss the correlation between irregularity and frequency. The interpretation of these results relies on the ability of our model to accurately capture regular structure in the inflectional systems of the languages that we study. For this reason, we make the conservative choice to exclude all languages whose average accuracy was below 75% from all further analyses below.
5.3 Irregularity across Languages
It is often observed that there are differences in the prevalence of irregularity across languages (Stolz et al., 2012). On one end of the spectrum, some languages have widespread (often suppletive) allomorphy in their marking of inflectional features. For example, Arabic marks plurality on nouns in one of more than a dozen different ways and these are idiosyncratic to the noun stem. Similarly, Georgian verbs often have different roots depending on their tense, aspect, or mood marking. On the other end of the spectrum, it is sometimes claimed that agglutinative languages like Turkish exhibit no irregularity whatsoever.
Figure 2 displays the average irregularity score per language for the 21 languages remaining after our 75% accuracy criterion. Recall from eq. 2 that the degree of irregularity is positive when the majority of predicted probability mass falls on forms that are not the correct target form (i.e., the form is irregular), and negative when the majority of probability mass falls on the predicted form (i.e., the form is regular). As can be seen from the figure, average irregularity is negative across languages. This is expected—most forms in these languages are predicted accurately by the model. However, there is wide variability in the average irregularity score between languages. In particular, in the most regular language, Portuguese, correct forms are about 25,000 times more likely on average than alternative forms. In the most irregular language, Hebrew, correct forms are only about 16 times more likely on average than alternative forms. We leave it to future work to validate and further study these cross-linguistic differences in irregularity predictions.
5.4 Irregularity and Frequency
In some morphological systems, such as the English past tense, there is a strong and well-known correlation between irregularity and frequency (Marcus et al., 1992; Pinker, 1999). In such systems, the most frequent past forms tend to be irregular and irregular forms tend to come from the most frequent verbs. Based on cases like this, it is widely believed in linguistics and psycholinguistics that there is an association between frequency and irregularity (Bybee, 1991; Haspelmath and Sims, 2010; Kiefer, 2000). However, to our knowledge, this relationship has never been explicitly tested quantitatively across many languages at once.
Recently, several authors have questioned the received wisdom that irregularity and frequency are related (Yang, 2016; Fratini et al., 2014).555But see Herce (2016). Thus, it has become important to test this relationship empirically. An example of such a challenge to the standard assumption comes from Yang (2016) who proposed an influential theory of morphological productivity known as the tolerance principle
. The mathematical derivation of the tolerance principle relies on the assumption that irregular forms are uniformly distributed throughout the frequency range(Yang, 2016).666Yang tentatively proposes that the correlation between frequency and irregularity might be accidental in languages such as English. He argues, however, that his theory is not contingent on this being the case (Yang, 2016, pp. 65).
Here we present the first study to probe the relationship between irregularity and frequency at scale. We first examine the relationship between the degree of irregularity and the frequency of individual word forms. To study this question, we examined the Pearson correlation between the log-transformed frequency of word forms in each language and their predicted irregularity scores . Because word occurrences fall into the class of large number of rare event distributions, finite samples will tend to underestimate the probability of infrequent words—word forms that appear times in some sample often differ by orders of magnitude in their true probability (Chitashvili and Baayen, 1993; Baayen, 2001). For this reason, we chose to exclude all frequency forms from our analyses.
The correlations for the 21 languages considered in this study are shown in Figure 3 with significant correlations () marked in blue. Overall, a slight trend towards a positive correlation between irregularity and frequency is discernible in this set of word forms. Following Mahowald et al. (2018), we tested this by fitting a mixed-effect model with irregularity as the dependent variable, language as a random effect (slopes and intercepts) and log count as a fixed effect (Gelman and Hill, 2007). The results give a positive coefficient of for the log count factor. The AIC-corrected log-odds ratio in favor of the model with a fixed effect of count (compared to a model with just random effects) is . A nested-model likelihood-ratio -squared test shows that the log factor is significant with .
An important question about irregularity is whether it is a property of individual forms, or rather whether it inheres to whole paradigms (Baerman et al., 2010; Stolz et al., 2012; Herce, 2016). To examine this question more closely, we ran an alternative correlational analysis examining the correlation between the sum of the counts of all forms associated with a lexeme and the average irregularity score for all forms associated with the lexeme (as in eq. 5). Figure 4 shows the results. Overall, a stronger trend towards a positive correlation between irregularity and frequency is discernible at the lexeme level than at the word-form level. We tested this by fitting a mixed-effect model with irregularity as the dependent variable, language as a random effect (slopes and intercepts) and log count as a fixed effect. The models gives a positive coefficient of for the log count factor. The AIC-corrected log-odds ratio in favor of the model with a fixed effect of count (compared to a model with just random effects) is . A nested-model likelihood-ratio -squared test shows that the log count factor is significant with . Thus, the correlation between irregularity and frequency is considerably more robust when considered at the lexeme level.
In this paper, we have introduced a measure of irregularity based on wug-testing a model of morphological inflection. In § 5.2.1, we showed that this measure produces results that are consistent with human judgements. Focusing on a subset of the languages for which the model was able to recover the correct inflected forms at a high rate (§ 5.2.2), we showed that average irregularity varies a good deal between languages. This result is consistent with the findings of Cotterell et al. (2018a) which gave large scale empirical evidence of a tradeoff between the size of morphological paradigms and the predictability of individual forms within each paradigm.
The main novel empirical result of our paper was presented in § 5.4 which showed that irregularity is correlated with frequency both at the level of individual forms as well as at the level of lexemes. To our knowledge, this is the first large-scale empirical demonstration of this piece of linguistic folk wisdom and provides evidence relevant to recent proposals questioning this generalization (Fratini et al., 2014; Yang, 2016).
Perhaps of greater interest than this positive result is the difference in the strength of the correlation between the level of individual forms and the level of lexemes. This difference appears to be driven by the fact that, in many cases, lexemes that contain high-frequency forms will also contain a few low frequency forms as well. Adopting the terminology of Yang (2002), we can say that low frequency forms free-ride on the higher frequency members of the lexeme.
This finding lends credence to models of linguistic structure which group words together by their lexeme or stem. Such models seem necessary to account for paradigmatic structure cross linguistically and to deal with phenomena such as the existence of defective paradigms—the phenomenon whereby certain inflected forms of a word seem to be impossible for speakers (Baerman et al., 2010). A canonical example is the past participle of stride (e.g., strode/stridden/strided). In these cases, the problem seems to be that the irregularity of the overall lexeme is known, but the particular word form has never been observed. Our results provide further support for the view that inflected forms represent surface exponence of common underlying morphological objects.
More generally, we observe that our wug-test techniques provides a general way of studying regularity and predictability within languages and may prove useful for attacking other difficult problems in the literature, such as detecting inflectional classes. By measuring which words or lexemes are most predictable from one another, a general picture of morphological relatedness within a language can be built in a bottom-up way.
The third author gratefully acknowledges support from the Fonds de Recherche du Québec—Société et Culture and the Natural Sciences and Engineering Research Council of Canada.
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