The fine implicative structure of European Portuguese conjugation

Recent literature has highlighted the extent to which inflectional paradigms are organisedinto systems of implications allowing speakers to make full use of the inflection systemon the basis of exposure to only a few forms of each word. The present paper contributesto this line of research by investigating in detail the implicative structure of EuropeanPortuguese verbal paradigms. After outlining the computational methods we use to thateffect, we deploy these methods on a lexicon of about 5000 verbs, and show how the morphological and phonological properties of European Portuguese verbs lead to the observed patterns of predictability.


Introduction
In the last fifteen years, the study of predictability relations in inflectional paradigms has become one of the central issues in theoretical morphology. Building on previous work on the distribution of stem allomorphs (Aronoff, 1994; Maiden, 1992; Stump, 2001, and many others) and on the predictive structure of affixal inflection classes (Carstairs McCarthy, 1994; Wurzel, 1984, and many others), Ackerman, Blevins, and Malouf (2009) showed that inflectional systems are structured in terms of implications between the sur face forms filling paradigms, and that the reliability of such implications can and should be assessed quantitatively.

1SG
2SG 3SG 1PL 2PL 3PL UTILIZAR 'use' utilˈizu utilˈizɐʃ utilˈizɐ utilizˈɐmuʃ utilizˈajʃ utilˈizɐw APRENDER 'learn' ɐpɾˈẽdu ɐpɾˈẽdәʃ ɐpɾˈẽdә ɐpɾẽdˈemuʃ ɐpɾẽdˈɐjʃ ɐpɾˈẽdɐj IMPRIMIR 'print' ĩpɾˈimu ĩpɾˈimәʃ ĩpɾˈimә ĩpɾimˈimuʃ ĩpɾimˈiʃ ĩpɾˈimɐj Table 1: The present indicative of three verbs, each representative of one of the three main inflectional classes For example, the present indicative forms of regular verbs in the three European Por tuguese conjugations, illustrated in Table 1, 1 displays complex implicative relations: the 2SG fully predicts the 3SG through a simple and general implication (the 3SG is identical to the 2SG minus the final /ʃ/); it also fully predicts the 3PL, although this time two more local generalizations must be observed (/ɐʃ/ corresponds to /ɐw/ while /әʃ/ corresponds to /ɐj/); it fails to fully predict the 1PL (the /әʃ/ ending is ambiguous between second and third conjugation) but still has some predictive value, as the opposition between /ɐʃ/ and /әʃ/ discriminates between the first and the two other conjugations. Crucially, the 1SG is an even worse predictor of the 1PL, completely failing to distinguish the three conjugations. Finally note that predictability is an asymmetric property: while the 2SG fails to fully pre dict the 1PL, it is fully predicted by it (/ɐmuʃ/ in the 1PL corresponds to /ɐʃ/ in the 2SG, /emuʃ/ to /әʃ/, /imuʃ/ to /әʃ/). Ackerman, Blevins, and Malouf (ibid.) argue that such networks of implicative rela tions are key to solving the Paradigm Cell Filling Problem, that is, the problem of assessing what information speakers of a language may rely on to infer unknown forms of a lexeme from forms they have already encountered. In later work, Ackerman and Malouf (2013) argued that the study of implicative structure is of high theoretical importance: the im plicative organization of paradigms reveal an aspect of morphological complexity that is orthogonal to "enumerative" indicators such as Greenberg's (1954) indices of synthesis and agglutination: while languages vary widely in the amount of information that is con veyed by inflected words and how it is conveyed, the inflectional strategies are organized in such a fashion that the forms filling a paradigm are not too hard to predict from one another. In parallel, Stump and Finkel (2013) demonstrated the feasability of evaluating implicative structure on a large scale using computational methods applied to large in flected lexica. These publications led to an expanding literature discussing empirical and computational refinements to the assessment of implicative structure (see e.g. Bonami and Beniamine, 2016; Boyé and Schalchli, 2019; Cotterell et al., 2019; Sims, 2015 as well as empirical studies of the distribution and sources of implicative structure in various sys tems (see notably Bonami  The present study has the double goal of contributing to both of these lines of re search. From a methodological standpoint, we elaborate on Bonami and Boyé (2014) and later literature in assuming a strictly wordbased approach (Blevins, 2006) to implica tive structure, where the shape of an unknown word is inferred from the surface shape of a known word, without any prior information on how that word could be segmented into stems and exponents. To do so, we improve on all the studies cited above by re lying on an algorithm for inferring alternations between surface forms that is not biased towards a particular morphological type, and hence is equally applicable to any language (Beniamine, 2018). From an empirical standpoint, we provide an in depth exploration of the morphological and morphophonological sources of unpredictability in European Por tuguese verbal paradigms: using computationally identified predictability values as our guideposts, we identify which properties of the system lead to such values. In this area we improve dramatically on Bonami and Luıś (2014) by relying on a much larger lexicon and properly taking into account stressconditioned vowel alternations.
The structure of the paper is as follows. In section 2, we briefly outline the struc ture of European Portuguese verbal paradigms, and present the dataset used in this paper. Section 3 guides the reader through the particular method for assessing predictability re lations used in this paper: we show how pairs of forms can be classified into alternation types, how these alternation types can be used to assess the probability of the form in a predicted cell from the form in a predictor cell, and how conditional entropy can then be used as a measure of average predictability. Section 4 applies this method to our dataset, and identifies two main sources and a few ancillary factors leading to low predictability in European Portuguese conjugation. Section 5 concludes the paper. Table 2 presents the general form of a verbal paradigm of European Portuguese. There are 5 tenses within the indicative mood (present, imperfect, simple past, pluperfect, future), and 3 tenses within the subjunctive mood (past, present, future), a conditional, and an imperative with only second person forms. 2 As to the nonfinite forms, there is the ordinary infinitive, in addition to a personal infinitive which agrees in person with the subject and whose 3SG form is syncretic with the ordinary infinitive; 3 there are also the gerund and past participle forms, the latter further inflecting in gender and number. Some transitive verbs distinguish two forms of past participle, e.g. ENCARREGAR 'put in charge' has participles encarregue and encarregado, and tradition holds that the two forms are used respectively for the formation of perfect and passive periphrases. It is however unclear whether this is a situation of overdifferentiation (Brown, 2007; Corbett, 2007, where the normal contexts of use of the participle are split and the two forms are in complementary distribution, rather than overabundance (Thornton, 2012), with an overlap in the distribution of the two forms. We leave this issue to future research.

1SG
2SG Traditionally, European Portuguese verbs are grouped into three classes distinguished by the theme vowels a, e or i (visible in the infinitive forms). Examples of this were already shown in Table 1 with the forms of the present indicative of three representative verbs 4 , where four distinct person markers can be identified, independent of class: 1SG /u/, 2SG /ʃ/, 1PL /muʃ/, 2PL /ʃ/. 5 The theme vowel is present in its distinctive form in the 1PL and 2PL, but can manifest itself in a reduced form depending on the case (/ɐ/ vs. /ai/, /e/ vs. /ɐi/). In the 2SG and 3SG, the distinction between the theme vowels of the second and third 5 For the purposes of this section we assume the following segmentation for 2PL present forms: (i) First conjugation: X+ai+ʃ (ii) Second conjugation: X+ɐi+ʃ (iii) Third conjugation: X+i+ʃ This segmentation, which follows Boyé's (2000) segmentation guidelines for Romance verbs, makes it possible to establish a consistent marker for 2PL forms, in the form of /ʃ/, including the few irregular verbs that have a 2PL form in /dәʃ/ (such as VIR in Table 1). It is worth noting that this segmentation is not to be taken as the only possible one: a plausible alternative would be to posit the contrasting theme vowels /a/ vs. /ɐ/ vs. /i/, followed by a marker /iʃ/ with coalescence of the two /i/ in the 3rd conjugation. Such alternative segmentation would treat the 2PL form of verbs as a suppletive inflected form (Boyé and Cabredo Hofherr, 2006a). However, deciding on the segmentation is of little consequence for this analysis: one of the virtues of having a wordbased (rather than a stembased) approach here is precisely that it does not require the choice of uniform segmentation. See Section 4 below on this point. conjugations is neutralised. In the 3PL, the theme vowel merges 6 with the person marker, giving us the realisation /ɐũ/ for the first conjugation and /ɐĩ/ for the other two. Finally, 1SG does not realise the theme vowel at all.
The situation found in the present tense is characteristic of the whole paradigm, where there are other cases of alternations in vowel quality of the theme vowel, partial neutraliza tion between classes 2 and 3, and fusional morphology. Overall, it is clear that the system of theme vowels contributes to increasing predictability of patterning, while their neutral isation (or even their absence in the PRS.IND.1SG) is a source of opacity: if a speaker has only been exposed to the PRS.IND.1SG form of a verb, they cannot reliably infer to which class this verb belongs. On the other hand, if they are exposed to a 2SG in /әʃ/, they can deduce unequivocally that the verb is not of the first conjugation (although uncertainty remains as to whether it belongs to the second or third).
The three major inflectional classes do not exhaustively model all conjugation pat terns found in European Portuguese.  Many studies of inflectional systems rely on a first classification of data in line with the pedagogical tradition, and often only take into consideration one wellbehaved paradigm for each identified inflectional pattern (see for example Bonami and Boyé 2003, Ackerman and Malouf 2013, Stump and Finkel 2013. This approach, while having the advantage of employing easily accessible data, has three major disadvantages for the study of implica tive relations between forms. First, it is common for the pedagogical tradition to gloss over some subtle contrasts in the data that can have important ramifications. Second, the analysis of only exemplary lexemes alone does not allow one to build up an image of the frequency distribution of the phenomena in question, which can lead to inadequate estimations of what should be considered regular. Third and finally, it prevents any de tailed study of the predictive value of phonotactic properties of lexemes that exemplify an inflectional pattern (Albright andHayes, 2002; Guzman Naranjo, 2019). The present work is therefore based on an extensive lexicon, 7 consisting of the full paradigm of about 5000 lexemes. We first used frequency lists provided by the AC/DC project 8 to select the 5000 most frequent verb lexemes in the CETEMPúblico corpus (San tos and Rocha, 2001). Fernando Perdigão then kindly provided us with paradigms for these verbs in phonemic transcriptions. These were obtained using pronunciation dictio naries and text to speech tools developed at the University of Coimbra (Candeias, Veiga, andPerdigão, 2015; Marquiafável et al., 2014) and corrected by hand. In the process, a handful of verbs had to be excluded.
The transcriptions used are surfaceoriented and standardised, 9 as there is of course variability in the pronunciation of European Portuguese words, conditioned on variety, register and speech rate, which affects in particular the realisation of unstressed vowels. The transcription used here corresponds to a possible realisation in a formal context with a relatively slow speech rate, which minimises the instances of fusion or coarticulation of vocalic sounds.
An important feature of the lexicon we used is that only one form is given per paradigm cell of each lexeme: overabundance (Thornton, 2012) is not taken into account. Notably, the lexicon records a single form of the past participle, in this case the most "regular" according to traditional description, and does not take into account variation in gender and number. The final lexicon consists of 4991 paradigms with 65 cells each, leading to a total of 324,415 inflected forms.

Identifying patterns of alternation
We now turn to the question of studying the implicative relations between paradigm cells. For paradigms with 65 word forms, there are 4160 pairs of cells to examine. 10 It is of course out of the question to work through them manually (let alone through a 4991 verb lexicon), and it is therefore necessary to automate the process.
The starting point of the process is to infer patterns of alternation from a set of pairs of 7 The lexicon can be consulted or downloaded at https://sbeniamine.gitlab.io/ europeanportugueseverbs and is permanently recorded on Zenodo under the DOI 10.5281/ zenodo.5121543. 8 https://www.linguateca.pt/acesso/contabilizacao.php\#listaPosCETEMPUBLICO, accessed on March 20, 2021 9 The transcriptions adopted are similar to those of Mateus and d'Andrade (2000), with four differences: (i) semivowels are not distinguished from high vowels; (ii) the nonlow central vowel is transcribed [ә] rather than [ɨ]; (iii) stress is marked by using the IPA symbol placed immediately before the stressed vowel ; (iv) diphthongs are written using the glides [j] and [w] after the initial vowel. 10 The study of implicative relations involves looking at each of the cases where one of the 65 cells predicts one of the 64 other cells, and 65×64 = 4160. Since patterns of alternation are bidirectional (the same pattern relating cell c to cell cʹ also relates cell cʹ to cell c), we can divide the size of the problem by two when looking for patterns, but not when examining asymmetric predictability relations based on these patterns.
forms. The algorithm we use to describe these relations is implemented as part of the free software toolkit Quantitative Modelling of Inflection (Qumín) 11 and described in detail in Beniamine (2018). This algorithm was inspired by Bonami and Boyé (2014) and Albright andHayes (2002, 2006). A concrete example is presented in Table 4, which shows the automatically detected alternations for the 20 most frequent lexemes of the corpus for the bidirectional relation between the 1SG present indicative and the 1SG present subjunctive. Note that the identified patterns are often (but not always) compatible with a segmental analysis that splits up stem and suffix. For verbs showing stem allomorphy (e.g. SABER 'know'), the difference between the two stems is encoded in the pattern of alternation: this situation is not problematic, since the purpose of this procedure is not to determine morpheme boundaries but to identify sets of lexemes that inflect in the same way.  The algorithm starts from a lexicon tabulating full paradigms, a table providing de compositions of each phoneme into minimal features. It then calculates alternation pat terns which describe bidirectional implicative relations between all pairs of cells, using the format shown in Figure 1, which reads: /u/ alternates with /ɐ/ at the end of the word, in a context consisting of an unbounded sequence of segments followed by a nonnasal, a nonnasal sonorant, and a nonlateral. 12 Figure 1: Structure of the alternation pattern instantiated by lexemes such as ter in Table 5 Notice that Qumín adapts notation from classical generative phonological rules (Al bright and Hayes, 2002; Chomsky andHalle, 1968); but, unlike classical rules, alternation patterns are bidirectional and can describe changes in more than one location in the word. The algorithm is agnostic as to the language or the type of morphological alternations it displays.
There are always many possible ways of describing how to transform one sequence of phonemes into another, which makes the task of inferring patterns difficult. To create good descriptions, the Qumín algorithm follows two criteria: patterns should be formally simple and they should be as general as possible. The algorithm attempts to satisfy them through three steps, as shown below, repeated for all pairs of cells (α, β) in the paradigms: (1) a. For each lexeme, formulate a set of formally simple hypotheses to describe the differences between the pair of forms in cells (α, β) b. Calculate phonotactic contexts for these hypotheses using a variant of minimal generalisation (Albright and Hayes, 2002). c. Choose the best hypotheses according to their generality. Table 5 illustrates the algorithm for the verbs from Table 4. Step one consists in gener ating one hypothetical alternation patterns for each pair of forms (see the column labeled "Finegrained pattern" in Table 4). This is done by aligning the phonological segments of each pair of forms in order to identify variable and stable material. Optimal alignments are chosen using a phonologically weighted scoring scheme 13 and the Needleman and Wun sch (1970) dynamic algorithm (also known as the Wagner-Fischer algorithm). As shown in Table 6 for the lexeme ENTRAR 'enter' between the PST.PFV.IND.1SG and the PER.INF.3PL, often there can be more than one optimal alignment for a given pair of forms. Since they all constitute possible hypotheses, they are all retrieved. In the case of the alternations from Table 4, none of the alignments are ambiguous, and the 20 pairs of forms lead to 20 finegrained patterns in Table 5. the alternation and the context contain placeholders in the form of underscore characters ("_") indicating how the context combines with the alternation: in the present instance, the alternation happens at the end of the context. Descriptions can make use of single characters representing phonological segments (u and ɐ in Figure 1), feature matrices representing natural classes of segments (e.g. [+son -lat]), sets of segments (e.g. {s, t}), variables over segments (e.g. X), and the Kleene star ("X*") and Kleene plus operator ("X+"), to represent unbounded sequence of segments. 13 The cost of a substitution between two phonemes a and b is set to 1 − −sim(a, b), where 'sim' is the similarity between two phonemes, calculated using the Jaccard similarity over their sets of natural classes, following Frisch, Pierrehumbert, and Broe (2004): for any two phonemes a and b, sim(a where C(x) is the set of natural classes to which x belongs. The cost of an insertion (or deletion) is a constant set to 0.4 times the median of all similarity costs among our inventory of phonemes.
The fine implicative structure of European Portuguese conjugation Isogloss 2021, 7/9 9 Lexeme Finegrained pattern Generalized pattern Count  The goal of the second step (column "Generalised pattern" in Table 5) is to merge contexts across lexemes to capture phonotactic constraints on the distribution of patterns. These are obtained using the Minimal Generalisation approach defined by Albright and Hayes (2002). First, all sets of patterns which share the same structural alternation are grouped together. Second, the most specific description which applies to all of these patterns is inferred, segment by segment. This differs from Albright & Hayes in two main ways: all patterns are merged at once, rather than merging them two by two. Second, no intermediate results are memorised. This is crucial to keeping generalization computationally tractable and hence applicable at a large scale.
As Table 5 shows, in the case of 1 st conjugation regular verbs such as PASSAR, we find a rather general pattern: the pattern constrains the forms to end with a consonant with a limited choice of place and manner of articulation, preceded by any nonempty sequence (noted 'X+'). For verbs of the 2 nd and 3 rd conjugations such as TER (the two conjugations do not differ with regards to the two forms of interest here), we can identify phonotactic restrictions based on the 7 examples present: all end with a sequence consisting of a non dorsal obstruent, a stressed oral vowel, and an nonapproximant. However, if we look at more data, neither of these restrictions hold anymore.   Table 7 reports the list of patterns computed over the whole lexicon. We can see that neither of the [u ⇌ ә] and [u ⇌ ɐ] alternations are associated anymore with phonotactic conditions. In addition, an interesting generalisation emerges concerning the PRS.IND.1SG of verbs in /o/ : the two verbs in question ( SER 'be' and ESTAR 'be') have stems ending in a voiceless anterior coronal obstruent. Finally, Table 5 and 7 both document some very specific patterns applicable to a single verb (e.g. IR) or a very small class of closely related verbs (e.g. QUERER and its derivative MALQUERER). Such patterns with very low type frequency for highfrequency verbs are the hallmark of irregular inflection in the traditional sense.
During this step, the program also attempts to generalize over alternations by recog nising simple phonological functions. For illustration of this process, we turn to the al ternation between the gerund and the indicative future 3SG. As exemplified in Table 8 Step three chooses the most general among the competing hypotheses. The generality of a pattern is assessed as its (type) frequency, that is, the number of lexemes to which it can be applied. Then for each lexeme, the most general applicable pattern is kept. In the present example, there are no competing patterns, so this step is trivial. 14 This procedure allows Qumín to find alternation patterns which are both simple and general. While the results do not reproduce usual morphemic segmentations, they are obtained by systematically applying the same principles over the entire lexicon. They are easy to examine, analyze and evaluate, and can be readily obtained from large lexicons.

Measuring predictability
Regardless of the exact method used to classify patterns of alternation between forms, one form can be informative in predicting the other. Let us consider again the data in Ta ble 7. There are clearly several generalisations. First, the three patterns compatible with an PRS.IND.1SG ending in stressed /o/ place complementary requirements on the phonotac tic context: one requires the last consonant to be /s/ or /t/, the second /v/, the third /d/. Faced with a form in /o/, one can therefore categorically predict what the PRS.SBJV.1SG will be. Second, if a verb ends in /u/, but doesn't end in either /ˈɐjʒu/ nor in /k ˈɛɾu/ in the PRS.IND.1SG, two endings are possible at PRS.SBJV.1SG: /ә/ or /ɐ/. As a consequence, we cannot categorically predict the ending of the subjunctive on the basis of the indicative. However, the two patterns contrast in terms of frequency, which gives us an indication of the probability of each alternative: given an unknown verb ending in /u/, but not in /ˈɐjʒu/ or /kˈɛɾu/, in the PRS.IND.1SG, the probability that its PRS.SBJV.1SG is in /ә/ is about 4166/(4166+816) ≈ 84%.
We can therefore see that knowledge of patterns of alternation and their distribution provides crucial information for prediting one form from another. However, there is rel evant information that is not explicit in Table 7. Let us take the case of verbs that end in /ˈɛɾu/ in the PRS.IND.1SG. In principle, three patterns could be satisfied by such verbs: the first or second pattern, which only requires that the form end in /u/; or the fourth pattern, which is compatible only with those forms ending more specifically in /ˈkɛɾu/. However, to assess how much uncertainty there is, we need to check what proportion of the verbs in the lexicon instantiate each of these possibilites. 15 To obtain this information, lexemes must now be classified not according to the pattern of alternation they display, but according to the patterns which could be satisfied given the phonological characteristics of their PRS.IND.1SG form. The lexicon must be examined once more, to identify all verbs compatible with the fourth pattern -i.e. all verbs with the PRS.IND.1SG form ending in /kˈɛɾu/. The result of this second search is that, while a total of 70 verbs end in /ˈɛɾu/, only 2 of those end in /kˈɛɾu/, namely the two verbs QUERER 'want' and MALQUERER 'wish ill'; hence there is in fact no uncertainty as to the subjunctive of a verb whose PRS.IND.1SG is in /kˈɛɾu/. Table 9 outlines the details of this search taking all lexemes in our dataset into ac count. The lexemes are categorized into classes based on the phonological shape of the PRS.IND.1SG form, that is, on the basis of the inventory of patterns that they could theoreti cally participate in. 16 The number of verbs in each class is recorded, as well as the number of lexemes that instantiate each of the possible patterns. In this example, only class c 1 leads to uncertainty. In all of the other classes, all lexemes instantiate the same pattern. It is important to remember that, even though the classification is entirely based on phonologi cal form, the resulting categories are not necessarily phonologically natural: for example, class c 1 is the class of verbs whose PRS.IND.1SG form ends in /u/, but not specifically in /kˈɛɾu/. This lack of phonological naturalness of the classes is expected: the categories are automatically determined according to the inventory of the patterns of alternation en countered, and not according to preconceptions on the sensitivity of the alternation to any phonological property. It also indicates that the classification in question firmly belongs to the domain of morphology, and cannot be reduced to purely phonological generalisations.

Class Size Sample lexeme
Pattern Example forms Count Having established and exemplified the methodology of the algorithm, let us examine the results of its application for some wellknown examples. First, it is important to re member that the implicative relations between two forms are generally directional: what is predictable in one direction may be unpredictable in the other, and vice versa. This is apparent in the task of predicting the PRS.IND.1SG from the PRS.SBJV.1SG. As shown in  In some extreme cases, there is perfect predictability between two cells in the paradigm in both directions. An example of such a situation is found between the infinitive and the PRS.IND.1PL form. Tables 11 and 12 show the patterns of alternation and their distribution in both directions. It can be seen that in both cases, there is no uncertainty: the first two patterns have mutually exclusive contexts of application while the other classes each contain a single verb.  The examples above have been chosen for their simplicity. On average, two paradigm cells X and Y are subject to a large number of patterns of alternation in either direction, along with a significant degree of unpredictability. Because of this, manually examining the patterns of alternation and their distribution one by one is a long and tedious task, which should only be undertaken if strictly necessary.

Conditional entropy as an overall measure of predictability
The distribution tables of patterns of alternation presented above give a fairly detailed picture of the implicative structure of only a small sample of the conjugation of European Portuguese. To assess the implicative structure of the system as a whole, itembyitem examination of these tables is not a viable method: with a 65 cell paradigm, one would need to examine 65 × 64 = 4160 tables. It is therefore essential to have quantitative measures that summarize the information in these tables. Following Ackerman, Blevins, and Malouf (2009), we use conditional entropy as a quantitative measure to obtain a global picture of the predictability of a cell in one paradigm from another. The rationale for choosing this measure goes beyond what can be elaborated upon in this article. We will therefore simply give an overview that will allow the reader to understand how conditional entropy is calculated. The initial idea is to give an explicitly probabilistic interpretation to the distributions of the patterns of alternation studied above. Let us first consider how likely it is that a random verb partake in a particular alternation pattern relating its PRS.IND.1SG and PRS.SBJV.1SG forms. Table 7 gives us type frequency counts for all the possible patterns, which we can use to estimate probabilities, as illustrated in (2).  Table 9). These allow us to estimate the probability that, given the PRS.IND.1SG of a random lexeme, it belongs to each of the classes in question. Finally, the numbers in the "Pattern frequency" column of Table 9 can be interpreted in probabilistic terms. These correspond to conditional probabilities: for example, if we restrict ourselves to class c 1 , 4166 out of 4982 ≈ 84% of verbs instantiate the pattern [u ⇌ ә / X+_] (first row on the table). Based on this percentage, we can infer that, knowing a verb belongs to class c 1 , the conditional probability for a random verb from class c 1 to instantiate the pattern [u ⇌ ә / X+_] is about 0.84. Table 13 shows the conditional probabilities that can be deduced in this way from Table 9. It should be noted that a zero value for conditional probability may occur for two reasons: either the pattern, although applicable in principle to this class, is not instantiated by any member of the class; or the pattern is simply not applicable to members of the class. The cells corresponding to the latter situation are greyed out in the table.   Table 13 provides an explicit probabilistic interpretation of the observations on pre dictability that we have deduced from Table 9. This interpretation has the advantage of allowing the use of tools from information theory to summarize the predictability of the relation between two paradigm cells. Given any random variable X, the entropy of this random variable, noted H(X), is defined as (4).
16 Isogloss 2021, 7/9 Beniamine, Bonami and Luís Intuitively, entropy captures the uncertainty inherent in selecting a value for a ran dom variable. Imagine a situation where a variable X 1 has two equally probable possible values. In this case, as shown in the calculation in (5), the entropy is 1.
( 1 2 log 2 1 2 + 1 2 log 2 1 2 ) = − log 2 1 2 = log 2 2 = 1 If one of the possible values is more likely than the other, the entropy decreases, as seen in (6): if we have two possible values, one is three times more likely than the other, the entropy drops below 1.
An extreme case is illustrated in (7): if one of the two values, although theoretically possible, is never attested in practice, the entropy falls to 0. 18 An entropy value of zero therefore corresponds to a situation of certainty.  = log 2 4 = 2 Let us look now look at a concrete example: from the data in (3), let us determine the uncertainty in ascertaining the class of a previously unseen verb. If we trust the informa tion in our dataset, there are 5 possibilities (one for each of the 7 classes). The calculation in (9) quantifies the entropy as being very close to zero, which corresponds to the intu ition that given a random verb, there is very little uncertainty about the kind of shape its PRS.IND.1SG could have.
Conditional entropy captures the dependence between the uncertainty associated with two random variables: if the value of X is strongly predictive of the value of Y , conditional entropy will be low; if the value of X only slightly predicts the value of Y , entropy will be higher. The calculation takes the form of a weighted sum of the entropies calculated for 18 Classes with a probability of exactly 0 are not taken into account in entropy calculations, since log 0 is undefined. Note though that this is innocuous, since lim We are now in a position to use entropy to provide a global measure of predictability within paradigms. Specifically, we can compute the conditional entropy of choosing a pattern of alternation given class membership of the predictor form. For example, from (3) and Table 9, we compute the conditional entropy of the patterns relating PRS.IND.1SG and PRS.SBJV.1SG given the class of the PRS.IND.1SG form: Applying the same calculation to the data in Table 10 results in a conditional entropy of zero for prediction of the PRS.IND.1SG from the PRS.SBJV.1SG): since each class contains only one pattern with a nonzero probability, each of the terms in the sum is zero. The difference between total and partial predictiveness is therefore well captured, and degrees of partial predictiveness can be differentiated quantitatively.
It is important to keep in mind that entropy can only provide a summary of a condi tional probability distribution and, like any summary, it provides less information than the original. Looking at the conditional entropy values is therefore always a shortcut, and it is necessary to examine the full probability tables to understand the linguistic factors respon sible for a certain value. This will be the approach of the following section: starting from a macroscopic view of the paradigms of European Portuguese induced by the examination of the distribution of conditional entropy values, we will examine in more detail specific implicative relations between pairs of cells and determine which properties of the system lead to a particularly high or particularly low predictability.

Empirical results
In this section we examine the results produced by the automatic analysis of the implicative relations between the 4160 pairs of cells in the European Portuguese paradigms. Since it is, of course, not possible to examine all cases one by one, we will be selective in the information we present.

Categorical predictability
Let us start by looking at examples of categorical implication, that is, implications with no exceptions. A first important observation is that out of the 4160 ordered pairs of cells, entropy is null in 818 cases, or about 19% of the cases. In other words, if two cells are randomly selected from the paradigm, in one out of five cases the shape of the second cell is completely predictable on the basis of that of the first. Likewise, there is categorical predictability in both directions for 382 out of 2080 unordered pairs of cells, or about 18% of the cases. This situation is markedly different from full interpredictability over the whole paradigm.
If we examine the situations that favour total interpredictability, it is clear that it is most often due to a situation that resembles canonical inflectional morphology (Corbett, 2007). In (12) we identify some sufficient conditions: (12) Sufficient conditions for a relation of absolute interpredictability a. The two paradigm cells have the stress in the same position, and b. there is regular demarcation throughout the lexicon between a constant string (a pseudostem) and a variable string (a pseudoending), 19 and c. theme vowels, where they are present, vary predictably from one cell to the other.
A closer look at the forms of the imperfect (Table 14) illustrates this point: stress is on the theme vowel for all forms. The theme vowel's quality is not constant, but it is nevertheless predictable: lexemes that have /a/ in the 1SG have constant vowel quality throughout the paradigm, and those that have /iɐ/ keep it throughout except in 3PL where they have a /iɐ/. And lastly, the forms show no stem allomorphy.   (12). A partic ularly enlightening exception is found in the perfective preterite (Table 15). Regular verbs have 5 directly analogous forms in this subparadigm, with stress on the theme vowel and a threeway vowel quality contrast. The 1SG is an exception, in that it neutralizes the dis tinction between the second and third conjugations; it is therefore not a good predictor of the rest of the subparadigm. If we examine irregular verbs however, both the 1SG and 3SG forms reveal a number of verbs that are exceptions to (13): the expected ending and/or stem are not used. It is interesting to note that these cases do not in fact reduce predictiveness because of their distinctive phonotactic properties. All these forms have a property that makes it possible to distinguish them from the standard inflectional pattern: an unusual ending in the 1SG and 3SG, the theme vowel /ɛ/ in the 4 remaining cells.
Another example of the same type can be found in the present indicative (Table 16). In this subparadigm, 2SG and 3SG are interpredictable. For these two cells, the vast majority of verbs satisfy the conditions in (12): stress is on the pretheme vowel, and there is a two way theme vowel contrast (/ɐ/ vs. /ә/) between first conjugation and other verbs, before invariant endings (/ʃ/ or zero). However, there is a small number of irregular verbs that use no suffix in the 3SG: 42 verbs of the regular 2nd and 3rd conjugations with an stem ending    Table 17, the paradigm space has been partitioned into 12 sets of cells or "zones of interpredictability", each labelled with a different label (a label appears multiple times when the zone it labels does not correspond to a contiguous area in the table). The boundaries of Table 17 highlight the morphomic character (as defined by Aronoff, 1994) of zones of interpredictability. Some areas correspond to a morphosyntactically natural class of forms: this is trivially the case for singlecell zones (Z1, Z5, Z7, and , as well as Z6 (indicative past imperfective) and one could argue that this is also true for Z9 (assuming that future and conditional have semantic properties in common).
On the other hand, the inventories of forms corresponding to the remaining zones are clearly disjunctive, as illustrated by the statement in (13) From the partition of Table 17 we can construct what Stump and Finkel (2013) call a distillation of the paradigm. A distillation is a set of cells in which each member corre sponds to a distinct interpretability zone. The purpose of a distillation is to allow the study of a subparadigm of a much more reasonable size without loss of information: given two zones Z and Z', we expect that any implicative relation of a cell of Z to a cell of Z' has an exact correspondent which connects any other Z cell to any other Z' cell. In principle, the choice of the cell representing each zone is inconsequential; among the hundreds of thousands of possibilities, 21 we choose the one indicated in Table 18.
Focusing on a distillation, it is possible to examine in detail the conditional entropy values linking the cells of the paradigm. Figure 2 provides Table 18 22 Isogloss 2021, 7/9 Beniamine, Bonami and Luís distillation. The remainder of this section will be devoted to identifying the what regu larities in the dataset lead to the predictability contrasts that this figure highlights. In this figure, the shade of the background of each cell shows the relative height of the entropy value found by the algorithm: the darker the background, the higher the value. A simple examination of the rows of Figure  If we now look at the columns of Figure 2, the structure is not so clearcut, although we can see that predictiveness is unevenly distributed: depending on the predictor form, low predictability is related to different parts of the paradigm. In the following paragraphs, we identify four main sources of prediction difficulty and show how these factors lead to the numbers in Figure 2.

Neutralisation of inflectional classes
A main source of unpredictability in paradigms is clearly the neutralisation of distinctions between conjugations in certain cells. Let us look again at the present indicative of regular verbs (Table 1). The 1PL and 2PL cells are the only ones to present a distinctive conjugation marker, in the form of an opposition between 3 theme vowels: /ɐ/ vs. /e/ vs. /i/ in the 1PL, /ai/ vs. /ɐi/ vs. /i/ in the 2PL. Three other cells neutralize the distinction between 2 nd and 3 rd conjugation: we have /ɐ/ vs. /ә/ only in the 2SG and 3SG forms, and /ɐũ/ vs. /ɐĩ/ only in the 3PL forms; this leads to a twoway instead of a threeway contrast. It follows from this observation that predicting the 1PL form from the 3SG, for example, is a problem: if the 2SG ends with /ә/, it is not clear whether /e/ or /i/ should surface in the 1PL. In the opposite direction, predicting the theme vowel is not a problem. The 1SG form is even more problematic since it neutralizes the distinctions between all three conjugations. According to the same reasoning, we therefore expect to have difficulty predicting the other forms: to predict the 3SG form, we do not know whether the unreduced form of the vowel is /ɐ/ or /ә/, and for the 1PL, whether it is /ɐ/, /e/ or /i/. All other things being equal, these observations lead to the expectation that predicting a cell with more theme vowel contrasts from a cell with fewer theme vowel contrasts should be harder than the other way around. In Figure 3, we present again the same numbers from Figure 2, but rearranging rows and columns to highlight the expected contrasts: the area set apart by a dashed line in the upper left part of the table is where we expect higher numbers. As the reader can check, our expectations are only partially met. All predictions from the fullyneutralised PRS.IND.1SG are maximally hard, as expected. Predictions from cells with a twoway contrast to cells with a threeway contrast are nontrivial: entropy values range from 0.141 to 0.314, and are an order of magnitude higher than those for pairs of cells that both exhibit a threeway contrast. (bottom right corner). However, these values are still lower than those found in some areas of the lower left part of the figure, where on the basis of the behaviour of theme vowels alone we would expect very low numbers.
We thus provisionally conclude that, while theme vowels alternations do contribute to explaining some contrasts in predictability in European Portuguese paradigms, they cannot be the only relevant factor.

Vowel alternations
While the number of theme vowel contrasts in each cell is the most obvious factor having an influence on predictability in European Portuguese, it is not the only one. Another major factor is the reduction of unstressed vowels. In European Portuguese, unstressed oral vowels are typically reduced according to a pattern that induces the neutralisation of distinctions visible only in stressed syllables (see for example Mateus & d'Andrade, 2000: 1723. 22 Table 19 lists the alternations in question. stressed i e ɛ a ɐ ɔ o u unstressed i ә ɐ u Although this reduction process is lexicallyconditioned and not systematic (Vigário 2003: 6873, see also Mateus & d'Andrade, 2000: 136), it is highly prevalent, and leads to neutralisations of contrasts with important consequences on the predictability relations between forms. Once again, let us focus on the present indicative to start. In the 1SG, 2SG, 3SG and 3PL, stress is on the prethematic vowel. In the 1PL and 2PL, it falls on the theme vowel. As a result, the prethematic vowel is not stressed, leading to neutralisation, which is illustrated in Table 20 with verbs from the first conjugation.  Because it results in opacity, vowel reduction increases unpredictability: when faced with a 1PL or 2PL form with the prethematic vowel /ә/, there is uncertainty about the quality of the prethematic vowel in related forms, as it can surface as either one of /e/ or /ɛ/. The same issue arises for the 1PL and 2PL forms with prethematic /ɐ/ and /u/.
To assess the importance of this phenomenon, we examined systematically vowel al ternations linked to stress shift in the relationship between PRS.IND.1SG and PRS.IND.1PL and found that 4909 of our verbs have the potential for such alternations. 23 Table 21 shows, for each unstressed prethematic vowel, what the options are for its stressed counterpart, 22 Depending on the dialect and speech rate, some unstressed vowels may not be realised at all. Taking into account such graded variation phenomena is beyond the scope of the methods and data used in this article. 23 The remaining 82 verbs either exhibit no stress shift, involve diphthongation, or implement some non trivial stem allomorphy such that it is not obvious which unstressed vowel in the plural form should count as the counterpart of the stressed vowel in the singular form. and how many verbs in the sample instantiate each possibility. For brevity, the last row lumps together all unstressed vowels that have a single stressed counterpart, so that stress shift cannot lead to opacity.  As the table shows, the welldocumented vowel reductions lead to a sizable amount of unpredictability; in addition, more situations of vowel alternation not listed in Table 19 emerge, that are less familiar but nevertheless contribute to making prediction hard. Over all then, vowel alternations are expected to lead to relatively high entropy values whenever a cell in which stress falls on the prethematic vowel, is predicted from a cell in which stress falls on the theme vowel. Figure 4 shows that this expectation is clearly met. We are again looking at the same numbers, but rows and columns have been reorganised to group to gether cells that exhibit the same stress pattern. As expected, the numbers in the lower left corner are all relatively high, between 0.39 and 0.5. In fact, these are the highest entropy values in the table if we leave  Another striking observation when looking at Figure 4 is that the numbers in the upper right corner, in which a word with stress on the theme is predicted from a word with stress on the prethematic vowel, are relatively high: while they are lower than those in the lower left corner, they are all higher than those in cells in the upper left corner and lower right corners. Why is that so? Again, we can point to stressconditioned vowel alternation as the source. We already discussed the fact that vowel alternations led to opacity when going from an unstressed to a stressed prethematic vowel. We now observe in Table 22 that the converse is also true: because there are exceptions to vowel reduction (Vigário, 2003), the stressed version of the vowel in the stem does not always fully determine the unstressed version. However, the amount of uncertainty is significantly lower (where there is opacity, one of the options is much more prevalent than the others), which is why the numbers in the upper right corner of Figure 4 are still much lower than those in the lower left corner.
The interaction between the two sources of unpredictability identified so far gives rise to an interesting situation. There is a correlation between the locus of stress and the number of theme vowel distinctions: the forms that distinguish 3 theme vowels are those that have stress on the theme vowel position (or on the postthematic in the case of the future and the conditional). These are therefore also those that have an unstressed prethematic vowel. As a result, the set of pairs of cells in which uncertainty is introduced by theme vowels doesn't overlap with the set of cell pairs in which uncertainty is introduced by alternating prethematic vowels. This situation has implications for the possibility of defining a system of principal parts for the conjugation of European Portuguese. Following Finkel and Stump (2007), a system of (static) principal parts is defined as a set of cells in the paradigm from which all other cells can be categorically inferred. Ideally, an inflectional system has a single principal part (see Albright's (2002) Uniform Base Hypothesis). This single principal part is a good candidate to be used as a citation form, as its knowledge is sufficient to derive the whole paradigm. In practice, however, as Stump and Finkel (2013) show in detail, it is rare that a single principal part is sufficient, if only because some irregular lexemes have unpredictable inflection.
In European Portuguese, the interaction between conjugation neutralisation and vowel reduction has the consequence that even if we limit ourselves to the subsystem of regular verbs of the three traditional conjugations, at least two principal parts are necessary: any form with stress on the prethematic vowel will be a poor predictor of forms with stress on the theme or postthematic vowel, and vice versa.

Local irregularities
We have now identified the two main sources of unpredictability in the system. This is shown visually in Figure 5, where rows and columns have been sorted by both stress place ment and number of contrasts in theme vowels, and areas where identified determinants of predictability have the same effect have been materialised by dashed lines. It is strik ing here that, as expected, we have very low entropy (lower than 0.1) for pairs of cells in the four areas where (i) there is no contrast in stress placement between predictor and predictee, and (ii) the predictor exhibits as many or more theme vowel contrasts than the predictee.
What has been left unaccounted for is the remaining variability within each of the areas: if we had identified all sources of unpredictability, the numbers within an area would all be exactly the same, and all four whiteish areas would contain only zeroes.
A clear remaining source of uncertainty is the existence of irregular allomorphy. Once again, the general situation can be illustrated by taking examples from the present indica tive. Consider the pair relationship between the 1PL and the 2PL. All patterns encountered are shown in Table 23. We see that there is a small set of 63 verbs that do not follow any of the three regular patterns of the three conjugations. It is natural to interpret this situation as a case of stem allomorphy: the strategy that uses the same stem for the 1PL and the 2PL is in opposition to two other strategies, which use the 2PL with a stem augment /dә/, with or without nasalisation of the theme vowel. The high frequency irregulars SER and IR present isolated alternations.
This situation of stem allomorphy leads to some uncertainty, insofar as some patterns of alternation are applicable in the same phonological contexts. Table 24 lists these situ ations. However, the relevant classes (c 3 , c 6 , c 7 , c 8 , c 9 ) make up only 10% of the lexicon, and the distribution of patterns within these classes is highly skewed. Remarkably, stem allomorphy does not lead to uncertainty in the other direction, because the form of the shorter stem is entirely predictable on the basis of that of the longer one. The prediction of PRS.IND.3SG from PRS.IND.3PL illustrates a different type of local ir regularity, this time due to irregular endings and not to stem allomorphy. The verbs of the 2nd and 3rd conjugation normally end in /ɐĩ/ in the 3PL and /ә/ in the 3SG. However, there are some verbs that do not have the expected ending in the 3SG, which we have already noted when commenting on Table 3. In most cases there is no uncertainty, because the class of verbs concerned has a distinctive phonological characteristic. For instance, all and only verbs with a 3PL ending in /jɐĩ/ have a 3SG form in /j/ instead of the expected /jә/; see e.g. VIR in Table 3. Nonetheless, the small set of unpredictable endings in the 3SG does contribute some very limited uncertainty (conditional entropy 0.007).
While commenting on the causes of each of the entropy values found for the 132 pairs of cells in the distillation individually may offer more insights about predictability,   it would extend our paper significantly. We believe we have illustrated the kind of factors that can determine entropy, and that other pairs of cells implement various combinations of the factors already identified.

Conclusions
This paper had both methodological and empirical objectives. On the methodological side, we presented an algorithmic solution to address the Paradigm Cell Filling Problem strictly inductively. The Qumín package takes as input raw lexical data with no morphological information other than an organization into lexemes and paradigm cell. Provided with information on the interpretation of phonemic symbols in terms of distinctive features, it allows for the automatic extraction of patterns of alternation between forms. Important design features of the algorithm are that it assumes a very general format for alternations, allowing for multiple points of variation, and does not encode any bias as to the direc tion of alignment between forms-hence any mode of affixation (prefixation, suffixation, circumfixation, infixation, or even rootandpattern) can be discovered from the data au tomatically. In the present study, these features are crucial to capturing the interplay of suffixation, theme vowel alternations, and stressconditioned vowel alternations that is the hallmark of European Portuguese conjugation. The classification of pairs of forms in terms of alternation patterns was then used to in fer relevant probability distributions describing dependencies between the shape found in some predictor cell and possible shapes in another predicted cell. We used these probabil ity distributions to compute conditional entropy values assessing the average predictability of one cell from another, but insisted that attention to the full distribution is necessary to get a more finegrained picture of the system. This proved very useful in our exploration of the sources of (un)predictability in European Portuguese conjugation: detailed exam ination of the distribution of patterns allowed us to go beyond a broad assessment of the difficulty of the Paradigm Cell Filling Problem and into a detailed discussion of the spe cific challenges posed by the fine structure of the European Portuguese system.
Moving on to empirical results, we showed that the predictive structure of European Portuguese verbal paradigms is conditioned by four factors which each lead to situations of opacity in the prediction of one cell from another. First, cells that neutralize the threeway distinction between theme vowels to a twoway or no distinction at all are poor predictors, because the speaker has to guess what the theme vowel is. Second, neutralizations of vo calic contrasts in unstressed positions lead to forms with an unstressed prethematic vowel being poor predictors of the stressed variant found in other forms. Third, although they are less prevalent than in other Romance languages, stem alternations do lead to unpre dictability, when there is uncertainty as to whether the form in the unknown cell uses the same stem as the predictor form or a different stem. Fourth, irregular suffixal exponence is a minor source of unpredictability.
There are many directions in which the research reported here can and should be ex tended. In the area of Romance morphology, the last three decades have witnessed an exhuberance of research on stem alternations, both in terms of detailed descriptions of syn chronic systems (see e.g. Boyé 2003 on French, Boyé andCabredo Hofherr 2006b on Spanish, Guerrero 2011 on Catalan, Montermini and Bonami 2013 on Italian) and of diachronic work on the emergence and maintenance of such systems (see Maiden 2018 for a recent and detailed assessment and guide to the literature). Stem alternations are an important contributor to unpredictability, but, as the present paper shows, by no means the only one. In addition, as Bonami and Boyé's (2014) selfcriticism highlights, largescale work on stem alternations forces the analyst into poorly justifiable segmenta tion decisions. The approach illustrated here avoids such decisions entirely, and hence would allow for a largescale reassessment of predictability in Romance conjugation on a quantitative basis.
From a theoretical standpoint, it is worth noting that the present paper adopts a some what ambiguous posture. On the one hand, the research is grounded in a purely inductive, wordbased approach to morphological structure, where no assumption is made as to a stable segmentation of words in subword constructs such as stems, theme vowels, and af fixes. On the other hand, these concepts were crucial to our investigation of the factors influencing predictability, where we reverted to the very traditional descriptive vocab ulary. We do not think there is a contradiction here, although there is a gap in current research that needs to be filled. The research tradition this paper participates in high lights the importance of an approach to the relations between forms in a paradigm that is both quantitative and inductive; and we insist that, when addressing prediction from form to form, we should avoid basing such predictions on a preconceived segmentation that speakers trying to solve the Paradigm Cell Filling Problem cannot be aware of. But that in no way entails that there is no other aspect to morphological structure. Quite to the contrary, we assume that the Paradigm Cell Filling Problem is paralleled by the Inflected Word Recognition Problem, the problem speakers face when trying to assess, from the shape of an unknown word, which paradigm cell of which lexeme that word belongs to (Bonami and Beniamine, 2021). We submit that traditional subword units capture some of those properties of words that play a role in addressing this problem, but do so in an ap proximate fashion, by insisting on discrete boundaries between exponents. To avoid this, such properties could be investigated with the same type of inductive, quantitative meth ods deployed in this paper. This still needs to be demonstrated in practice though. A full picture of the structure of European Portuguese conjugation awaits that demonstration.