Luciano Da Fontoura Costa
email: luciano@ifsc.usp.br

Comparing Probability Densities: A Comparative Approach

Probability densities, characterized by non-negativity and normalization, are among the most important concepts underlying analysis and modeling the physical sciences from both theoretical and applied perspectives. The great relevance of this topic is readily accounted, among other facts, by the several approaches that have been used to compare two given densities. In this work, we report a comparison of several approaches for comparing statistical densities, including Kolmogorov-Smirnov, normalized Euclidean distance, Pearson, Kendall, and Spearman correlations, as well as the real-valued interiority, Jaccard and coincidence similarity indices, while considering several respective parametric variations. The comparison indices are then compared themselves by using the coincidence method for translating datasets into networks. Several interesting results are described regarding the comparison of two 1D normal densities, one with parameters kept fixed and the other with varying mea and standard deviation. Interesting results are obtained that not only confirm the highest selectivity and sensitivity of the coincidence index, but also indicate that all the three multiset-based indices tend to be more selective and stable.

Introduction

Developing and evaluating models of nature constitute the basic approach to science, often known as the scientific method. Given that several theoretical and applied situations involve intrinsic variations that can not be accurately predicted, statistical modeling approaches (e.g. []) have been often adopted. Given an experiment or data characterized in terms of a random variable X, the respective density probability function, or probability density for short, provides all the statistical information that can be obtained regarding the characteristics of X. As such, probability densities provide effective models of the studied phenomenon, through the respectively associated random variable X.

Given the particular importance of probability densities for scientific modeling, it frequently occurs that two given data X and Y need to be compared in terms of their respective density probabilities p(X) and p(Y ). Several interesting approaches have been developed for this finality, including but not being limited to the measurements as the Pearson (e.g. [START_REF] Degroot | Probability and Statistics[END_REF][START_REF] Kreyszig | Advanced Engineering Mathematics[END_REF]), Kendall [START_REF] Kendall | A new measure of rank correlation[END_REF][START_REF] Kendall | The treatment of ties in rank problems[END_REF] and Spearman (e.g. [START_REF] Daniel | Spearman rank correlation coefficient[END_REF]) correlation coefficients , Kolmogorov-Smirnov (e.g. [START_REF] Marsaglia | Evaluating kolmogorov's distribution[END_REF][START_REF] Birnbaum | One-sided confidence contours for probability distribution functions[END_REF]), as well as the Euclidean distance (possibly in some normalized form, e.g. [START_REF] Da | Multiset neurons[END_REF]).

More recently, the concept of coincidence similarity was proposed as an extension of the Jaccard similarity coef-ficient (e.g. [START_REF] Jaccard | Distribution de la flore alpine dans le bassin des dranses et dans quelques régions voisines[END_REF][START_REF] Jaccard | Étude comparative de la distribution florale dans une portion des alpes et des jura[END_REF][START_REF] Samanthula | Secure multiset intersection cardinality and its application to jaccard coefficient[END_REF][START_REF]Jaccard index[END_REF][START_REF] Schubert | A note on the Jaccardized Czekanowski similarity index[END_REF]), that has been most frequently used for comparing sets, to consider real-valued data and to incorporate quantification of the interiority between the two datasets being compared [START_REF] Da | Further generalizations of the Jaccard index[END_REF][START_REF] Da | On similarity[END_REF]. This similarity index has been found to be characterized by implementing particularly selective comparison, while being highly sensitive when comparing similar patterns and robust to localized perturbations [START_REF] Da | Multiset neurons[END_REF][START_REF] Da | On similarity[END_REF]. These welcomed properties have allowed the coincidence similarity index to be applied to several areas and problems (e.g. [START_REF] Da | Coincidence complex networks[END_REF][START_REF] Da | On similarity[END_REF]).

In the present work, we develop a comparative approach to several measurements for comparing probability densities, including Kolmogorov-Smirnov, Pearson, Kendall and Spearman correlation coefficients, a normalized version of the Euclidean distance, as well as three multisetbased similarity indices for real-valued densities, namely the interiority, Jaccard and coincidence [START_REF] Da | Further generalizations of the Jaccard index[END_REF][START_REF] Da | On similarity[END_REF].

We start by briefly revising the three adopted multiset similarities and then proceed to comparing the several considered measurements respectively to varying instances of 1D normal densities.

Multiset Similarities

While similarity indices such as the interiority (or overlap [START_REF] Vijaymeena | A survey on similarity measures in text mining[END_REF])and Jaccard have been typically employed for comparing two sets, it is also possible to derive multiset (e.g. [START_REF] Hein | Discrete Mathematics[END_REF][START_REF] Knuth | The Art of Computing[END_REF][START_REF] Blizard | Multiset theory[END_REF][START_REF] Blizard | The development of multiset theory[END_REF][START_REF] Mahalakshmi | Properties of multisets[END_REF][START_REF] Singh | Complementation in multiset theory[END_REF]) versions of them that are capable of dealing with real-valued arguments, including negative numbers and even continuous functions (e.g. [START_REF] Da | Further generalizations of the Jaccard index[END_REF][START_REF] Da | On similarity[END_REF]).

Given that the densities to be used in the present work are non-negative, the multiset versions of the three considered indices -namely the interiority I(f (x), g(x)), Jaccard J (f (x), g(x)) and coincidence C(f (x), g(x)) similarities between two real-valued 1D functions f (x) and g(x), can be expressed in simplified form as:

I(f (x), g(x)) = ´S min {f (x), g(x)} dx min ´S f (x)dx, ´S g(x)dx (1)
J (f (x), g(x)) = ´S min {f (x), g(x)} dx ´S max {f (x), g(x)} dx (2)
C(f (x), g(x)) = I(f (x), g(x)) J (f (x), g(x)) ( 3
)
where S is the common support of f (x) and g(x). We also have that 0 ≤ I(f (x), g(x)) ≤ 1; 0 ≤ J (f (x), g(x)) ≤ 1; and 0 ≤ C(f (x), g(x)) ≤ 1.

Comparing Approaches: Varying Mean

Henceforth, we will consider the 1D normal probability density function, which can be expressed as:

g(x, σ, µ) = 1 σ √ 2π e -0.5( x-µ σ ) 2 (4)
In this section we compare the comparison methods respectively to 1D normal densities with mean displacements. More specifically, one of the densities remains with respective parameters fixed as µ = 0 and σ = 1, while the other has σ = 1 but means varying from 0 to 3. The resulting measurements are shown in Figure 1.

The enhanced selectivity of the multiset similarity indices can be readily observed from Figure 1, with the coincidence index implementing the most strict comparison between the varying probability densities.

The relationship between the several considered comparison methods can be effectively quantified and visualized by application of the coincidence methodology to translate datasets, where each entry is characterized by respective features, into complex networks [START_REF] Da | Coincidence complex networks[END_REF]. More specifically, we take the values shown in Figure 1 as features characterizing the performance of each of the considered comparison approaches. The results obtained by adopting α = 0.5, T = 0, and exponent [8] D = 4 are depicted in Figure 2. The obtained result reflects effectively the similarity between the curves in Figure 1, with the normalized Euclidean distance resulting markedly different from the other indices. In this case (mean shifting), the Kendall and Spearman correlation coefficients resulted markedly similar, while the Pearson correlation resulted surprisingly similar to the interiority index. The Kolmogorov-Smirnov approach resulted particularly central in the coincidence network. The coincidence similarity index re-sulted well-separated as a consequence of its markedly enhanced selectivity and strictness.

Comparing Approaches: Varying Standard Deviation

In this section we report on an experiment similar to that described in the previous section, but here the standard deviation of the second function is varied, while its mean is kept as µ = 0. The results are presented in Figure 3. The respective coincidence network obtained with the same parameters as before is shown in Figure 4.

Several interesting results can be observed. First, we have again that the normalized Euclidean distance resulted very distinct from the others, as in the previous case and as could be expected. Also as before, the Kendall and Spearman coefficients resulted quite similar, as well as the interiority and Pearson indices. Perhaps the main surprising aspect concerns the great similarity observed between the Kolmogorov-Smirnov and Jaccard indices, suggesting that they perform very similarity in this specific comparison situation where one of the normal densities undergoes a change in standard deviations. The coincidence similarity resulted again well apart from the other methods as a consequence of its pronounced selectivity.

Concluding Remarks

Comparing probability densities constitutes a recurrent issue in theoretical and applied physical sciences. In the present work, we compared several approaches frequently employed for that finality, namely the Kolmogorov-Smirnov, and Pearson, Kendall and Spearman correlation coefficients, as well as three multiset-based similarity approaches corresponding to the interiority, Jaccard and coincidence indices extended to real-values.

The considered indices were compared respectively to comparing two 1D normal densities, one kept fixed while the other underwent variations of mean and standard deviation.

The obtained results are particularly interesting in which they confirm the enhanced selectivity and sensitivity of the multiset indices, with the coincidence similarity resulting in the most strict comparison. It should be born in mind that neither of the considered approaches have absolute advantages. Rather, each of them have specific interesting features that can best suit specific requirements and applications. In the present work, for instance, focus is placed on the selectivity of the approaches in the sense of performing more strict comparisons between the probability densities.

It can also be hypothesized with basis on the presented visualizations that the multiset indices seem to provide a quantification of similarity between the considered densities that is closer to human intuition. This is an interesting possibility worth further investigation.

Figure 1 :

 1 Figure 1: Comparison of the several considered measurements for comparing two probability densities: one with fixed µ = 0 and σ = 1, and the other with varying means. The results obtained by the Kolmogorov-Smirnov approach are shown to 10 times their original values, and have been complemented by 1 (i.e. 1 -ks) in order to become congruent with the other measurements. The three multiset-based approaches resulted markedly more selective than the other methods.

Figure 2 :

 2 Figure 2: The coincidence network obtained by taking into account the comparison values obtained by each of the considered methods as features, respectively to varying means.

Figure 3 :

 3 Figure 3: Comparison of the several considered measurements for comparing two probability densities: one with fixed µ = 0 and σ = 1, and the other with varying standard deviations. The results obtained by the Kolmogorov-Smirnov approach have been complemented by 1 (i.e. 1 -ks) in order to become congruent with the other measurements. The three multiset-based approaches resulted markedly more selective than the other methods.

Figure 4 :

 4 Figure 4: The coincidence network obtained by taking into account the comparison values obtained by each of the considered methods as features, respectively to varying standard deviations.

Acknowledgments.

Luciano da F. Costa thanks CNPq (grant no. 307085/2018-0) and FAPESP (grant 15/22308-2).


M Jouitteau

M Rezac

From mihi est to 'have' across Breton dialects *

We address the syntax and parametric variation underlying the verb 'have' from the empirical domain of Breton diachronic and dialectal variation. We introduce the Celtic agreement system characterized by Complementarity Effects and show why Breton 'have' is special among the Celtic language and at the same time unique among Breton verbs. We propose syntactic tests for decomposing the structure of 'have'. With these tools in hand, we show that 'have' across Breton dialects appears to trace an ordered path in UG parameter space from a strictly analytic mihi est to a fully lexicalized 'have'. We develop the predictions our proposal makes internally to each dialect, and list predictions to be checked in future data collection.

From mihi est to I have

The verb 'have' has unique properties in the grammatical system of Breton. All other verbs are subject to the Complementarity Effect on agreement:

(1) Complementarity Effect: Agreement morphology is restricted to crossreferencing pro-dropped subjects of transitives and intransitives, whereas in the context of phonologically overt subjects the 3SG form of the verb shows up (Jouitteau & Rezac 2006 and references there).

, spells out phi-Agree and/or pro itself (3b).

(3) a. T uphi=3SG

… 3SG [ vP … subject …] (overt subject) b. T uphi=i + pro i … 3SG [ vP … t pro …]
(null subject)

J&R take the 3SG intervener to be specifically the boundary of the extended vP, transitive and intransitive alike, which has nominal properties in Breton. This extended verbal structure could be identified as a small clause. J&R show that overt subject Case licensing occurs low within it. The Case licensing of the direct objects of transitives occurs still lower in the vP. Case appears overtly only on postverbal weak pronouns. In Table 1b, objects appear as verbal proclitics, and subjects as pro-drop agreement on the verb, for example (we return to other variants). The verb 'have' in Table 1c is a morphological anomaly in Breton. Its agreement is a prefix, not a suffix, and it recalls strongly the direct object proclitics of transitive verbs (1b) although not synchronically identifiable with them (1d). This prefix is tacked onto a root whose initial segment varies, recalling again the variation of the initial segment of the root torr-/dorr in function of the object proclitics, followed by eus. Eus is independently the existential form corresponding to French il y a "there is" but lit. "there has" and English "there is". The rest of the paradigm, not given here, makes it clear that 'have' is mostly built by prefixes on the same root on which be is built by suffixes, e.g. am bez 'I have (habitually), bez-an 'I am (habitually)'. Moreover, 'have' simply recruits the participle bet and mostly the infinitive bout/bezañ of be for its own participle and infinitive1 . That at least is the situation in the more conservative varieties. As Table 1d shows, innovation in others has gone a long way towards regularizing the morphology to suffixal exponence.

More importantly still for us, the verb 'have' is also anomalous syntactically: it is the sole verb in Breton that does not obey the Complementarity Effect, agreeing with an overt subject, (4)a. J&R present an account where 'have' is built on an applicative structure that introduces the possessor above the 3SG intervener for agreement with other subjects. The point of departure is the hypothesis that the structure underlying the verb is the copula be and a prepositional element (Benveniste 1960[START_REF] Freeze | Existentials and other locatives[END_REF][START_REF] Kayne | Towards a modular theory of auxiliary selection[END_REF][START_REF] Guéron | Le verbe avoir. In Forme (Hommages à Mitsou Ronat)[END_REF], 1995[START_REF] Guilliome | Possession and the double object construction[END_REF]. On this view, there is an intimate connection between be in (4)b, with a small clause complement relating the possessor and the theme similar to a theme and a location, and 'have' in (4)a.

(4) a. Daou varc'h-houarn o deus ar merc'hed, … two horses-iron have.3PL the girls 'The girls have (own) two bicycles, b. … met hemañ 'zo gante bremañ 'zo din. but this.one is with.them now is to me but the one that they have (lit. is with them) now is mine.'

The analytic copular constructions (4)b are attributed the structure (5)a, and they are the only possibility in the other Celtic languages for expressing possession. Breton too uses them for accompaniment and possession or attribution in (4)b. For possession however, Breton may also use the synthetic verb 'have' (4)a. According to the proposals cited, this is another way of distributing the same basic elements, suggested internally to Breton by the use of the root of be in 'have' noted above. The typical execution relates analytic 'be' to synthetic 'have' through movement, deriving (5)b from (5)a, corresponding to the derivational theory of applicative constructions where double object (6)b originates as prepositional ( 6)a (Baker 1988).

(5) a. J&R propose rather to base-generate the elements of the analytic construction in a different configuration to give the synthetic one, (5)c. Doing so takes cue from the current nonderivational theory of applicative structures, which relates prepositional (6)a and double object (6)b by basegenerating the same elements in different configurations, rather than by movement [START_REF] Pylkkänen | Introducing arguments[END_REF]. Starting with the applicative (5)c permits basegenerating the possessor outside the extended vP with its 3SG intervener, explaining why it is the sole argument immune to the Complementarity Effect, insofar as 'have' is the sole verb with this structure. However, we may leave it open whether (5)c can first derive from (5)a, along the lines of Ormazabal and Romero (1998); it is only important that prior to T phi-Agree the structure (5)c obtains. J&R suppose that the same structure is also a possibility for the auxiliary 'have', replacing possessor by agent and the complement of be by the past participle phrase; this converges with the same parallelism proposed in [START_REF] Mahajan | Universal Grammar and the typology of ergative languages[END_REF], [START_REF] Kayne | Towards a modular theory of auxiliary selection[END_REF][START_REF] Guéron | Le verbe avoir. In Forme (Hommages à Mitsou Ronat)[END_REF]. 2,3 2 We note that Guéron (1986: 166f.) proposes essentially the same two structures, one for possesor have with have+P selecting the thematic possessor in [Spec, IP], one for relational have where be+P select a small clause to the nonthematic subject of which P assigns a possessor reading. While we can connect Breton have to her proposal about theta-roles, we cannot do so for another hallmark of the relational construction for her, indefiniteness of the object. Among the uses of English have, Breton has only the possession and recipient ('get') reading, and the perfect auxiliary function; unavailable are the locational reading and as in French the experiencer and causative readings. However, the definiteness restriction seems to be about the same as in French or English, The town has the resource / the fame #(to do this), with the attributive X is to Y construction used for definites when infelicitous here. 3 A reviewer asks if our hypothesis predicts that all ditransitive verbs like 'give' should pattern with 'have', due to the presence of an Applicative structure. This is not the case. We predict that only ditransitive verbs whose applicative structure is higher than the extended vP

In this paper, we turn to the variation of 'have' across Breton dialects, which goes far beyond the morphological regularization note in J&R and indicated in Table 1d. The variation instantiates key points in the development from the analytic construction of the other Celtic languages to a fully transitive 'have' where full assimilation to regular transitive verbs has taken place, tracing an evolutionary path in UG parameter space. We follow this route, noting that it maps reasonably onto a procession among Breton dialects from the conservative peripheral regions to the innovative centre. The starting point is a surprisingly conservative variety where 'have' is essentially an intransitive applicative, the ending point a full transitive, and the syntactic stage proposed in J&R in between.

(7) Evolution of Breton 'have' a. Analytic prepositional 'be': theme nominative subject + PP possessor b. Synthetic applicative 'be': oblique applicative possessor as subject + nonagreeeing nominative theme as object (Gwenedeg). c. Unique transitive: nonoblique possessor subject above be-vP, unique in visibility to phi-Agree + regular transitive object. (KLT / Central) d. Regular transitive: possessor becomes agent in [Spec, vP]. (Marginal) 2 Gwenedeg: Applicative intransitive

The origin of the Breton synthetic 'have' is in an applicative construction regularly and literally expressing X is to Y, where the theme X is the subject of the intransitive copula be and the possessor Y is the indirect object. As a pronoun, the indirect object could be coded as an infix enclitic on an obligatory preverbal particle, glossed R. In (8), we see this with the indirect object of give, and in (9), that of be. Here the theme clearly behaves as the intransitive subject in controlling the agreement morphology of be. Servants were to me = 'I had servants.' (Middle Welsh, Black Book of Carmarthen 96.2)

In Breton a reanalysis of this regular be + indirect object changed its structure to 'have'. In the first step, the theme ceased to be treated as the subject of the verb for agreement, and be froze in the 3SG form. It is tempting to link this to the Complementarity Effect, whereby phonologically overt subjects do not agree, for it was fully in vigor by Middle Breton when synthetic 'have' appears and the agreement seen in Middle Welsh (9) would be impossible. 4 As the theme lost its subjecthood for agreement, the possessor gained it, by coming to double the indirect object clitic. Thus (10) has two parses, as be with a theme subject, with no agreement because of the Complementarity Effect, or 'have' with a possessor theme. In (11), the latter structure is favored, because the possessor controls a covariant verbal morphology or agreement, a typological typical correlate of subjecthood. (11) extends doubling to a configuration where it was not available to indirect objects, of the independent pronoun in the preverbal focus position, creating a new possessor subject. The switch in agreement controllers from theme to possessor was a watershed in the transition from 'be' to 'have ' (cf. Even 1987:125). From the earliest robust attestations of synthetic 'have' in Middle Breton, the case morphology and word order make it clear that the possessor is the subject and the theme is not. However, 'have' now comes with a unique agreement pattern in the language: an overt subject that agrees, and an agreement / prodrop morphology that is prefixal rather than suffixal.

The Gwenedeg dialect shows what appears to us to be pretty much this system (11), and establishing this point permits us to present diagnostics of the intransitive applicative structure. The possessor is the subject, albeit with the unique property of not obeying the Complementarity Effect. As a lexical noun or focused pronoun, it has direct case rather than the prepositional shell of indirect objects; as a weak pronoun, it is coded by the prefixal agreement of 'have' which has become distinct from particle + object proclitic, while in the meanwhile indirect objects have lost the ability to be coded by proclitics entirely; and it occurs in the structural position of the subject, preceding the theme. The theme, however, has not adopted the manner of transitive objects at all. Direct objects in Gwenedeg are coded by proclitics (O) on the finite verb, the er of ( 12)a. In contrast, the objects of 'have' are marked by a distinct series, the absolute enclitics (A) like ean of (12)b, and moreover they are restricted to 3 rd person. The absolute pronouns had historically the same distribution as DPs, but by Middle Breton came to be used (i) in the preverbal focus position for focalized subject or object, (ii) in the postverbal position only for subjects, (iii) as a partly distinct echoic pronominal series to double clitics and pro-drop agreement [START_REF] Hemon | Yezhadur istorel ar brezhoneg[END_REF]for echoics, Stump 1989). 5 The object of 'have' is the unique non-subject that uses the independent absolute pronouns rather than the clitic/agreement-dependent echoics in the postverbal field. In this, it looks on the surface like a subject, which it continues historically. Unlike all other absolute and echoic pronouns however, it is restricted to third person. So the theme of 'have' presents two mysteries: a subject-like morphological marking not found elsewhere for objects, and its restriction to 3 rd person.

(12) a. M' er/hur guel b. M' em es ean / *te I him/us(O) see.3SG

I have.1SG him(A)/you(A)

5 An example of a postverbal non-echoic, absolute subject is ( 16)a; evit bout me koh 'despite being me old' is another. Postverbal absolutes have some other uses, mostly subject-like (setu huy 'there you are'), except for objects of imperatives. Objects of imperatives fit the same story as for objects of have below, as discussed in [START_REF] Rezac | Elements of cyclic syntax: Agree and Merge[END_REF] for both Breton and Finnish, but we note that they are unexpectedly not always constrained to 3 rd person (in Gwenedeg). Independently of this constraint, Central Breton [START_REF] Humphreys | Phonologie et morphosyntaxe du parler breton de Bothoa[END_REF] and Low-Gwenedeg [START_REF] Cheveau | Approche phonologique, morphologique et syntaxique du breton du Grand Lorient (bas-vannetais[END_REF]) use a still further pronominal series for objects of imperatives.

(Gwenedeg, based on [START_REF] Guillevic | Grammaire Bretonne du Dialecte de Vannes[END_REF] This behavior of the objects of 'have' is brought into relation with the distribution of and restrictions on "nominative objects" in Finnish in Rezac (2004: 305-17). Finnish expresses 'have' by the be construction in ( 13), with an oblique (allative) possessor as the structural subject and the theme as the object [START_REF] Kiparsky | Structural Case in Finnish[END_REF]). The theme is nominative if and only if it is 3 rd person, and accusative like regular transitive objects otherwise. This personbased nominative-accusative alternation is general in Finnish for a certain class of structures. They fit the cross-linguistic Person Case Constraint, by which Case assignment and phi-agreement are out across an oblique intervener, ( 14) (Anagnostopoulou 2003). In Finnish the constraint restricts Case-licensing of the theme by T to 3 rd person if there is a c-commanding oblique subject, which in ( 13) is the possessor. So ( 14) is instantiated as (15). Finnish has an alternative way of Case-marking the theme if it is 1 st /2 nd person, by the accusative lower down in the structure; other languages that show the same restriction may not, like Icelandic [START_REF] Rezac | Escaping the Person Case Constraint: Referential computation in the φ-system[END_REF] The Gwenedeg 'have' is strikingly reminiscent of Finnish. The historical origins of 'have' are basically (13)/(15), and here and only here in the language do we find a non-subject postverbal absolute pronoun, and an absolute pronoun restricted to 3 rd person. We take this to diagnose the constraint ( 14) and attribute to Gwenedeg the structure (15). The possessor is a Quirky subject that receives inherent direct Case. The Case of T thus can go to the theme. Restriction to 3 rd person results from the possessor that intervenes between T and the pronoun that receives direct Case. 6Our approach thus stands on the property of the origin of the possessor as an indirect object to treat it as the oblique intervener in the structure (15). This is a theoretical prediction of our account for a grammar with "nominative" objects restricted to 3 rd person, since a nonoblique element would absorb the Case licensing capacity of T and leave the object with only the Case of v, like transitive objects. Within Gwenedeg itself, the behavior of the possessor is consistent with its being oblique, unlike in the later stages of the development of 'have'. The verb 'have' in Breton is the signal exception to the Complementarity Effect in having an agreeing subject, but in fact in Gwenedeg this is not quite so: the possessor only controls prefixal morphology on the verb if 1 st /2 nd person. A 3 rd person DP co-occurs with the 3SGM prefix instead of the proper 3SGF or 3PL prefix, which shows up only under pro-drop, (16). Gwenedeg varieties differ in treating independent 3 rd person pronouns as 1 st /2 nd person or as 3 rd person lexical DPs. Person and pronominality based restrictions are familiar characteristics of clitic doubling, not of the phi-Agree underlying phi-agreement in English and French. Clitic doubling may be unselective, but it frequently picks targets high on the scale of personhood and pronominality [START_REF] Poletto | The higher functional field: Evidence from Northern Italian dialects[END_REF] 7 . Clitic doubling differs from phi-Agree in another way: it can occur with arguments with or without structural Case, while phi-Agree is restricted to the latter [START_REF] Rezac | Φ-Agree and θ-related Case. In Φ theory: Φ-features across interfaces and modules[END_REF]. We conclude that the morphological covariance of 'have' with an overt possessor in Gwenedeg is not due to phi-Agree but to clitic doubling, making it possible for the possessor to be analysed as an agreement with the possessor of have, below analysed as clitic doubling, arguably preempting the pro-drop agreement of the theme. Second, it is cross-linguistically common for 3 rd person intransitive subjects to control agreement if postverbal rather than preverbal, as is indeed the case in Finnish independently of oblique-subjects (Kiparsky 2001). 7 There is no variation in this matter according to the lexical or auxiliary status of have, and the preverbal or postverbal position of the subject.

oblique. The clitic that doubles the subject is 3SG and incorporates into the verbal complex. Being an oblique, it is invisible for Agree.

Historically, clitic doubling may represent a very early stage in the transition of 'have' to a possessor subject: a point where agreement of be with the theme has been lost, but its agreement with the possessor is relatively close to what is available to proclitics. This step may well have been aided because the proclitic in all uses (direct and indirect object, oblique subject of be) could be doubled by a following 'echoic' pronoun: am boe me (Fleuriot 2002: 19(2), 23(3)). Such postverbal "echoic" pronouns, non-argumental but of the same form as absolute pronouns, could have helped the rise of possessor-'have' agreement (16b). 8 In Gwenedeg we seem to glimpse the first step of possessor agreement by clitic doubling of an absolute pronoun 9 .

Clitic doubling for 'have' agreement is suggested by another phenomenon unique to Gwenedeg in Breton: agreeing infinitives. If the agreement of 'have' arises from the clitic doubling of oblique arguments, it has nothing to do with phi-Agree and is not expected to be restricted to the canonical domain of phi-Agree, finite structures. 10 We briefly note three points about Gwenedeg agreeing infinitives here. First, they are obligatory in some varieties, in alternation with a nonagreeing infinitive in others, and absent in some other (Gwenedeg izel). Second, they occur in all contexts where other Breton dialects use an infinitive, including full CP control complements, reduced complements of modals, and even the X° infinitives fronted by long head movement in the ober conjugation (q.v. [START_REF] Borsley | Long head movement in Breton[END_REF][START_REF] Jouitteau | La syntaxe comparée du breton[END_REF]. Third, they co-occur with absolute objects. We get:

(17) a. Me / hi garehe am bout / hi devout bara/ean.

8 Echoic doubling is sometimes restricted to 1 st /2 nd person, [START_REF] Trépos | Grammaire bretonne[END_REF], [START_REF] Kervella | Yezhadur bras ar brezhoneg[END_REF]. 9 We find a pronoun doubled by a clitic in two contexts: (i) clitic doubling of a preverbal independent pronoun by the oblique proclitic of 'be', and (ii) doubling of such proclitics by a following 'echoic'. The two cases could have a common origin in the doubling of independent absolute pronouns in preverbal and postverbal position alike by the proclitics. In Modern Breton however, they are to be distinguished: the preverbal pronouns have yielded the absolute series that are generally preverbal (minor examples mentioned in note 5) and only ever doubled by the prefix (< proclitic) of 'have', while the postverbal pronouns have become the echoics that double a variety of preceding agreement morphemes (including the proclitics). Clitic doubling need not be subject to the above illustrated 1 st /2 nd person restriction11 . Extension of clitic doubling in Gwenedeg to all possessors would set up a situation identical to what phi-Agree would produce, creating a point of transition to a system where the possessor is analysed as a nonoblique goal of phi-Agree. As a consequence, the theme loses the ability to be Case-licensed by T which must license the possessor, and can only survive by become a regular transitive object. This will be the next stage in the evolution of 'have'.

Gwenedeg provides a second point of transition towards a transitive structure: auxiliary 'have' is found beside lexical 'have' in Gwenedeg and as far back in the history of Breton as synthetic 'have' goes. Analytic tense with 'have' auxiliary provide the sole context where 'have' can code the theme either by absolute pronouns, ean, or by the same proclitic series as regular direct objects, er, (18) (cf. partly Finnish (13)), and suggests free variation in the presence of the transitive Case-licensing system above the theme: Case-licensing by T is available only to 3 rd person theme, by v to all. ( 18 The proclitic object option is available for all lexical verbs (cf. above or 'see' in (12)a). 'Have' stands out in the paradigm once again and cannot host the proclitic. Le Roux (1957: 202) and Kervella (1995: §428(k)) attribute complementary distribution of object proclitics and the prefix agreement morphology of tensed 'have' to their common origin. We do not disagree but it leaves unexplained the lack of object proclitics in non-finite lexical 'have'.

We note that historically it is the participle in auxiliary construction that licenses the direct object proclitic. The participle either does or does not distribute case to its internal argument. In Gwenedeg, 'have' is intransitive and licenses no internal argument12 . Gwenedeg therefore has a synthetic 'have' which is still close to the original analytic X is to Y structure. It is an applicative construction rather than a prepositional construction, mapping the possessor above the theme. The theme is still Case-licensed by the system Case-licensing subjects rather than direct objects, although the latter system has already obtained its foothold in auxiliary 'have'. The possessor shows an agreement pattern that makes 'have' the unique agreeing verb in Breton, but for now it is a clitic doubling rather than a phi-Agree pattern and it extends to infinitives.

In the next stage, the traits of an intransitive applicative vanish and a full if unique transitive has arisen, for which we have identified two points of transition.

KLT: Special transitive

In other Breton dialects, the transition of 'have' to a full transitive structure incipient in Gwenedeg has been completed. The result is a verb identical in case marking and infinitive formation to any other transitive, yet distinct in the syntax of agreement by displaying no Complementarity Effect: overt subjects agree. This is the pattern of the standard language, but not an artifact of it. It seemed earlier on to be the dominant pattern across the three non-Gwenedeg dialects which form a unit beside it for other isoglosses: Kerneveg, Leoneg, and Tregerieg (or KLT). In the central area at the Kerne-Treger boundary a new system has arisen and expanded across much of Kerne and Treger, relegating the older pattern to the periphery. We call the latter the KLT pattern, beside the newer Central pattern. 13When we left Gwenedeg, neither the possessor nor the theme of be behaved like the arguments of a transitive. In KLT, the theme has become a plain transitive object, ( 19). KLT dialects do not code weak pronoun objects as proclitics like Gwenedeg does. The older system survives in parts of Leon, beside a new system that replaces them elsewhere. The new a-forms consist of a preposition a-'from', on which the pronoun incorporates. Contrary to the proclitics, the a-forms have no restrictions. The absolute and echoic pronoun series exist but are not used for objects of 'have'. The treatment of the possessor also differs from Gwenedeg. All possessors, overt or null, control the same prefix agreement morphology, (20). On the surface, this makes 'have' more different from other verbs of the language than in Gwenedeg, because for the other verbs no overt subjects agree. However, in a deeper way the possessor comes closer to the plain subject of transitives: all possessors are treated alike regardless of person and type, and all control one of the typologically common traits of transitive subjects, phi-agreement, even if other subjects do not by the Complementarity Effect. Like other verbs, 'have' also has no agreeing infinitives. ( 20 It is this stage that is the focus of [START_REF] Jouitteau | Deriving the Complementarity Effect: Relativized Minimality in Breton agreement[END_REF]. They interpret it as a modification of the applicative structure (15) of Gwenedeg by the loss of oblique case on the possessor. It becomes a plain DP and as such participates in structural Case assignment and phi-Agree relations of T, (21). This simple change derives the dramatic differences of KLT from Gwenedeg. Loss of oblique case assimilates the possessor to transitive subjects and it gets the Case they get. However, its origin in the specifier of the applicative head P/Appl places it in a position where it avoids the intervention of the 3SG (extended) vP boundary that creates the Complementarity Effect for other subjects, which are within it. The result is a unique configuration where overt DPs control phi-Agree. The structural Case of the possessor has repercussions farther down for the theme, because the possessor absorbs the Case of T and it is not available for the theme. The latter must be Case-licensed by the same mechanism as transitive objects, so its absolute form disappears (it resorts to an a-form). Consistently with the idea that infinitives are not the locus for phi-Agree, in dialects where the agreement morphology of 'have' comes from phi-Agree, we find no 'agreeing' infinitives for 'have'. There are two telling variations on the KLT pattern that fit into this proposal. The first is a regularization in agreement morphology of 'have'. Its historical origin (opaque in synchrony) as an indirect object proclitic makes it a unique proclitic subject agreement. In some KLT varieties a partial regularization to that used by other verbs occurs. In ( 22) for example, the 3PL future def-i-nt [have-FUT.3PL] replaces o def-o [3PL have-FUT.3SG] by taking on to the perceived stem def-of 'have' the regular future 3PL suffix -i-nt. Such regularization never seems to occur within a Gwenedegtype system were prefixes reflect clitic doubling, not phi-Agree14 . The second change within much of KLT is the loss of gender distinctions in 3SG. Within Breton, pronouns but not verb agreement make the M(asc)/F(em) distinction in 3SG. 15 Supposing phi-Agree to be insensitive to gender distinctions, J&R suggest that the lack of gender in verb agreement comes from its spelling out phi-Agree with pro rather than pro itself. On 'have', the Gwenedeg systems shows gender distinctions, reflecting the nature of the prefix as an object proclitic. If in KLT, 'have' agreement with overt subjects is phi-Agree, it should neutralize gender distinctions. Many varieties do so, (23): some only for agreement with overt subjects, others for pro-dropped subjects of 'have' as well. 16 However, contrary to the proposal, it seems relatively common for a KLT system to reflect gender distinctions even with overt subjects of 'have', suggesting that phi-Agree does make gender distinctions and they do not appear on the pro-drop agreement of regular verbs by diachronic accident. We leave this for future research. 4 The Central innovation

In the "Central" dialectal area at the Kernev-Treger-Gwened border, the principal source of innovation in Breton dialects, the (arguably antecedent) KLT pattern is modified by one that at first sight looks like a mixture of pro rather than the subject pro itself, so it is unsurprising that phi-Agree with the overt subject of 'have' undergoes phi-Agree starts using this morphology, although we will see a different road for a dialect to take in the next section.

16

KLT and Gwenedeg properties. Mostly, it is KLT: the object is coded fully as the object of transitives (mostly with a-forms in this area), and there are no agreeing infinitives. As in KLT, a regularization of the agreement morphology of 'have' occurs, but generally goes considerably further than in KLT. The Gwenedeg-like characteristic is the apparent restriction of agreement with overt subjects to 1 st /2 nd person, as in the paradigm in Table 2. Morphologically, we see that the old prefix paradigm has been reinterpreted to show only person distinctions (1 m, 2 p, 3 n), and plurality in 1 st /3 rd person is marked by recruiting the suffixes of the regular verb (1PL m(p), 3PL int). However, the 3PL marking is only used for pro-dropped 3PL subjects; overt subjects use the 3SG form neus. Although this looks like Gwenedeg agreement with 1 st /2 nd person, the Central dialects do not show the properties of Gwenedeg that go with oblique possessors: absolute objects and agreeing infinitives.

There is another interpretation of the pattern in Table 2: phi-Agree occurs but does not affect suffix morphology. Other dialects in this area furnish evidence for it. In them, the 3PL overt/pro alternation i neus -pro neuint extends to 1PL as ni beus/meus -pro meu(so)mp/neu(so)mp (Favereau 1997: §424), showing that the prefix and suffix reflect distinct systems that correlate with overt and pro-dropped subjects respectively. We posit that the prefix reflects phi-Agree, and because it tends to make only person distinctions, 3 rd person is neutralized, but 1-2-3 contrasts remain. Rarely, phi-Agree is also shown by the suffix, as in Table 2 ni meum. Typically however, the suffix reflects only pro, alone or in combination with the prefix for phi-Agree: 1PL meu(so)mp/neu(so)mp, 2PL peuc'h, 3PL neuint.

Plain transitive

The final stage in the evolution of Breton takes the KLT system one step further by removing the sole syntactic difference between it and regular verbs. Agreement with all overt subjects disappears, and 'have' falls under the Complementarity Effect, whereby it agrees only with pro-dropped subjects. All other properties of the KLT 'have' remain, which already are those of plain transitives. The system appears rarely. 18 (24) ni/te/huy/ey (a) neus I/you(SG)/you(PL)/he R have.3SG 'We/You have.' or 'He has.' This is the logical conclusion of the transitivization of the verb, the creation of a regular transitive. We suppose at this step to occur simply the loss of the applicative structure and thus to result a plain transitive like Czech mít, or Basque eduki 'have, hold'. Chiefly possessive, in combination with a small clause they also express the perfect (25) and so could serve as the perfect auxiliary. For Breton, this stage remains to be documented and investigated.

(25)

Mám tu chybu opravenou (Czech) I.have the error corrected 'I've got the error corrected'  'I have the error which is corrected.'

Conclusion and prospects

We have followed the development of the verb 'have' from an applicative intransitive to a plain transitive, tracing the path of its diachronic evolution through its incarnations across the spectrum of Modern Breton dialects. 18 While the pattern is commonly cited, it seems hard verify. The loci classici give eighteenth century Treger sources where 2 nd /3 rd persons use the 3SGM form: Le Roux (Le Roux 1957: 186), Hemon (2000: §174). Le Roux (1977: maps 82ff.) finds (only) nearly complete regularization only at St. Fiacre (Treger). We note however that the posited evolution of 'have' has happened in the history of Breton to another, much rarer verb: teurvezout 'daign', which started out with the same history of the proclitic indirect object to become subject agreement and finished by full regularization before the modern period (Hemon 2000: §151).

Along the road, various tools have been used to probe the 'have' construction in order to ascertain its syntactic character: There are many theoretical predictions here to be set against further data from Breton and elsewhere. The various aspects of the syntax of 'have' do not define a bi-polar opposition but a collection of combinable properties, some of which have not been met with. Gwenedeg has oblique possessors with restricted clitic doubling and absolute themes Case-licensed by T; eliminating the restriction on clitic doubling would produce a Gwenedeg system with fully "agreeing" possessors, while adding the transitive v would produce a Gwenedeg system with themes optionally and obligatorily marked as direct objects. However, other aspects of the system are fixed; absolute objects restricted to 3 rd person indicate Case-licensing by T and a ccommanding intervener, rather than Case licensing by v or free Case licensing by T. Beside predictions, partial and partly contradictory syntaxmorphology correlations emerge, whose status is unknown and that are to be explored in future work. Gwenedeg-type systems with clitic doubling show no tendency to regularize the prefix morphology of 'have', which might be explained if the suffix morphology of regular verbs indicates phi-Agree with pro rather than incorporated pro itself. However, in the Central area the suffix morphology seems recruited specifically to spell out pro and the prefix the person dimension of phi-Agree, in contrast.

Turning to diachronic development, the forms of 'have' across Breton instantiate key stages along the path of transformation from an analytic X is to Y to transitive Y possesses X construction (Le Roux 1957), along with the likely points of transition between adjacent stages. Gwenedeg proves surprisingly archaic, arguably a simple intransitive applicative, accounting for most of the properties of its 'have' construction that diverge from Standard Breton: quirky subjects, absolute objects, their restriction to 3 rd person, non-agreement with 3 rd person overt subjects, agreeing infinitives. Yet already as in the earliest Breton sources for 'have', the verb is in use in auxiliary constructions where the object of the participle has regular transitive marking. It facilitates transition to the KLT pattern with its loss of inherent case on the possessor and a consequent unique pattern of agreement with an overt subject, revealing its syntactic origin in a high applicative above the extended vP barrier that envelops all other arguments. In KLT the anomalous prefixal agreement morphology tends to get fixed, and nowhere more so than in the innovative central zone whose pattern is rapidly spreading out across Breton and has relegated the classical KLT pattern to the western periphery, meeting the archaic Gwenedeg system at its South Eastern edge. At the end of the evolutionary road, in some North Eastern Treger varieties 'have' seems to be on its way to become a regular transitive with no applicativity to it anymore.

We have barely touched here on the factors facilitating the first steps of this transition and the difference between Breton (and Cornish), with stable synthetic 'have' as possession and auxiliary verb, and Old Irish and Middle Welsh which began but aborted its formation. The Complementarity Effect seems to us one answer: present from the beginning of Middle Breton and earlier, it is far more finicky in Middle Welsh for postverbal subjects and absent for preverbal ones [START_REF] Willis | Historical syntax[END_REF], and arises in Modern Irish by convoluted routes during the Middle Irish period [START_REF] Mccone | The Early Irish Verb[END_REF]). As we have noted, the Complementarity Effect would have made it easy in Breton to reanalyse the theme as not a canonical subject for agreement, giving it its structural object status. Case also has a role, at least in Irish where dative distinguishes indirect objects and might have prevented their reanalysis as subjects. We leave the details for future work.

( 8 )

 8 y gwr a-m roddes i gwin the man R-me gave the wine (Middle Welsh, Evans 1964: 57, Fleuriot 2002: 23, Book of Taliesin 40

  peh en des ind / er sent groeit what have.3SG they / the saints done (Larboulette, Buhe er sent) b. m' am-eus me glinet I have.1SG I slipped 'I have slipped.' (Herrieu, Kammdro an Ankoù)

  have known them(O)

  PP poss iphi Appl/P [ vP=3SG be [ vP v trans … theme]]] --phi-Agree/Case-------------

  regular o defo] my sons have-FUT-3PL 'My sons will have…' (Kerne, Ernault 1888: 265 < Barzaz Breiz)

  ket ' vara … Marharid 3SG.M has told to-me NEG is NEG P bread 'Marharid told me there is no bread.' (Leon, Amañ 'z eus plijadur, p. 66) 17

  and agreement of the theme as canonical intransitive subject, intransitive subject in object position, and regular transitive object. b. The Person Case Constraint restriction on the theme, revealing a ccommanding oblique intervener for T-theme Case licensing. c. Sensitivity to person and pronominality for agreement with overt subjects, indicating clitic doubling rather than phi-Agree. d. Agreeing infinitives, indicating clitic doubling rather than phi-Agree. e. Loss of gender distinctions, perhaps indicating phi-Agree.

  . 2003. The syntax of ditransitives. de Gruyter: Berlin. Benveniste, E., 1960. 'Etre et Avoir dans leurs fonctions linguistiques', In: Bulletin de la société Linguistique de Paris, LV, reproduced In: E. Benveniste, 1966, Problèmes de linguistique générale. Gallimard, Paris, 187-207.

Table 1 :

 1 Agreement in Breton

	variant)

X

a: X knows b: breaks X c: X has d: X has (

  [ VP be [ SC possessor theme]] b. [ TP possessor k (…) [ vP be+P i [t k t i theme]]] c. [ PP possessor [P/Appl+be i [ vP t i theme]]]

(6) a. she [√gave [apples [P/Appl John]]] = She gave apples to John b. she [John [P/Appl [√gave apples]]] = She gave+P John apples

  .

				Finnish
	(13)	Hänella	on kirja / se / *sinä	// *kirjan / *sen / sinut
		him.ALL is book/ it /*you.NOM	// *book / *it /√you.ACC
		'He has the book / it / you'.
	(14)	AGR/Case … (*oblique) … XP	[Person Case Constraint]
	(15)	possessor-oblique

i T-AGR/Case NOM … t i … theme=3 Finnish

Table 2

 2

		: 'have' paradigm in Poher (Kerne uhel), Trevidig (1987)
		SUBJ 'have' ãnoued 'a cold' 'have' + pro SUBJ
	1SG	mé meus	ãnoued meus
	2SG	té peus	ãnoued peus
	3SGM,F hẽw, hi neus ãnoued neus
	1PL	nĩ(m) meum ãnoued meum
	2PL	houi peus	ãnoued peus
	3PL	i/hè neus	ãnoued nẽũint

The infinitive of 'have' can in some varieties also be kaout, coopted from kavout 'find', but with a restricted syntactic distribution: it cannot be used for the perfect auxiliary have, where the infinitive of be always steps in. Like the verb 'be', 'have' can also not be targeted by 'long head movement'.

Notably the Complementarity Effect comes into force much later in Irish, and it is often violated with postverbal subjects in Middle Welsh and systematically with nonclefted preverbal subjects in the 'abnormal sentence' seen in (9)(Willis 1998, vs. clefted 'mixed sentence'). These languages did not develop syntetic have although they have the antecedent analytic structure.

Weak subject pronouns are generally required to incorporate into T/V to trigger pro-drop agreement. The object of have does not have this option, along with the other postverbal absolutes of note 5. This is unsurprising, although it needs more work. First, there is already

Dialectal variation in subject doubling is not surprising. SeePoletto (2002) for the range of subject doubling variation across related dialects.

Proclitics of tensed verbs and proclitics of non-tensed verbs were morphologically distinct in earlier Breton, where the proclitics of the participle and infinitive belong to the distinct possessive series(Hemon 2000: §53(3)). Gwenedeg no longer keeps this distinction(Guillevic and Le Goff 1986: 154).

The traditional nomenclature of Breton dialects follows the old diocesal lines: Gwenedeg in theSouth-East (uhel 'low' in E and izel 'low' in W), Kerneveg SW (uhel inland, izel maritime), Leoneg NW, Tregerieg NE. Overlaid on it is a newer pattern where the innovative center at the Kerne-Treger boundary creates a Central continuum spreading out chiefly SW and NE, leaving Gwenedeg and Leoneg as peripheral conservative regions, beside over very distinctive though non-conservative pockets like Bigouden in Kerne.

J&R propose that the pro-drop morphology of regular verbs is reflecting phi-Agree with

Independent and clitic pronouns, as well as the prosodically fused ones that surface as suffixes on prepositions with the behavior of strong pronouns:[START_REF] Jouitteau | Deriving the Complementarity Effect: Relativized Minimality in Breton agreement[END_REF].

In Le Roux (1977: map 84), the latter occurs in most of Kerne but is only rare in Leon and Treger.

Jean-Pierre le Goff, Plounévez-Lochrist, Amañ 'z eus plijadur:66.