Urtzi Etxeberria

Ricardo Etxepare

Number agreement in Basque: Counting vs. Measuring 1

. The distinction we make between the measuring field (in Measure Phrase position) and the counting field (in NumP position) affects referentiality; in fact, agreement and reference only become relevant upon reaching NumP -not before that position, i.e. not in Measure Phrase position. We also show that non-agreeing quantifiers are sensitive to the nature of the predicates they associate with.

1

Introduction: the phenomenon Usually, Basque plurality denoting nominal expressions trigger obligatory agreement in number with the inflected verb. 2

(1) Anek liburu-ak erosi ditu/*du Ane.erg book-D-pl buy have.pl/have.sg

However, it has been noted (see [START_REF] Rotaetxe | Numeral eta aditzen arteko konkordantziaz[END_REF]Txillardegi 1977Txillardegi , 1978;;EGLU 1985;[START_REF] Etxepare | Asko-ren zenbait alderdi[END_REF] that so called 'vague' weak quantifiers in Basque only optionally agree in number with the inflected verb (2a-d). 3,4 (2) a.

Bezero asko etortzen da/dira halako egunetan customer many come-hab is/are such days-in 'A lot of customers come on such days' b.

Bezero gehiegik eskatu du/dute arrain zopa customer too-many-erg asked aux-sg/aux-pl fish soup 'Too many customers asked for fish soup' c.

Maiak lagun gutxi ikusi du/ditu gaur Maia.erg friend few seen aux.sg/aux.pl today 'Maia has seen few students today' d.

Gure bezero ugari aurkitu dut/ditut beste denda horretan our customer big-number found aux-sg/aux-pl other shop that-in 'I found a big number of our customers in that shop'

The notion of what we mean by 'vague' weak quantifier can be intuitively grasped by means of the following contrast:

(3) a.

Mila ikasle etorri dira/*da thousand student come aux-pl/aux-sg 'One thousand students came' b.

Milaka ikasle etorri dira/da thousand-suffix student come aux-pl/sg 'Thousands and thousands of students came' Whereas (3a), which involves a definite quantity, triggers plural agreement in the inflected verb, (3b), which involves a non-definite quantity (equivalent to thousands of in English), only optionally triggers agreement. Cardinal quantifiers, in the varieties of Basque we focus on here, always trigger plural agreement. Vague quantificational expressions constructed out of them, on the other hand, may not.

This phenomenon is general in the Basque area, with some interesting dialectal variation that we will not be able to address here (see Etxeberria and Etxepare, to appear). The present paper offers a preliminary analysis of the phenomenon. We claim that non-agreeing quantificational expressions are not counting expressions, but measure phrases. Measures constitute the other quantificational domain in Basque that presents an agreement alternation in number: 5 (4) Hiru litro ardo edan du/ditu three liter wine drunk aux-sg/aux-pl 'He/she drank three liters of wine'

We may wonder at this point what the agreement alternation is: is it an alternation between plural number features and singular ones? Or is the singular agreement form just a default, selected in the absence of any number feature? It is not easy to answer to this query directly in the context of the inflected forms. However, if we move to other syntactic contexts, the answer seems to favor the conclusion that third person singular agreement, in the context of vague quantifiers in Basque, is just a default, with no correspondence with actual number features. One such context is provided by secondary predication, which requires agreement in number (see [START_REF] Artiagoitia | Verbal projections in Basque and minimal structures[END_REF]. The example in (5) gives an illustrative example with a Small Clause complement:

(5) Liburuak hondatu(*-ak) ikusi ditut book-D.pl worn-out.pl seen aux.pl 'I've seen (the) books worn-out'

The sentence (5) contains a Small Clause predicate hondatuak 'worn-out' which obligatorily agrees in number with the subject liburuak 'books'. Now consider the contrast in (6):

(6) a.

Liburu asko hondatuak ikusi ditut book many worn-out.pl seen aux.pl 'I've seen many books worn-out' b.

*Liburu asko hondatua ikusi dut book many worn-out.sg seen aux.sg 'I've seen many books worn-out' Whereas a vague quantifier that agrees in the plural with the inflected verb licenses a secondary predicate with a plural suffix -k on it, a vague quantifier that does not agree in the plural cannot license singular agreement in the secondary predicate either. The conclusion seems to be that agreement in the singular with the quantifiers that do not agree in the plural with the verb is impossible, and that therefore, the relevant quantifier forms must lack number features, either plural or singular. 6 That the problem is in number agreement and not, say, in the ability of non-agreeing quantifiers to license a secondary predication is shown by the following fact: if we allow for a secondary predicate that does not have number, secondary predication with vague quantifiers becomes possible. One relevant configuration involves the [-ta] adverbial ending for participles, which does not agree in number in Basque. When the participial substitutes for the [determiner+number] suffix, secondary predication with vague quantifiers becomes possible (7).

(7) Liburu asko hondatu-ta ikusi dut/ditut book many worn-out.part seen aux.sg/aux.pl 'I've seen many books worn-out'

The main hypotheses we defend in this paper are the following: first, we will argue that non-agreeing quantifiers are conceptually measures. Basque shows that measures head their own functional projection in the expanded structure of the Noun Phrase. This functional projection is placed in-between the Classifier Phrase (where division occurs), and the Number Phrase (where counting occurs, following [START_REF] Borer | name only[END_REF]. We also show that certain referential properties, such as the possibility of establishing a discourse variable, and the potential for the enumeration of individuals crucially require the projection of the counting number head. Besides projecting a dedicated functional structure, we also show that non-agreeing quantifiers are sensitive to the nature of the predicates they associate with. Measure Phrases seem to measure both individuals and events/states, as long as the latter denote non-trivial part-whole structures. The predicate sensitivity of measuring quantifiers, we claim, has two sources: one is the monotonicity constraint proposed by [START_REF] Schwarzschild | The grammar of measurement[END_REF] as holding of measure functions universally; the other one is a homomorphism relation [START_REF] Krifka | Nominal reference, temporal constitution, and quantification in event semantics[END_REF][START_REF] Filip | Quantification, Aspect and Lexicon[END_REF][START_REF] Nakanishi | Domains of measurement: Formal properties of non-split/split quantifier constructions[END_REF][START_REF] Nakanishi | Measurement in the nominal and the verbal domain[END_REF] which maps the denotation of a noun phrase into the denotation of the predicate. The predicate sensitivity of non-agreeing quantifiers can thus be viewed as the result of this mapping relation. The paper is organized as follows: in section 2 we present the received analysis concerning the agreement alternation in Basque. Section 3 provides arguments against this view. Section 4 shows that non-agreeing quantifiers must be interpreted distributively. This imposes certain restrictions on the class of predicates they can co-occur with. Section 5 discusses the nature of the quantifiers involved in the agreement alternation. It is shown that the relevant quantifiers are so-called degree-quantifiers [START_REF] Doetjes | Quantifiers and selection: On the distribution of quantifying expressions in French, Dutch and English[END_REF]): quantifiers which combine with any syntactic constituent as long as it can be interpreted cumulatively. Section 6 suggests some cross-linguistic analogues of the Basque alternation. Section 7 discusses the semantic basis of predicate sensitivity. Section 8 proposes a syntactic structure for non-agreeing quantifiers. Section 9 presents the conclusions of this paper.

A previous view: non-agreeing cases as masses

The descriptive grammar of Euskaltzaindia (1985: 223-224) assimilates the absence of number agreement with weak quantifiers to the absence of number in mass terms. Take for instance the contrast in (8).

(8) a.

Haragi asko jaten du meat much eat-hab aux.sg 'He eats a lot of meat' b.

Haragi asko jaten ditu meat many eat-hab aux.pl 'He eats many types of meat'

The presence of number agreement in (8b) triggers a count interpretation of the mass term haragi 'meat', which comes to denote a set of individualized meat types. The grammar of the Academy suggests that the absence of number agreement with count terms has the opposite effect: it converts count terms into mass terms. The grammar comments on the following sentences in (9).

(9) a.

Liburu asko erosi dut book many bought aux-sg 'I bought many books' b.

Liburu asko erosi ditut book many bought aux-pl 'I bought many books'

According to the Academy's grammar, (9a) and (9b) do not have the same interpretation: whereas "in the first case we consider a mass of books; in the other case we consider one book and then another one, and another one, and so on" (1985: 223). To make things clearer, the grammar presents the following case.

(10) a.

Harri asko bota dute stone much thrown aux-sg 'They threw a lot of stone' b.

Harri asko bota dituzte stone many thrown aux-pl 'They threw many stones'

In (10a) harri 'stone' is taken to be non-count, as a big quantity of stone. In (10b) it refers to a big quantity of stones (as a count term). The Academy's grammar does not go beyond the intuition above. Although we will not pursue this line of analysis, we share the intuition that (10b) offers more opportunities for an individualized treatment of the stone than (10a). For instance, (10b) would be more appropriate to describe a situation where demonstrators attack the police by throwing stones to them. This implies the existence of individualized pieces of stone, and a multiplicity of stone-throwing events. (10a) on the other hand, would be more appropriate to describe loads of stone being dumped during some road construction. It can be shown however that non-agreeing quantifiers are not mass terms. As a starting point, we consider Pelletier's well known thought experiment (1975) to characterize mass terms. He proposes the existence of two imaginary machines, that he calls the Universal Grinder and the Universal Objectifier. For the Universal Grinder, we are to imagine a device which can grind anything, no matter how big or small. Into one end of the device "is inserted an object of which some count expression is true, and from the other end spews forth the finely-ground matter of which it is composed. So a hat is entered into the grinder and after a few minutes there is hat all over the floor" (from Pelletier and Schubert 1989:342). This is so despite the fact that we could also have said that there is felt all over the floor, using a mass expression. Examples of this type "show that many count expressions can be seen to already have within them a mass sense or a mass use" (ibidem: 343). Taking the word sagar 'apple' as our putative count term, we could take (13) to involve the mass coming out of the Universal Grinder:

(13) Entsaladak sagar pixkat dauka salad-D-erg apple bit has 'The salad has a bit of apple in it' Take, however, something like (14), with a non-agreeing vague quantifier:

(14) Ikasle asko ikusi dut gaurko batzarrean student a lot of seen I-have today's meeting-D-in 'I have seen a lot of students in today's meeting'

The sentence in ( 14), with a non-agreeing quantifier, does not involve a mass term, in Pelletier's sense: what I have seen in ( 14) is not scattered pieces of student, but a number of students, all of them of a piece. True, the force of this argument against a mass-approach to non-agreeing quantifiers depends on the force of Pelletier's metaphor to characterize mass terms as a whole. We know that in this sense, the metaphor is not comprehensive enough. Other mass terms appear to reflect objects that we would better locate in the entering side of the machine. This is the case of mass terms like furniture or crockery (Chierchia 1998): ground-up furniture and furniture do not mean the same, despite the mass status of the term. In any case, even with simple ambiguous nouns such as apple, the mass-approach falls short of accounting for the range of interpretations that non-agreeing cases have. Consider a sentence like (15):

(15) Plater honetan sagar asko ikusten dut dish this-in apple many see aux-sg 'I see a lot of apple in this dish' or 'I see a lot of apples in this dish' As shown by the translations, non-agreeing quantifiers can be interpreted in two ways: either as mass terms, referring to a quantity of apple, or as referring to a plural set of (whole) apples. In other words: the sentence in (15) can be interpreted as making reference to, say, a dish containing a set of piled-up entire apples. The mass-approach has nothing to say about this second interpretation.

Other properties distinguishing mass terms from non-agreeing cases lead us to reject the mass approach to non-agreeing quantifiers. [START_REF] Lonning | Mass Terms and Quantification[END_REF] shows that masses cannot entertain a predication relation with non-homogeneous predicates. Homogeneous predicates are those that are both cumulative and divisive. The examples in (16) involve a non-homogeneous predicate (to weigh more than 300 kilos). Whereas mass quantifications can not be the subject of the nonhomogeneous predicate (16a), non-agreeing quantifiers with a count noun can (16b).

(16) a.

*Ur askok 300 kilo baino gehiago pisatzen du water a lot of 300 kilo than more weight-hab aux '*A lot of water weights more than 300 kilos' b.

Zaldi askok 300 kilo baino gehiago pisatzen du horse a lot of 300 kilo than more weight-hab aux 'A lot of horses weight more than 300 kilos'

The distributive character of non-agreeing quantifiers

One of the characterizing properties of non-agreeing quantifiers (and which further distinguishes them from mass terms) is their distributive nature [START_REF] Etxepare | Asko-ren zenbait alderdi[END_REF]. They can only be interpreted distributively, and this sets certain restrictions on the kind of predicate they can attach to.

Distributive readings

Consider for instance the contrast between ( 17) and ( 18).

(17) Azkenean gazte askok altxatu behar izan zuten harria finally young many-erg lifted must have aux-pl stone-D 'Ultimately, many youngsters had to lift the stone' √ collective √ distributive (18) Azkenean gazte askok altxatu behar izan zuen harria finally young a lot of-erg lift must have aux-sg stone-D 'Ultimately, many youngsters had to lift the stone' * collective √ distributive ( 17) involves an agreeing vague quantifier. This yields two possible readings for the predicate: a distributive one, where each of the youngsters lifts the stone, and a collective one, where the entire set of youngsters lifts the stone. ( 17) also allows intermediate readings, where the set of youngsters divides in small groups to lift the stone. The range of distributive readings in ( 17) is typical of count plural entities (see [START_REF] Krifka | Thematic relations as links between nominal reference and temporal constitution[END_REF]. Unlike ( 17), ( 18) only allows a strict distributive reading, where youngsters individually lift the stone, and several stone-liftings (as many as there are youngsters) occur.

Predicate classes

Non-agreeing quantifiers are incompatible with collective predicates (predicates that do not allow event distribution). The examples in ( 19)-( 21) all contain a predicate that does not naturally allow atomic distribution (distribution down to the atomic entities making up a plurality). Whereas agreeing quantifiers can be combined with those predicates (a), non-agreeing ones cannot (b):

(19) a.

Ikasle ohi askok festa horretan topo egin zuten student ex many-erg party that-in meet done aux-pl 'Many ex-students met at that party' b.

*Ikasle ohi askok festa horretan topo egin zuen student ex a lot of-erg party that-in meet done aux-sg 'A lot of ex-students met at that party' (20) a.

Lantegian, langile asko batzartu dira factory-in worker many met are 'At the factory, many workers had a meeting' b.

?*Lantegian, langile asko batzartu da factory-in worker a lot of met is 'At the factory a lot of workers had a meeting' (21) a.

Jonek liburu asko ordenatu ditu Jon-erg book many arranged aux-pl 'Jon arranged many books' b.

??Jonek liburu asko ordenatu du Jon-erg book a lot of arranged aux-sg 'Jon arranged a lot of books' Having a meeting, reaching an agreement or arranging books in a certain order denote relations that require more than one individual and give rise to collective readings. Predicates that denote such a relation are incompatible with non-agreeing quantifiers.

Once-only predicates

Consider ( 22):

(22) a.

Polizi askok kolpatu dute manifestaria policemen many-erg beat aux-pl demonstrator-D 'Many policemen have beaten the demonstrator' b.

Polizi askok kolpatu du manifestaria policemen a lot of-erg beat aux-sg demonstrator-D 'A lot of policemen have beaten the demonstrator' A predicate like manifestaria kolpatu 'beat the demonstrator' does not, unfortunately, make reference to a unique event: it is something that can happen more than once, even with the same demonstrator (leaving aside fatal events). In this context both the agreeing and the non-agreeing quantifier are possible. Now take (23a), a sentence that contains the predicate putrea hil 'kill the vulture'. This is something that can only occur once, if the same vulture is involved. Let us call this type of predicate a 'once-only predicate'. Once-only predicates cannot combine with non-agreeing quantifiers, as shown in (23b). The reason must be the same that precludes the occurrence of non-agreeing quantifiers with collective predicates. Although once-only predicates are not collective, they don't license a distributive relation, by definition. But non-agreeing quantifiers must be interpreted distributively.

(23) a.

Baserritar askok hil zuten putrea farmer many-erg kill aux-pl vulture-D 'Many farmers killed the vulture' b.

*Baserritar askok hil zuen putrea farmer a lot of-erg kill aux-sg vulture-D 'A lot of farmers killed the vulture'

5

What do these quantifiers quantify over?

Non-agreeing quantifiers show certain restrictions with regard to the predicate they combine with. In abstract terms, we can talk of their 'predicate sensitivity'. Those constraints must at least include the impossibility of combining with (i) collective predicates (section 4.2); and (ii) onceonly predicates (section 4.3).

The predicate sensitivity shown by those quantifiers indicates that their domain of quantification includes events, not only objects. We propose that an appropriate paraphrase for a sentence with a non-agreeing quantifier (24a) is something like (24b):

(24) a.

Ikasle asko etorri da gaur student many come is today 'Many students came today' b.

Ikasle etorrera asko egon da gaur student come-Nom many been is today 'There has been a lot of student-coming today'

This view of the non-agreeing quantifiers approaches them to so-called 'event-related readings' of weak quantifiers, as presented in [START_REF] Krifka | Four thousands ships passed through the lock: Object-induced measure functions on events[END_REF] and [START_REF] Doetjes | The semantics of event-related readings: a case for pairquantification[END_REF]. We compare the Basque structures with event-related readings in section 6.4. Together with their vagueness, there is a further property that characterizes the quantifiers entering into the agreement alternation: they seem to operate across a large class of domains. The set of domains that the relevant quantifiers operate on includes plural nouns, with and without agreement: [START_REF] Doetjes | Quantifiers and selection: On the distribution of quantifying expressions in French, Dutch and English[END_REF][START_REF] Doetjes | Degree quantifiers[END_REF]) calls "degree-quantifiers": Degree Quantifiers are insensitive to the categorial properties of the phrase they combine with, as far as the latter can be interpreted cumulatively. Cumulativity can be defined in the following terms:

(25)
(28) Cumulativity [START_REF] Krifka | Scope inversion under the rise-fall contour in German[END_REF]:

∀X ⊆ UP [CUMP(X) ↔ ∃x,y [X(x)∧X(y) ∧ ¬x=y] ∧ ∀x,y [X(x)∧X(y) → X(x⊕y)]]
(X is cumulative iff there exist y, x with the property X (and x distinct from y) such that for all x and y, if x, y have the property X, then X is a property of the sum of x and y) [START_REF] Doetjes | Quantifiers and selection: On the distribution of quantifying expressions in French, Dutch and English[END_REF] argues that Degree Quantifiers measure their domain of quantification; in other words, they are measures. We conclude that this naturally applies to Basque non-agreeing quantifiers and that non-agreeing quantifiers in Basque are conceptually measures.

Crosslinguistic connections

The predicate sensitivity shown by non-agreeing quantifiers in Basque finds interesting correlates in other languages. Predicate sensitivity is, in those languages, also the result of a structural alternation that involves measuring quantifiers. We point out two cases here: split quantification as described by [START_REF] Nakanishi | Domains of measurement: Formal properties of non-split/split quantifier constructions[END_REF][START_REF] Nakanishi | Measurement in the nominal and the verbal domain[END_REF]; and so-called Quantification at a Distance [START_REF] Obenauer | Une quantification non canonique: La quantification à distance[END_REF][START_REF] Doetjes | Quantifiers and selection: On the distribution of quantifying expressions in French, Dutch and English[END_REF]). 7

Split measure phrases

An alternation in meaning analogous to the Basque one arises in Japanese, with what Nakanishi calls Measure Phrases (MPs). For [START_REF] Nakanishi | Domains of measurement: Formal properties of non-split/split quantifier constructions[END_REF], Measure Phrases in Japanese are all weak quantifiers, including cardinals, and they can occur under two different configurations: the measuring element can show up close to the noun it measures, or it can show up detached from it (so-called split MPs) [START_REF] Nakanishi | Domains of measurement: Formal properties of non-split/split quantifier constructions[END_REF][START_REF] Nakanishi | Measurement in the nominal and the verbal domain[END_REF]. In (29a) and (30a), the measure quantifier and its nominal restriction are adjacent to each other, and occur inside the quantificational phrase. In (29b) and (30b), the nominal restriction occurs in the left periphery as a topic, and the measuring quantifier appears adjacent to the verbal predicate. Note however, that in (29b) the measuring quantifier is followed by a classifier. The presence and the shape of a classifier depend on the presence and the nature of the following noun. (29b) shows that the topicalized nominal in (30b) is somehow present inside the measure phrase. For present purposes, we can think of this presence as a copy of the topicalized noun.

(29) a.

[Gakusei san-nin]-ga ie-ni kaet-ta [student three-CL]-nom home-to went 'Three students went home' b.

Gakusei-ga ie-ni san-nin kaet-ta student-nom home-to three-CL went 'Three students went home' (30) a.

[Mizu san-rittoru]-ga tukue-nouede kobore-ta [water three-liter]-nom table-in spilt 'Three liters of water were spilt on the table' b.

Mizu-ga tukue-nouede san-rittoru kobore-ta water-nom table-in three-liter spill-past Non-split MPs and split MPs give rise to a certain number of asymmetries: whereas nonsplit cases can occur in any context, split cases show certain restrictions with regard to the predicate. First, split measures cannot occur with once-only predicates.

(31) a.

[Gakusei san-nin]-ga kinoo Peter-o korosi-ta [student three-CL]-nom yesterday Peter-acc kill-past 'Three students killed Peter yesterday' b.

??Gakusei-ga kinoo san-nin Peter-o korosi-ta student-nom yesterday three-CL Peter-acc kill-past (32) a.

[Gakusei san-nin]-ga kinoo Peter-o tatai-ta [student three-CL]-nom yesterday Peter-acc beat-past 'Three students beat Peter yesterday' b.

Gakusei-ga kinoo san-nin Peter-o tatai-ta student-nom yesterday three-CL Peter-acc beat-past (31), as opposed to (32), contains a once-only predicate. In this context, the split MP is impossible. Split MPs, on the other hand, are possible in contexts like (32), which do not involve a once-only predicate.

Another property shown by split MPs is that they go with distributive predicates but not with collective ones.

(33) a.

[Otokonoko san-nin]-ga kinoo isu-o tukut-ta [guy three-CL]-nom yesterday chair-acc do-past 'Three guys made up chairs/a chair yesterday' √ collective √ distributive b.

Otokonoko-ga kinoo san-nin isu-o tukut-ta guy-nom yesterday three-CL chair-acc do-past * collective √ distributive (34) a.

[Tomodati huta-ri]-ga kyonen kekkonsi-ta [friend two-CL]-nom last-year married 'Two friends got married last year'

√ collective = a single couple √ distributive = two couples b.

Tomodati-ga kyonen huta-ri kekkonsi-ta friend-nom last-year two-CL marry-past * collective = a single couple √ distributive = two couples Nakanishi observes the same meaning effects for split Quantifier Phrases in German, Greek and Catalan. He concludes that split quantifiers quantify over both individuals (denoted by their nominal restriction) and events (provided by their verbal argument).

Quantification at a distance

Quantification at a Distance (QaD) refers to a construction that contains a Degree Quantifier that can alternatively have both adverbial and adnominal properties [START_REF] Obenauer | Une quantification non canonique: La quantification à distance[END_REF][START_REF] Doetjes | Quantifiers and selection: On the distribution of quantifying expressions in French, Dutch and English[END_REF]).

(35) Quantification at a Distance: a. Jean a lu beaucoup de livres Jean has read many of books 'Jean read a lot of books' b.

Jean a beaucoup lu de livres Jean has many read of books 'Jean read a lot of books' c.

*Jean a lu de livres Jean has read of books (35a) is an ordinary sentence with a quantified object. ( 35b) is a construction that involves a split quantifier phrase. The Degree Quantifier beaucoup 'much/many' is in an adverbial position, to the left of the past participle. The form of the direct object de livres corresponds to the one we find in the context of an adnominally used Degree Quantifier, as in (35a). The use of de NP is excluded in the absence of the quantifier, as shown in (35c), suggesting that there is some relation between the Degree Quantifier and the de NP in QaD constructions.

As in the case of split MP in Japanese, QaD is predicate-sensitive. It is impossible with once-only predicates (example from [START_REF] Doetjes | Quantifiers and selection: On the distribution of quantifying expressions in French, Dutch and English[END_REF]):

(36) a.

*En soulevant le couvercle il a beaucoup trouvé de pièces d'or lifting the lid he has a-lot found of coins of-gold b.

En soulevant le couvercle il a trouvé beaucoup de pièces d'or lifting the lid he has found a-lot of coins of-gold 'Lifting the lid he found a lot of gold coins' And it must have a distributive interpretation (Obenauer 1983: 83):

(37) a.

La délégué a salué beaucoup de militants the delegate has salute a-lot of militants 'The delegate greeted many militants'

√ collective √ distributive b.
La délégué a beaucoup salué de militants the delegate has a-lot salute of militants 'The delegate greeted many militants' * collective √ distributive

Differences between Split MPs / QaD and Basque Non-agreeing Quantifiers

Despite the common features of the Basque agreement alternation and the split quantification cases, the Basque non-agreeing quantifiers show important differences with regard to both split MPs and Quantification at a Distance. First, Basque non-agreeing quantifiers differ from Split-MP cases in that cardinal quantifiers in Basque do not show the same alternation: cardinals always agree in number in Basque (cf. (3)).

(38) Hiru lagun etorri dira/*da three friend come aux-pl/*aux-sg 'Three friends came'

Second, non-agreeing quantifiers differ from QaD, which only affects incremental themes [START_REF] Tenny | Aspectual role and the syntax-semantics interface[END_REF], in that the phenomenon extends to all arguments of the verb: transitive subjects (39a), indirect objects (39b) and objects (39c).

(39) a.

Azkenean gazte askok altxatu behar izan zuen harria finally young a lot of-erg lift must have aux-sg stone-D 'Ultimately, many youngsters had to lift the stone' b.

Ugazabak langile askori eskatu dio aparteko orduak egiteko boss-erg worker a lot of-dat ask aux-sg extra hours do 'The boss asked a lot of workers to work overtime' c.

Mirenek liburu asko ikusi du liburutegian Miren-erg book a lot of see aux-sg library-in 'Miren has seen a lot of books in the library' Finally, non-agreeing quantifiers differ from both split MPs and Quantification at a Distance cases in that the quantifiers are not adjuncts (or adverbs), unlike the floated quantifiers in Split-MPs and Quantification at a Distance (see [START_REF] Doetjes | Quantifiers and selection: On the distribution of quantifying expressions in French, Dutch and English[END_REF][START_REF] Nakanishi | Measurement in the nominal and the verbal domain[END_REF] for arguments in this regard). The sequence [Noun Phrase+Non-agreeing Quantifier] behaves as a constituent for intents and all purposes. Non-agreeing quantifier phrases show morphological Case, and they condition the selection of the appropriate auxiliary (which varies depending on the intransitive, transitive or ditransitive status of the VP). Syntactically, the sequence behaves as a constituent, and splitting its terms is not possible under any circumstance. ( 40 *Ikasle gaur etorri da asko student today came aux-sg many A 'floating' or adverbial approach to the Basque cases therefore does not seem appropriate. We explore the possibility that predicate sensitivity in Basque is the result of a semantic mapping process affecting Measure Phrases in any position (see section 7).

A note on event-related readings

The predicate-sensitivity of non-agreeing quantifiers in Basque led us to conclude that their measure function applies not only to their nominal restriction, but also to the verbal predicate. That is, non-agreeing quantifiers in Basque quantify over both individuals and events. (41a, b), repeated below, give an intuitive paraphrase of the meaning we have in mind:

(41) a.

Ikasle asko etorri da gaur student many come is today 'Many students came today' b.

Ikasle etorrera asko egon da gaur student come-Nom many been is today 'There has been a lot of student-coming today'

The paraphrase in (41b) is reminiscent of what [START_REF] Krifka | Four thousands ships passed through the lock: Object-induced measure functions on events[END_REF] has called 'event-related readings'. Event-related readings are illustrated in (42), from [START_REF] Krifka | Four thousands ships passed through the lock: Object-induced measure functions on events[END_REF]:

(42) Four thousand ships passed through the lock last year

The sentence in ( 42) is ambiguous between two different readings: in the first one, the number of different ships that passed through the lock last year amounts to four thousand. In the other reading, it is the number of passing events that amounts to four thousand. The latter reading is still true even if less than four thousand different ships did the passing, and this is the case if there are ships that have passed through the lock more than one time a year. It is natural to ask whether the Basque agreement alternation is related to the ambiguity above. The analogy would go in the following sense: we have shown that non-agreeing quantifiers are predicate-sensitive, and this follows if non-agreeing quantifiers quantify over events. We would therefore expect that event-related readings would correspond only to those cases where weak quantifiers do not agree in number with the verb. This is not the case: first, cardinal quantifiers do not enter the alternation (they always agree in plural). On the other hand, agreeing cardinal quantifiers give rise to event-related readings, as shown in ( 43).

(43) 4000 itsasontzi sartu dira aurten kanalean 4000 ship got-into aux-pl this-year lock-in '4000 thousand different ships entered the lock today' or 'There have been 4000 thousand events of ship-passing this year' This is the case for the rest of the agreeing weak quantifiers: all of them allow event-related readings:

(44) Untzi asko sartu dira gaur kanalean ship many got-into aux-pl today lock-in 'Many ships got into the lock today' or 'There have been many events of ship-passing through the lock today'

The ambiguity therefore targets in the same way agreeing and non-agreeing quantifiers, and cannot be linked to non-agreeing cases. Summarizing: we have seen the properties of non-agreeing quantifiers in Basque and its crosslinguistic connections. We have also seen that there are some clear and important differences between the Basque cases and the crosslinguistic connections (e.g. Split MPs in Japanese or German, French QaD). In what follows, first, we provide a semantic basis of the predicate sensitivity shown by Basque non-agreeing quantifiers ( §7); and second, we propose a new syntactic analysis for nominal expressions in general and for Basque non-agreeing quantifiers in particular ( §8).

7

A semantic approach to predicate sensitivity

Monotonicity in the nominal domain

It is known that measures (in general) show some semantic restrictions on the nominal expression:

(45) a. three litres of wine b.

*three degrees of wine According to [START_REF] Schwarzschild | The grammar of measurement[END_REF][START_REF] Schwarzschild | The role of measurement in the syntax of noun phrases[END_REF] the relation between measure functions -volume in (45a) and temperature in (45b)-and measured nouns is not uniform and in order to create grammatical structures the measure function must be monotonic with respect to the noun it combines with.

(46) a measure function µ is monotonic relative to domain I iff: (i) there are two individuals x, y in I such that x is a proper subpart of y, and (ii) µ(x) < µ(y)

As expressed in (46), being monotonic for a measure function means that it tracks the part-whole structure of the denotation of the noun. 8 A common way to represent that the denotations of nouns have part-whole structures is by means of a lattice structure [START_REF] Link | The Logical Analysis of Plurals and Mass Terms: A Lattice Theoretical Approach[END_REF].

(

) x∪ 1 y∪ 1 z x∪ 1 y x∪ 1 z y∪ 1 z x y z 47
Now, Schwarzschild argues that if we assume this to be the structure of the denotation of a noun it is possible to explain the contrast in (45). The measure function Volume in (45a) is monotonic with respect to the noun wine because if a quantity of wine has a certain volume, then every proper subpart of it will have a lower volume, and superparts of it will have larger volumes. On the other hand, the measure function temperature in (45b) is non-monotonic with respect to the noun wine because if the wine has a certain temperature, it is not necessarily true that proper subparts of it will have a lower temperature and that superparts of it will have a higher temperature.

Monotonicity in the verbal domain

Basque non-agreeing quantifiers do not show only semantic restrictions on the nominal domain: as we have already seen, they also show certain restrictions on the verbal domain, i.e. they are 'predicate sensitive'. Those constraints must at least include the impossibility of combining with (i) collective predicates (cf. section 4.2); and (ii) once-only predicates (cf. section 4.3).

In order to account for these restrictions, we adopt the idea that predicates (as is the case for nouns) can also be represented by a part-whole structure [START_REF] Nakanishi | Domains of measurement: Formal properties of non-split/split quantifier constructions[END_REF][START_REF] Nakanishi | Measurement in the nominal and the verbal domain[END_REF]. To do so, we assume that the denotation of a verb contains an event argument e [START_REF] Davidson | The logical form of action sentences[END_REF] and that what a verb denotes can be expressed by a lattice of events, as in (48) (see [START_REF] Landman | Events and Plurality[END_REF].

(48) e 1 ∪ E e 2 ∪ E e 3 e 1 ∪ E e 2 e 1 ∪ E e 3 e 2 ∪ E e 3
e 1 e 2 e 3

Then, the measure function that applies to the VP will have to be monotonic with respect to the part-whole structure denoted by this VP.

(49) a measure function µ is monotonic relative to domain E of events iff:

(i) there are two events e 1 , e 2 in E such that e 1 is a proper subpart of e 2 , and (ii) µ(e 1 ) < µ(e 2 )

The monotonicity constraint in the verbal domain can explain why non-agreeing quantifiers cannot combine with once-only predicates, and with collective predicates (cf. section 4). Note that (i) once-only predicates do not denote part-whole structures since they make reference to a single event and something like break the sand castle will not have proper subparts of breaking the sand castle; (ii) collective predicates, as opposed to distributive ones, denote a single event, and again, there will be no part-whole structure of events. Now, if this is the case, non-agreeing quantifiers (being measure functions) will not be able to apply to these predicates in a monotonic fashion. This is the reason why non-agreeing quantifiers will only be able to combine with predicates that denote a non-trivial part-whole structure. Agreeing Basque weak quantifiers (which we argue not to be measure functions, see section 8), on the other hand, can combine with any predicate because they show no restriction on the verbal domain and do not have to apply to predicates monotonically.

Homomorphism

As we just mentioned (cf. also section 6), non-agreeing quantifiers quantify over both nouns and verbs, but how can this property be explained? One possibility is to create a homomorphism function between individuals and events allowing measures to measure both. This homomorphism function is based on [START_REF] Krifka | Nominal reference, temporal constitution, and quantification in event semantics[END_REF], where he argues that temporal adverbials like for an hour in John slept for an hour cannot directly measure the sleeping event because events have no measureable temporal extent. Instead, he argues that for an hour indirectly measures the sleeping event by measuring its run-time, that is, he proposes a homomorphism function from events E to event run-times T. What a homomorphism function does is preserve some structural relation defined on its domain in a similar relation defined in its range, as in h(e 1 ∪ E e 2 ) = h(e 1 )∪ T h(e 2 ). Krifka claims that, given a measure function for run-times and a homomorphism function from E to T, it is possible to build a derived measure function which will be used for a domain different from the original domain of application (i.e. a measure function that is for run-times but is used to measure events). In (50), the measure function for events µ' is defined by µ and h: for all events, the amount of the event e measured by µ' in E equals the amount of h(e) measured by µ in T.

(50) ∀e [ µ'(e) = µ(h(e)) ] (Krifka 1989: 97) [START_REF] Nakanishi | Domains of measurement: Formal properties of non-split/split quantifier constructions[END_REF][START_REF] Nakanishi | Measurement in the nominal and the verbal domain[END_REF] extends the homomorphism analysis proposed by Krifka for events to split-MP (cf. section 6.1). What we do here is extend somewhat Nakanishi's analysis to Basque non-agreeing quantifiers, which we argue are conceptually measures. Nakanishi argues that there is a homomorphism function from events in E denoted by the VP to individuals in I denoted by the NP. Then, given a measure function for individuals and a homomorphism function from E to I, it is possible to derive a measure function µ' for events. If we take this proposal as correct, it would follow from there that Basque non-agreeing quantifiers could measure both individuals and events.

In (52), a measure function applies to individuals mapped from events by a homomorphism function h. Following (50), the derived measure function µ'(e) in ( 51) is equal to µ(h(e)) in (52) (a measure function applying to individuals mapped from events). By mapping events to individuals and measuring the range of that mapping, Basque non-agreeing quantifiers will be able to measure at the same time individuals (since µ applies to the output of h(e)) and events (since the derived µ' applies to e). In this way, non-agreeing quantifiers indirectly measure events by measuring individuals. 9 This analysis captures the observation that a non-agreeing quantifier operates both on the VP denotation and on the denotation of the host NP, measuring individuals.

(51) A measure function associated with non-agreeing quantifiers

E

µ' e µ'(e) (52) A measure function associated with non-agreeing quantifiers

E h I µ e h(e) µ(h(e))
If this is correct, non-agreeing quantifiers will have to be monotonic relative not only to the partwhole structure of the VP, but to the part-whole structure of a nominal domain mapped from a verbal domain. The incompatibility of non-agreeing quantifiers with once-only predicates can be explained as before: these predicates have no part-whole event structure and as a consequence there will be no homomorphism function that can be applied to the domain of events. Now, we also know that non-agreeing quantifiers force distributive readings (see section 3).

(53) Mutil askok mahai bat egin zuen boy a lot of-erg table one make aux.sg 'A lot of boys made a table '  * collective √ distributive

In order to obtain this distributive reading let us suppose that a verb like make a table can be pluralized and can form a lattice of make-a-table events, if so, there can be a homomorphism from the event lattice to a lattice of boys (individuals) (cf. [START_REF] Nakanishi | Domains of measurement: Formal properties of non-split/split quantifier constructions[END_REF][START_REF] Nakanishi | Measurement in the nominal and the verbal domain[END_REF]. Note that a measure function can apply monotonically to the range of the homomorphism function, that is, the lattice of boys, because the homomorphism function preserves the part-whole structure of the lattice of events.

( Thus, in the distributive reading the non-agreeing quantifier in (53) measures events as many (assuming that e 1 ∪ E e 2 ∪ E e 3 are many events), and this is mapped into the individuals x∪ 1 y∪ 1 z. The individual x∪ 1 y∪ 1 z consists of x, y, z, each of whom will be taken to be an agent of an atomic make-a-table event e 1 , e 2 , e 3 . On the other hand, the collective reading is ungrammatical due to the fact that there would only be a single make-a-table event e and the boys would also form a single agent. Then, there will be no possibility of applying a measure function monotonically because the single event will have no part-whole structure. As we showed in section 5, the quantifiers entering into the agreement alternation seem to operate across a large class of domains. The set of domains that the relevant quantifiers operate on includes plurals nouns, with and without agreement:

(56) Plural agreement: a.

Ikasle asko etorri dira gaur student many come are today 'Many students came today'

No agreement: b.

Ikasle asko etorri da gaur student many come is today 'Many students came today'

It also includes mass nouns:

(57) Jonek garagardo asko edan du gaur Jon-erg beer many drunk aux today 'Jon drank a lot of beer today' And it also extends to the verbal domain. They can be used as adverbial quantifiers:

(58) Jonek asko dantzatu du Jon-erg many danced aux-sg 'Jon danced a lot'

In the analysis that we propose next, we put this adverbial use aside and concentrate on the nominal uses of these vague quantifiers.

Our analysis builds on [START_REF] Borer | name only[END_REF] where it is argued that all nouns are unspecified for any properties (including the mass/count property) and that as a result of the absence of any grammatical specification and unless more syntactic structure is added, nouns denote masses (as the default case). In order to interact with the count system the denotations of nouns need to be portioned-out. This portioning-out function is realized by means of a classifier, but classifiers, Borer argues, are not exclusive to languages like Chinese (which possess a complex classifier system) but can also be found in other languages. In English, for example, what accomplishes the portioning-out function will be the plural marker -s, which Borer takes to play exactly the same role as Chinese classifiers.

With all this in mind, the syntactic structure proposed by Borer for nominals is the one in (59): first we have an NP (which will be mass by default), above the NP we have the Classifier Phrase (ClP) where the portioning-out function takes place, dominating ClP we have the Quantity Phrase or Number Phrase (NumP) which is responsible for the assignment of quantity to stuff (i.e. masses) or to divisions of it (i.e. where the counting occurs), and lastly, the highest projection is the DP projection.

(59) [ DP [ NumberP [ ClassifierP [ NP ]]]] According to Borer, both ClP and NumP may be missing from the structure. When the ClP is absent, the noun is interpreted as mass. This is basically what we have in (60) where we provide an example of the structure of a mass noun combined with a weak quantifier. So, we start with the NP money and since we want this NP be interpreted as a mass term, there will be no portioning-out function, that is, there will be no ClP present in the structure. Then, in order to quantify the stuff denoted by the NP money the NumP must be present and this is where the English quantifier much is placed. The same applies to the Chinese example shenme qian 'much money'. In (61) we have an example of a count term (combined with a weak quantifier); just because we want to interpret the noun as count, more structure than that in (60) will be needed. As was the case in (60) with the noun money, we start with an NP person which is taken to be a mass term by default. However, in order to interact with the count system the NP needs to be portioned-out, i.e. we need a ClP present in the structure, and this portioning-out function is fulfilled by plural inflection -s in English. Once the classifier has divided the stuff, the portionedout stuff can be counted, and this is exactly what the quantifiers many or three (and their Chinese counterparts) do in NumP position.

However, Borer's analysis faces a number of problems when we consider Basque data: it would make no distinction between agreeing and non-agreeing Basque quantifiers because they would both appear in NumP position (despite the clear and important differences existing between them). Furthermore, recall that among the agreeing Basque quantifiers we have numerals, and numerals always agree with the predicate in number as shown in (62), in opposition to what happens with vague quantifiers in Basque.

(62) Hiru ikasle berandu iritsi dira/*da three student late arrive aux.pl/aux 'Three students arrived late' Thus, the same syntactic position, i.e. NumP, would be home for elements with very different properties: agreeing quantifiers, non-agreeing quantifiers and numerals.

We think that -taking into account the differences between agreeing and non-agreeing quantifiers in Basque-we have evidence enough to conclude that agreeing quantifiers are counting quantifiers while non-agreeing quantifiers are measures (cf. section 5). What we propose then is that measures appear in a different syntactic position and that they head their own functional projection in the expanded structure of the Noun Phrase: the Measure Phrase (MP). As expressed in (63), the MP is placed just in-between the ClP (where division occurs) and the NumP (where counting occurs). Now, the distinction we make between the measuring field (in MP position) and the counting field (in NumP position) affects referentiality; in fact, we think that agreement and reference (i.e. establishing discourse variables and permitting enumeration) become relevant upon reaching NumP (and by extension, DP) -not before that position, i.e. not at MP. That referentiality comes once you get to NumP is borne out by anaphora cases as well as by the referential possibilities that agreeing and non-agreeing quantifiers show. We provide anaphora cases in (64): agreeing quantifiers create grammatical sentences and can be antecedent to anaphora (in (64a)) -which is not allowed for non-agreeing quantifiers, as the ungrammaticality of example (64b) clearly shows.

(64) a.

Ikasle In (65a), we see that agreeing Basque quantifiers allow the enumeration of individuals, i.e. it is possible to make reference to the members of the set we are talking about. The enumeration of individuals denoted by the NP combined with non-agreeing quantifiers is not possible, (65b).

(65) a.

Politikari askok, alegia A, B, C, D, ez dute lotsarik iritziz aldatzeko politician many-erg that-is neg aux.pl shame opinion change-fut 'Many politicians, to name A, B, C, D, are not ashamed of changing their opinion' b.

*Politikari askok, alegia A, B, C, D, ez du lotsarik iritziz aldatzeko politician a lot of-erg that-is neg aux.sg shame opinion change-fut 'A lot of politicians, to name A, B, C, D, are not ashamed of changing his/her opinion'

Assuming the structure we propose in (63) as correct, let us see now how the different uses of a vague quantifier like asko 'many/much' would fit in this structure. 10 In combination with mass terms, the structure will be the one in (66), i.e. a full DP. The noun garagardo 'beer' in (66) will be interpreted as a mass term due to the fact that there is no ClP in the structure, and hence, no portioning-out of the stuff. Above NP we will have MP (the position where the vague quantifier asko will appear in this case), its function being that of measuring the quantity of beer. Above all, we will have the DP projection. We saw in the initial sections that non-agreeing quantifiers need the NP they combine with to have atomic structure (cf. sections 2 and 3). It follows from there that non-agreeing quantifiers do not measure masses and therefore the portioning-out function is needed; in other words, ClP must be present in the structure. We assume that there is a covert classifier head in Basque (represented as ∅ in ( 67)) that portions-out stuff. 12 According to Borer, once you portion-out stuff there is no other possibility but to ennumerate it by means of a counter (numerals, quantifiers, etc.) which would appear in the NumP position. Non-agreeing Basque quantifiers show that this is not necessarily so and that it is possible to not be in NumP position and still need the stuff to be portioned-out in order to measure it. Furthermore, recall that we argue that it is upon reaching NumP position that referentiality and agreement appear, and non-agreeing Basque quantifiers do not show any of these properties (see examples (64-65)). Thus, the structure we propose for non-agreeing quantifiers is the one in (67): first we have the noun ikasle 'student' which enters the structure as a default mass term; it must be portioned-out in order to combine with non-agreeing quantifiers (which do not measure masses) which will be placed in MP position. It is exactly at the functional projection MP that the structure stops, going no higher than that (i.e. neither NumP nor DP will project). 13 The structure we propose for non-agreeing quantifiers in (67) is the one that allows the application of the homomorphism function permitting these elements to measure individuals denoted by the NP and events denoted by the VP. Finally, agreeing Basque quantifiers are considered simple counters and as a consequence they will appear in NumP. Of course, these quantifiers quantify over portioned-out stuff and the presence of ClP with a covert classifier head will also be necessary in these cases. Thus, first in (68) we have the noun ikasle 'student' which, as in (67), enters the structure as a default mass term; it must be portioned-out in order to combine with the counting system, hence ClP is necessary. Above the ClP we will have NumP where the agreeing quantifier appears assigning quantity to the portioned-out stuff. And lastly, the DP projection.

Conclusions

In this paper we have shown that non-agreeing quantifiers in Basque are conceptually measures. Furthermore, based on the differences between agreeing and non-agreeing quantifiers and observing that the latter do not behave as counters (i.e. they can not appear in NumP position) we have proposed a new syntactic structure for NPs (building on [START_REF] Borer | name only[END_REF] where measures head their own functional projection. This functional projection is placed between the Classifier Phrase and the Number Phrase. We have also shown that non-agreeing quantifiers are sensitive to the nature of the predicates they associate with and that Measure Phrases seem to measure both individuals and events, as long as the latter denote non-trivial part-whole structures. The predicate sensitivity of measuring quantifiers has been explained using the monotonicity constraint as expressed in [START_REF] Schwarzschild | The grammar of measurement[END_REF] and a homomorphism function [START_REF] Krifka | Nominal reference, temporal constitution, and quantification in event semantics[END_REF][START_REF] Nakanishi | Domains of measurement: Formal properties of non-split/split quantifier constructions[END_REF][START_REF] Nakanishi | Measurement in the nominal and the verbal domain[END_REF].

(

  60) Masses: [ DP [ NumberP much [ ClassifierP [ NP money]]]] (English) [ DP [ NumberP shenme [ ClassifierP [ NP qian]]]] (Chinese)

(

  61) Counts: [ DP [ NumberP many/three [ ClassifierP -s [ NP person]]]] (English) [ DP [ NumberP san [ ClassifierP ge [ NP ren]]]]

(

  63) [DP [Number Phrase [Measure Phrase [Classifier Phrase [NP ]]]]

  11

( 66 )

 66 Mass nouns: garagardo asko beer much [ DP [ MP asko [ NP garagardo]]]

( 67 )

 67 Non-agreeing Qs: ikasle asko [-agr] student a lot of [ MP asko [ ClP ∅ [ NP ikasle]]] ⇒ homomorphism

  NumP asko [ ClP ∅ [ NP ikasle]]]

3 Are non-agreeing quantifiers mass?

	(11) Mass:
	a.	Ardoa edan dut
		wine-D drunk aux
		'I drank wine'
	Plural count:
	b.	Ardoak edan ditut
		wine-D-pl drunk aux-pl
		'I drank wines'
	(12) Plural count:
	a.	Ikasle asko ikusi ditut
		student many seen aux-pl
		'I have seen many students'
	Mass:
	b.	Ikasle asko ikusi dut
		student many seen aux
		'I have seen much student'
		For argument's sake, if
	we were to reformulate the Academy's proposal slightly, it could be stated as saying that number
	morphology coerces masses into counts (11), whereas absence of number morphology coerces
	count nouns into masses (12):

  And it extends also to the verbal domain. Simple vague quantifiers like asko 'much/many', gutxi 'few/little', ugari 'abundant', gehiegi 'too much' can be used as adverbial quantifiers:

	Plural agreement:
	a.	Ikasle asko etorri dira gaur
		student many come are today
		'Many students came today'
	No agreement:
	b.	Ikasle asko etorri da gaur
		student a lot of come is today
		'A lot of students came today'
	Mass nouns:
	(26) Jonek garagardo asko edan du gaur
	Jon-erg beer	much drunk aux today
	'Jon drank a lot of beer today'
	(27) Jonek asko dantzatu du
	Jon-erg much danced aux.sg
	'Jon danced a lot'
	In this sense, vague weak quantifiers in Basque correspond to what

  ) e 1 ∪ E e 2 ∪ E e 3

			h		x∪ 1 y∪ 1 z
	e 1 ∪ E e 2	e 1 ∪ E e 3	e 2 ∪ E e 3	x∪ 1 y	x∪ 1 z	y∪ 1 z
	e 1	e 2	e 3	x	y	z

  i asko presaka etorri dira, student many hurry-in come aux.pl eta _ i mahaia altxatu ondoren _ i alde egin dute and table lift after go do aux.pl 'Many students came in a hurry, and after lifting the table they left.' b. *Ikasle i asko presaka etorri da, student a lot of hurry-in come aux.sg eta _ i mahaia altxatu ondoren _ i alde egin du and table lift after go do aux.sg 'A lot of students came in a hurry, and after lifting the table he/she left.'

The research conducing to this paper has benefited from the Basque Government projects GIC07/144-IT-210-07 and Hm-2008-1-10, from the project FR2559 Fèderation Typologie et

Universaux Linguistiques, from the project TSABL (ANR-07-CORP-033) from ANR, from the project FFI2008-00240 from MCE, from the project (MICINN)/FFI2011-29218 from MCE, from the project FFI2001-26906 from MCE as well as from the project ISQI from ANR. Thanks also to the two anonymous reviewers for useful comments and suggestions. Usual disclaimers apply. 2 The reader is referred to Artiagoitia (2000, this volume), or [START_REF] Etxeberria | Quantification and domain restriction in Basque[END_REF]Etxeberria ( , 2010, in prep) , in prep) for possible analyses of the various readings the Basque definite article can force. 3 The phenomenon extends to all arguments of the verb: transitive subjects (ergative case), indirect objects (dative case), and direct objects (absolutive case); cf. section 6.3 for examples. 4 Right now, we are unable to create a parallelism between so-called vague quantifiers in Basque and other crosslinguistic constructions. It would seem that there exist similar constructions in Spanish and Catalan where some weak quantifiers (e.g. mucho 'much' vs. muchos 'many' can both combine with 'apparent' count terms like estudiante 'student') can show agreement alternation with the verbal predicate, however, a more fine-grained study is needed in order to confirm whether we would be talking about the same phenomenon or not; cf. Etxeberria & Etxepare (in prep). 5 For measure expressions and their syntax in Basque, cf. Etxeberria & Etxepare (in prep) and Goenaga (2008, this volume). 6 Despite the fact that non-agreeing quantifiers lack number features and show no agreement with the verbal predicate (i.e. the inflected verb shows default third person singular agreement), we will continue using 'aux-sg' in the glosses for ease of exposition. 7 Russian also possesses nominals that show agreement alternation with the verb. These nominals (small nominals in [START_REF] Pereltsvaig | Small nominals[END_REF] V e`tom fil'me igralo [pjat' izvestnyx akterov]. in this film played.neuter five famous actors 'Five famous actors played in this film.' One difference between the non-agreeing Basque cases and the Russian cases is that Russian numerals can also enter into agreement alternation, something that is not allowed in Basque. For space reason this case will not be considered in this paper. 8 An alternative (and previous) formulation to the monotonicity constraint is the one proposed by [START_REF] Krifka | Nominal reference, temporal constitution, and quantification in event semantics[END_REF] where he argues that measure functions must be extensive with respect to the noun. One of the requirements for extensivity is that the measure function be additive.

(i) µ is an extensive measure function for a given part structure iff: µ is additive: If ¬x⊗y, then µ(x⊕y) = µ(x) + µ(y) [The sum of the measure of non-overlapping elements is the measure of their sum] 9 The homomorphism function need not be from events to individuals and could also be applied the other way around, that is, from individuals to events. This is actually what [START_REF] Filip | Measures and indefinites[END_REF] does when analyzing the Russian verbal prefix na-in its cumulative sense of approximately a relatively large quantity (of) and the attenuative/delimitative po-. It could be the case that the homomorphism function needed by non-agreeing Basque quantifiers to measure both individuals and events is implemented by homomorphically mapping the part-whole structure denoted by the NP to the part-whole structure denoted by the VP. Note in fact that nonagreeing quantifiers first apply to the NP they combine with and are not to be treated as adverbials (cf. section 6). 10 Note that Basque is a head final language. However, for the structures that we will be proposing in this paper we leave directionality aside. 11 A reviewer wonders why mass terms combined with quantifiers such as asko 'much' create full DPs, in contrast with what happens with non-agreeing quantifiers. The reason why this is so is related to referentiality. 12 A question that comes to our mind is the following: Does Basque posses a plural marker that can portion-out stuff just like the plural marker -s does in English? The answer to the question is 'yes' and 'no'. 'Yes' because Basque possesses a plural marker, realized morphologically as -k; and 'no' because this plural marker is categorically and phonologically dependent on D, that is to say, the plural marker never appears in Basque unless D is present (see [START_REF] Etxeberria | Quantification and domain restriction in Basque[END_REF]Etxeberria , 2010, in prep), in prep). Furthermore, note that the plural marker -k does not appear with the weak quantifiers we are considering in this paper. If -k were behaving as a classifier (i.e. portioning-out stuff) it should have appeared with both agreeing and non-agreeing quantifiers, but it does not as shown by the examples used in the whole paper. 13 Borer (2005: ch.4, ch.8) argues that measure expressions head a quantity phrase (NumP) and that the DP fails to project resulting in the absence of referential reading. Similar claims have been made by [START_REF] Ritter | Two Functional Categories in Noun Phrases: Evidence from Modern Hebrew[END_REF] or [START_REF] Li | Argument determiner phrases and number phrases[END_REF]. See also [START_REF] Pereltsvaig | Small nominals[END_REF] where what she calls Small Nominals are argued not to project a whole DP (but a QP (NumP)) and to lack individual reference. Non-agreeing Basque quantifiers would show that reference appears with the presence of overt number morphology (agreement with the inflected verb in this case) and upon reaching NumP, not below, i.e. not in MP. Note, however, that the presence of NumP makes the presence of DP obligatory (cf. fn. 8).


Vincent Padovani
email: padovani@pps.univ-paris-diderot.fr

Decidability of All Minimal Models (Revised Version -2012)

This unpublished note is an alternate, shorter (and hopefully more readable) proof of the decidability of all minimal models. The decidability follows a proof of the existence of a cellular term in each observational equivalence class of a minimal model.

The first proof I gave of the decidability of all minimal models [START_REF] Padovani | Decidability of All Minimal Models[END_REF] was far from being easy to understand. I was only an inexperienced student at the time, struggling to solve the problem of the decidability of Higher Order Matching. I was trying to generalise to order five my decidability result at order four ( [START_REF] Padovani | Filtrage d'Ordre Supérieur[END_REF], [START_REF] Padovani | Decidability of fourth-order matching[END_REF]) when I realized that every solution of an atomic matching problem (a problem whose right-members are constants of ground type) could be transformed into a cellular term (the so-called "transferring" terms in [START_REF] Padovani | Decidability of All Minimal Models[END_REF]), a term of very simplified structure. The decidability of atomic matching followed immediately from this key-result. A few months later, at the open-problem session of TLCA 1995, Ralph Loader pointed out that another immediate consequence of this decidability result was the existence of a computable selector for the observational equivalence classes of the minimal models of simply-typed lambda-calculus.

Because I was so immersed in the Matching Problem and wanted to prove the decidability of atomic matching at each order, it seemed very natural to prove the existence of cellular representatives by induction on the order of terms. Unfortunately this choice was probably the worst I could make, and resulted in a long, tedious and obfuscated proof. Two years later, using the same techniques, Ralph Loader proved the decidability of Unary PCF [START_REF] Loader | Unary PCF is decidable[END_REF]. Loader's proof was then drastically simplified by Manfred Schmidt-Schauß [START_REF] Schmidt-Schauss | Decidability of Behavioural Equivalence in Unary PCF[END_REF] who gave a clever, simple and beautiful algorithm to compute a selector for Unary PCFa fortiori for every minimal model. This is the point where I realized that something was probably wrong with my own proof : even if the decidability of all minimal models followed from the existence of cellular representatives at each order, the latter property was actually independant from the first, and clearly required a proof by induction on the length of terms.

This unpublished note -written a few years ago -presents a short and simple proof of the existence for each term of an observationally equivalent cellular term, followed by a proof of decidability of each minimal model. The proof considers only one ground type, but can be easily extended to finitely many ground types (if you feel it is really necessary, you can try to read [START_REF] Padovani | Decidability of All Minimal Models[END_REF], or even [START_REF] Padovani | Filtrage d'Ordre Supérieur[END_REF] if you can read French).

We consider the λ-calculus with a single ground type •, a typing à la Church, and finitely many constants of type •. All terms are assumed to be in η-long form. The notation λy 1 . . . y n .u is a shorthand for (λy 1 . . . (λy n u) . . .) and implies that u is of ground type.

Let t, t ′ be closed terms of the same type. We say that t and t ′ are observationally equivalent if and only if the following property holds:

• t, t ′ : • and t = β t ′ , or
• t, t ′ : A → B, and for every closed u : B, we have (t u) ≡ (t ′ u).

The following lemma is a well-known result:

Lemma 1 (Context Lemma) If u ≡ u ′ then for all t, (t u) ≡ (t ′ u).
The following property is false in a simply-typed λ-calculus dealing with higherorder constants, true if all constants are of ground type :

Proposition 1 (Stretching lemma) Let t = λy 1 . . . y n .M [u] be a closed term,
where M is a context with a hole of ground type. Then t is observationally equivalent to λy 1 . . .

y n .M [M [u]]. Proof. Indeed, M [M [u]] = M [x][M [u]/x], and for all closed t, M [M [u]][t/y] = M [t/y][x][M [u][t/y]/x], with either • M [t/y][x] = β x and M [M [u]][t/y] = β x[[M [u][t/y]/x] = M [u][t/y], or, • M [t/y][x] = β a and M [M [u]][t/y] = β a[M [u][t/y]/x] = a

Cells and cellular terms

Definition 1 A cell is a context of the form Σ[ ] 1 . . . [ ] K = (y (C 1 [ ] 1 . . . [ ] K ) . . . (C p [ ] 1 . . . [ ] K ))
where all [ ] i are holes of ground type and each

C i [ ] 1 . . . [ ] K is a context with no free variables.
By definition each cell contains a unique (head) occurrence of a unique free variable.

Definition 2 A term t is cellular if and only if it is of the following form:

• λy 1 . . . y n .a where a is a constant, or,

• λy 1 . . . y n .Σ [w 1 ] 1 . . . [w k ] K where:
-Σ is a cell whose free variable is amongst y 1 , . . . , y n , -each λy 1 . . . y n .w j is a cellular term. where each λy 1 . . . λy n u i is a cellular term.

Clearly every cellular (resp. semi-cellular) term is a closed term. Note that in the definition of a semi-cellular term, u i is not necessarily of ground typee.g. if u i = λx 1 . . . x p .t, then t may contain cells with head variables amongst y 1 . . . , y n , x 1 , . . . , x p .

The introduction of cells will simplify our proofs below, but the following alternate definition of a cellular term will probably be easier to grasp. Let t = λy 1 . . . y n .u where no variable is simultaneously free and bound in t. The term t is cellular if it is closed, and if for every subterm w = (y i v 1 . . . v n ) of u, the free variables of w are amongst y 1 , . . . , y n . In other words, if some variable z = y 1 . . . y n is bound in t, then no y i is allowed to occur between λz and an occurrence of z. For instance, the indentity

λy 1 y 2 .y 1 (λz.y 2 z) : ((• → •) → •) → ((• → •) → •)
is not cellular, whereas λy 1 y 2 .y 1 (λd.y 2 (y 1 λz.z))

is a cellular term... observationally equivalent to the first. The following proposition is easily proven by induction on the length of t:

Proposition 2 Let Σ be a cell whose free variable is amongst y 1 , . . . , y n . If

λy 1 . . . y n .M [Σ[w 1 ] . . . [w K ]] is cellular, (resp. semi-cellular) then λy 1 . . . y n .M [w k ] is cellular (resp. semi-cellular).
Note that for all w 1 , . . . , w

K , Σ[w 1 ] . . . [w K ] = Σ[x 1 ] . . . [x k ][w 1 /x 1 . . . w K /x K ]
λy 1 . . . y n .M [Σ[w 1 ] . . . [w k-1 ][N [Σ[v 1 ] . . . [v K ]][w k+1 ] . . . [w K ]]
is observationally equivalent to

λy 1 . . . y n .M [Σ[w 1 ] . . . [w k-1 ][N [v k ]][w k+1 ] . . . [w K ]]
(the names "stretching", "shrinking" were found by Thierry Joly around 1996).

Decidability of all minimal models

For any finite set of constants C, we write ≡ C the restriction of observational equivalence between closed terms whose constants belong to C.

Theorem 2 There exists a comptable function R such that for all types A, and for any finite set of constants of ground type C, R(A, C) is a finite list of terms containing a representative of each ≡ C -class.

Proof. Let A = A 1 . . . A n → •, where A i = B i 1 . . . B i pi → •. Let y 1 : A 1 , . . . , y n : A n . For each i ∈ [1 . . . n]
, let K i be any integer greater1 than the number of ≡ C -classes of type A i . For each j ∈ [1 . . . p i ], let W j i be a complete set of representatives for the pair (B i j , C ∪ {d i 1 . . . d i Ki }), where the d i k 's are fresh constants of type •. We let V i be the set of all terms of the form (y i w). where w ∈ Π pi j=1 W j i . We define R(A, C) as the least set of terms R satisfying: • λy.a ∈ R for all a ∈ C.

• if λy.w 1 , . . . , λy.w Ki ∈ R, and if v ∈ V i is not used in the construction of these terms, then

λy.v[w 1 /d i 1 ] . . . [w Ki /d i Ki ] ∈ R.
Clearly, R is a finite set of cellular terms. Let t be any closed term. We shall prove that there exists a term in R equivalent to t. By the preceding theorem, we can assume that t is cellular. We proceed by induction on the length of t.

Suppose

t = α λy 1 . . . y n .Σ[s 1 ] . . . [s L ]
where Σ is a cell of head variable y i . The number of holes appearing in the normal forms of all Σ[u i /y i ] where u i is a closed term, is bounded by K i . If [ ] l does not appear in these normal forms, then we can replace s l with an arbitrary constant, yielding a term of same class. As a consequence, we can assume that L ≤ K i . Then there exists in V i a term v Σ such that λy

i .Σ[d i 1 ] . . . [d i L ]
≡ λy i .v Σ By induction hypothesis there exists r 1 , . . . , r L such that λy.s 1 ≡ λy.r 1 ∈ R, . . . , λy.s L ≡ λy.r L ∈ R. Then:

t ≡ λy 1 . . . y n .v Σ [r 1 /d i 1 . . . r L /d i L ]
The conclusion follows from the fact that, as in proposition 3, we do not need to use v Σ in the construction of λy.r 1 , . . . λy.r L , that is to say,

λy.v Σ [r 1 /d i 1 . . . r l-1 /d i l-1 , M [v Σ [r ′ 1 /d i 1 . . . r ′ L /d i L ]]/d i l ,

From

  now on, by C[ ] 1 . . . [ ] K we mean a multiple-hole context whose holes are amongst [ ] 1 , . . . , [ ] K .

  r l+1 /d i l+1 , . . . , r L /d i L ]is equivalent toλy.v Σ [r 1 /d i 1 . . . r l-1 /d i l-1 , r ′ l /d i l , r l+1 /d i l+1 , . . . , r L /d i L ]Remark. Call hereditary cellular every cellular t such that for each celly(C 1 [ ] 1 . . . [ ] K ) . . . (C n [ ] 1 . . . [ ] K ) in t, each λx 1 . . . λx K C j [x 1 ] . . . [x K] is hereditary cellular. Note that all terms returned by the algorithm are hereditary cellular. It it not too difficult to prove that all terms returned by a restriction of Schmidt-Schauß' algorithm to a minimal model (see[START_REF] Loader | An algorithm for the minimal model[END_REF] for Loader's presentation of Schmidt-Schauß' algorithm) are also hereditary cellular.

If |R(B i j , C)| = k j then we can take K i = |C| (k 1 ×...×kp i ) .

Existence of cellular representatives

Lemma 2 Every semi-cellular term t is observationally equivalent to a cellular term.

Proof. By induction on the length of t. Assume t = λy 1 . . . λy n .u is in normal form, with u = (y i u 1 . . . u p ). Let M be the minimal context such that

• each t k is a term of ground type, of the form (y . . . ) with y ∈ {y 1 . . . y n }.

For each k, t k is a subterm of some u j , and the closure of this latter term is cellular. Hence, there exists a cell Σ k and terms w k 1 . . . w k L k such that:

Now, for each (k, l) let

By proposition 2, for all (k, l), λy 1 . . . y n .N k l is semi-cellular. By induction hypothesis, there exists a term u k l whose closure is cellular and equivalent to the closure of N k l . We define u ′ as

Clearly, t Corollary 1 For all types A, there exists a term t : A → A such that for all terms w : A, the normal form of (t w) is a cellular term equivalent to w.

Proof. Define t as the cellular equivalent of the η-long form of λx x.


Rousseau Sandra

Jutten Christian

Congedo Marco

DESIGNING SPATIAL FILTERS BASED ON NEUROSCIENCE THEORIES TO IMPROVE ERROR-RELATED POTENTIAL CLASSIFICATION

Keywords: BCI, single-trial classification, errorrelated potential, spatial filtering

In this paper we present an experiment enabling the occurrence of the error-related potential in high cognitive load conditions. We study the single-trial classification of the errorrelated potential and show that classification results can be improved using specific spatial filters designed with the aid of neurophysiological theories on the error-related potential.

INTRODUCTION

The error-related potential (ErrP) is an event-related potential (ERP) which is generated when a subject commits or observes the commitment of an error. It was first reported in 1991 by Falkenstein et al. [START_REF] Falkenstein | Effects of crossmodal divided attention on late ERP components.II. Error processing in choice reaction tasks[END_REF] and has been since the subject of growing interest. This potential is time-locked to the observation of the error and is mostly characterized by a negative deflection (Ne) [START_REF] Gentsch | Dissociable medial frontal negativities from a common monitoring system for self-and externally caused failure of goal achievement[END_REF], followed by a large positivity (Pe) [START_REF] Steinhauser | Performance monitoring and the causal attribution of errors[END_REF]. There exists different kind of ErrPs depending on the agent commiting the error. In BCIs we generally observes the interaction ErrP which occurs when a subject observe an external device commiting an error and the feedback ErrP which is observed when a suject commits an error but becomes aware of it only after an external feedback. Lately several authors have become interested in its integration in BCI systems as a control loop. The integration of the ErrP in BCIs involves two main operations: its single trial detection and the use of this information to on-line modify the system. Since its signal to noise ratio is very low, as any ERP, the ErrP can easily be seen by summing up several trials but is much less detectable on a single-trial basis. The single-trial detection of the ErrP is a crucial point for its integration in BCIs, thus learning its characteristics to design optimal filtering methods is a key point. In this paper we first present an experiment we designed to obtain ErrP data in high cognitive load conditions and study the occurrence of ErrP. Then we propose to apply spatial filtering methods to enhance ErrP signal to noise ratio in order to improve its single-trial detection. We present three different theories on ErrP origin and use them to design different types of filter. These filters are then applied to our data be-fore classification is performed. Filters are then compared in terms of their classification accuracy and the reliability of the corresponding theories is discussed.

DATA

Experiment

The experiment we designed involved a memory game where performance feedback was given after the subject answered questions. No specific reward was given and the experiment was designed so as to induce a high cognitive load. Thus the subject had to focus on other things than just the feedback. The experiment involved two sessions that lasted together approximately half an hour. Each session consisted of six blocks of six trials, for a total of 6×6×2 = 72 trials. Stimuli were presented on a computer screen in front of the subjects and consisted of digits displayed in square boxes. Nine square boxes were arranged in circle on the screen. Each trial started with the display of the score for 3000 ms followed by a fixation cross, which was also displayed for 3000 ms. Then the memorization sequence started, each memorization consisted in a random sequence of two to nine digits appearing sequentially in random positions, with each digit of the sequence randomly assigned to a different box for each sequence. Subjects were instructed to retain positions of all digits. At the end of the sequence the target digit (always contained in the previous sequence) was displayed and subjects had to click on the box where it had appeared. Once the subject had answered, the interface waited for 1500 ms in order to avoid any contamination of ErrP by beta rebound motor phenomena linked to mouse clicking [START_REF] Salmelin | Spatiotemporal characteristics of sensorimotor neuromagnetic rhythms related to thumb movement[END_REF], [START_REF] Pfurtscheller | Central beta rhythm during sensorimotor activities in man[END_REF]. Then, if the answer was correct, the chosen box background color turned into green ("correct" feedback), otherwise it turned into red ("error" feedback). Subjects were then asked to report if the feedback (error/correct) matched their expectation by a mouse click ("yes"/ "no"). Following the answer a random break of 1000 ms to 1500 ms preceded the beginning of the new trial (see Figure 1).

In order to keep the subjects motivated throughout the experience, the accumulated score was computed at the beginning of each trial. When subjects localized correctly the Fig. 1: Time course of one trial target digits their score increased, otherwise, it remained unchanged. The number of digits in the sequence was fixed within blocks (between two and nine) and updated, according to performance, at the beginning of each block. The first block started with four digits for all subjects. The number of digits in the sequences adaptively increased or decreased of one digit at each block, according to subjects performance. The number of digits was adapted with an algorithm tuned to allow about 20% of errors for all subjects, regardless the working memory ability and adapting to fatigue as well as other possible nuisance intervening during the experiment. Between the two sessions the screen was shut down to allow a rest break of 2 -3 minutes. The use of the adaptative algorithm allowed us to tune the level of the experiment according to each subject in order to have a similar cognitive load for every subject.

Participants

25 healthy volunteer subjects participated, 14 males and 11 females. Subjects were informed of the procedure before the experiment and filled an information form. All subjects were BCI-naifs at the time of the experiment and none of them reported neurological or psychiatric disorders in the past. Due to the presence of artifacts, four subjects were excluded from analysis. The age of participants ranged from 20 to 30 with a mean (standard deviation) of 24 (2.5). The mean error rate (standard deviation) was equal to 18 (4.6)% of the trials.

Acquisition and preprocessing

EEG recordings were made from 31 sensors using the extended 10/20 system. Both earlobes, digitally linked, were used as electrical reference. The ground sensor was positioned on the forehead. The impedance of each sensor was kept below 5kΩ. The EEG was band-pass filtered in the range 0.1-70 Hz and digitized at 500 Hz using the Mitsar 202 DC EEG acquisition system. Data were bandpass-filtered between 1-40 Hz using an order 4 Butterworth filter with linear phase response. Eye blinks were extracted using ICA (independent component analysis ( [START_REF] Comon | Handbook of Blind Source Separation: Independent component analysis and applications[END_REF])). One EOG source (or more when necessary) was suppressed for each subject. It was manually selected using both the temporal shape of the source and its topography.

Observation of the ErrP

In Figure 2 we plot the event-related potential averaged over subjects for correct trials and for error trials for one second post-stimulus (observation of the error). In this figure we see that for error trials the ERP is characterized by a sharp negativity followed by a small positivity which is consistent with previous reports ( [START_REF] Yeung | ERP correlates of feedback and reward processing in the presence and absence of response choice[END_REF]; [START_REF] Hajcak | It's worse than you thought: The feedback negativity and violations of reward prediction in gambling tasks[END_REF]). For correct trials, a negativity is also observed but with a much lower intensity. As we said before, after an error trial one can observe the occurrence of the error-related potential which is mostly characterized by a sharp negativity (ERN). However, another event-related potential also appears after correct trials called the correct-related negativity (CRN) [START_REF] Falkenstein | ERP components on reaction errors and their functional significance: A tutorial[END_REF], [START_REF] Luu | Medial frontal cortex in action monitoring[END_REF], [START_REF] Vidal | Error negativity on correct trials: A reexamination of available data[END_REF]. This potential is also characterized by a negativity occurring at the same time as the ERN (but less intense) and localized at the same electrodes. There exists different theories concerning the link between these two potentials and their neurophysiological origins. Here we present three possible theories: H1: The ERN and the CRN are two completely different phenomena which occur only in one of the two conditions (error or correct). This means that these two potentials are generated by different sources and could be separated by specific spatial filtering. These are two uncorrelated activities.

H2: The ERN and the CRN are generated by the same source which is simply modulated by the value of the outcome. Thus, the differentiation of the ERN and the CRN can only be done by observing the intensity of this source. This theory is supported by many studies which localized, using blind source separation techniques or fMRI, the source responsible for the ERN and the one for the CRN at the exact same location [START_REF] Roger | Rostral Cingulate Zone and correct response monitoring: ICA and source localization evidences for the unicity of correct-and error-negativities[END_REF], [START_REF] Vidal | Error negativity on correct trials: A reexamination of available data[END_REF].

H3: The ERN is the sum of two phenomena, one which is linked to the observation of a result or of a conflictual response (and which is common to the CRN), another one which is only linked to the occurrence of an error [START_REF] Ullsperger | Subprocesses of performance monitoring: A dissociation of error processing and response competition revealed by eventrelated fMRI and ERPs[END_REF]. Thus we have ERN = CRN + another potential. This means that these two potentials share common neural circuits but that in addition to these, other independent sources are activated only for error trials.

Using these different theories we can build different spatial filters. Here we will develop three different spatial filters corresponding to each of these theories. Theories will then be tested and compared using the classification rate we obtain using each filter.

Filtering methods

In order to optimize our classification results we have tried to spatially filter our data so as to improve the SNR. Filtering was done using xDAWN algorithm [START_REF] Rivet | P300 speller brain-computer interface: Enhancement of P300 evoked potential by spatial filters[END_REF], [START_REF] Rivet | xDAWN algorithm to enhance evoked potentials: Application to brain computer interface[END_REF]. In this algorithm we consider that the signal is the sum of one target evoked potential plus other possible superimposed evoked potentials (but non-target) and noise. Thus the signal can be written as:

X = D 1 A 1 + D 2 A 2 + N (1)
where X: represents the signal, D i : is a Toeplitz matrix whose first column entries are set to zero except for those that correspond to the stimuli of type i. A i : represents the responses synchronized with the stimuli of type i, N : represents the noise. Evoked responses A i are estimated as:

A 1 A 2 = (D T D) -1 D T X (2)
with

D = [D 1 , D 2 ].
The aim of xDAWN is then to find the filter U which maximizes the signal to signal plus noise ratio (SSNR):

U = argmax T r(U T 1 U ) T r(U T X U ) (3)
with 1 = (D 1 A 1 ) T (D 1 A 1 ) and X = X T X.
Here D 1 and D 2 will be defined differently according to the theories we use. H1: ERN = CRN . We have two types of stimuli: "errors" and "corrects". D1 will be constructed using the stimuli corresponding to erroneous responses and D2 will be constructed using the stimuli of correct responses. A 1 will be an estimation of the mean potential for error trials (ERN) and A 2 will be an estimation of the mean potential for correct trials (CRN).

H2: ERN = CRN . We have only one type of stimuli which corresponds to the reaction to an outcome of performance. D1 will be constructed using the stimuli corresponding to correct and erroneous responses. There will be no D2. A 1 will be an estimation of the mean potential for both error and correct trials.

H3: ERN = CRN + P 1 . We have two types of stimuli "errors" and "reaction to an outcome of performance". D1 will be constructed using the stimuli corresponding to erroneous responses. D2 will be constructed using the stimuli corresponding to correct and erroneous responses. A 2 will be an estimation of the mean potential for both error and correct trials. A 1 will be an estimation of the remaining mean potential after A 2 has been subtracted from error trials.

Moreover, here we want to classify both correct and error trials, both might be considered as target trials. Thus for all these theories two types of filters were designed, one to improve the SSNR of D 1 A 1 (F 1 ) and one to improve the SSNR of D 2 A 2 (F 2 ). This means that in equation (3) we will use 1 to design F 1 and 2 to design F 2 .

Preprocessing and classification

Data were first band-pass filtered between 1-10 Hz and then spatially filtered using the previously described methods. Data were then classified using Bayesian LDA classifier [START_REF] Mackay | Bayesian interpolation[END_REF]. Since xDAWN algorithm returns filters classified in descending order of performance we used the first two components as features for our classifier. In order to avoid over-learning data were subsampled at 32 Hz. Most studies on ErrP singletrial classification did not perform any spatial filtering but simply selected FCz and Cz signals (based on prior physiological knowledge) [START_REF] Artusi | Performance of a Simulated Adaptive BCI Based on Experimental Classification of Movement-Related and Error Potentials[END_REF], [START_REF] Ferrez | EEG-based braincomputer interaction: Improved accuracy by automatic single-trial error detection[END_REF]. Thus, classification was also performed on raw data (from electrodes FCz and Cz) in order to compare classification results obtained with and without spatial filtering. Classification was performed on 21 subjects. For each subject a leave-one method was used, which means that spatial filters and classifier were learned on the whole data except one and then tested on the remaining data. Classification was performed using F 1 filters, F 2 filters and both.

RESULTS

In figure 3 we plot the mean (and standard deviation) results of classification over the 21 subjects for error trials and correct trials using the different methods. First we can see that classification accuracy is higher for correct trials than for error trials, this is also mostly the case in other studies on the ErrP single-trial classification [START_REF] Ferrez | Simultaneous real-time detection of motor imagery and error-related potentials for improved BCI accuracy[END_REF], [START_REF] Seno | Online detection of P300 and error potentials in a BCI speller[END_REF], [START_REF] Goel | Cortical Current Density vs. surface EEG for Event-Related Potential-based Brain-Computer Interface[END_REF]. Moreover we get, for the reference method, an average classification accuracy of 67% for error trials and 70% for correct trials, which The height columns correspond to the different classification method used, the first one corresponds to the reference (no spatial filtering), then we have results for the first theory (H1) using filters for A 1 (F1), A 2 (F2) and both (F12), H2 corresponds to results for the second theory (since there is only target signal we have only one type of filter). The three last columns correspond to results for theory 3 (H3) using filters for A 1 (F1), A 2 (F2) and both (F12) is a little bit lower than most results obtained in other studies [START_REF] Chavarriaga | To err is human: Learning from error potentials in braincomputer interfaces[END_REF], [START_REF] Ferrez | You are wrong!automatic detection of interaction errors from brain waves[END_REF], [START_REF] Iturrate | Single trial recognition of error-related potentials during observation of robot operation[END_REF], but still very good in regard of the low number of trials available (72 trials with only 20% of errors). We can see that for error trials all filters seem to perform equivalently and provide results similar as those obtained with the reference method, i.e. without spatial filtering. Thus it seems that for detecting errors no spatial filtering is needed or at least that xDAWN filtering method is not optimal, still it does not deteriorate results. For the classification of correct trials it is clear that method H1 and H3 are the most performant ones. Moreover we can also see that the best classfication results are obtained using only the first filter (F1), i.e. when using the ERN as the target potential. H3-F1 slightly outperforms H1-F1 with an average classification accuracy of 81% for H3-F1 against 78% for H1-F1. Both methods clearly out-perform the reference method (70% of accuracy). A one-way ANOVA was performed on the classification results to assess the relevance of the observed effect (F = 8.21, p ≤ 0.01). Post-hoc multiple comparison tests showed that both H3-F1 and H1-F1 lead to significantly better results than the reference method. However no significant difference was found between H3-F1 and H1-F1. On the contrary one can see that H2 method gives the same kind of results (69% for H2) as the reference method. In figure 4 we plot the results for each subject for correct end error trials for method H1-F1, H3-F1 and reference. We can see that results for error trials are highly different from one subject to another and that depending on the subject each method can highly outperform the other ones. For correct trials the reference method leads to better results than H1-F1 for only 6 subjects out of 21 ones and than H3-F1 for only 3 subjects out of 21. Thus it seems that H3-F1 is a robust method. Moreover it has to be noted that when reference method outperforms one of the two other methods the difference is always very low (with a maximum difference of 5% only) while when H3-F1 or H1-F1 outperforms the reference method the difference might go up to 24%. When comparing H3-F1 and H1-F1 there does not seem to be any clear difference between both, indeed H1-F1 outperforms H3-F1 for 8 subjects while H3-F1 outperforms H1-F1 for nine subjects, thus there does not seem to be a method that is more reliable than the other.

In figure 5 we plot the topographic maps for four different subjects corresponding to the first filter component obtained using the different strategies. We can see that H3 gives more focused and more stable topographic maps. H2 gives highly variable maps which are not always consistent with the litterature on ErrP localization. H1 gives better results than H2 but maps are more spreaded and less stable than those obtained with H3 except for the fourth subject. These observations are consistent with the classification results.

DISCUSSION

In this paper we presented a high cognitive load experiment which allowed the generation of ErrP. We obtained satisfying classification results in regard of the very small number of trials available (72 trials) with no particular spatial filtering. In a second time we have proposed to develop spatial filters in order to improve this classification accuracy. The xDAWN method allows us to estimate spatial filters optimized for a given target stimulus on which other undesired stimuli might be superimposed. We developed three different spatial filters based on three different existing theories on the nature of the error-related potential and its link to the correct-related potential. These three theories lead to different estimations of the target stimulus and thus to different spatial filters. First, results showed that classification accuracy could be greatly improved using well defined spatial filters, at least for correct trials. This is very encouraging for improving ErrP single-trial detection since more specific filters could be constructed. Two methods clearly outperformed the reference method (which used no spatial filters but only the selection of relevant electrodes), improving classification results for the ErrP up to 10%. One interesting point is that in neurophysiological studies the most supported theory is the one corresponding to H2 i.e. the one corresponding to the fact that the ERN and the CRN reflect the same phenomenon while in our study it is the one which lead to the poorer results. H3 and H1 lead to similar results with H3 slightly outperforming H1, this can be explained by the fact that, in the case where H3 would be the right theory, H1 might still give us a slightly good estimate of A 1 . Indeed the difference between potentials estimated in H1 and H3 is just that in H3 we first subtract the average global potential (i.e. corresponding to both correct and error trials) before estimating A 1 . This average global potential can be seen as noise. Thus, H1 will simply lead to a noisy estimation of A 1 but, if this noise is not too high we will get similar results. This paper allowed us to enlight two important points, first that ErrP classification could be improved by spatial filtering and that neurophysiological theories could be used to design these filters and that, in return, these filters and their corresponding classification accuracies could bring information on the relevance of these theories. Further studies should be made in this sense by using other neurophysiological knowledges to develop even more optimized filters.

Fig. 2 :

 2 Fig. 2: Mean potential averaged over 21 subjects at electrode FCz. Blue thick line corresponds to error trials and red dashed line corresponds to correct trials

Fig. 3 :

 3 Fig. 3: Classification results averaged over the 21 subjects. (a) For error trials. (b) For correct trials. Blacks bars correspond to the standard deviation. The height columns correspond to the different classification method used, the first one corresponds to the reference (no spatial filtering), thenwe have results for the first theory (H1) using filters for A 1 (F1), A 2 (F2) and both (F12), H2 corresponds to results for the second theory (since there is only target signal we have only one type of filter). The three last columns correspond to results for theory 3 (H3) using filters for A 1 (F1), A 2 (F2) and both (F12)

Fig. 4 :

 4 Fig. 4: Classification results for the 21 subjects. (a) For error trials. (b) For correct trials. Black bars correspond to the reference method and dark gray bars to the H1-F1 method and light gray bars to the H3-F1 method

Fig. 5 :

 5 Fig. 5: Topographic maps for four different subjects using the different theories. (a) Filtering is done using H1, we plot the filter obtained for the errors. (b) Filtering is done using H2. (c)Filtering is done using H3, we plot the filter obtained for the errors.