Integration of Syntactic and Lexical Information in a Hierarchical

Home , Dependency grammar

I Integration of syntactic and lexical information in a hierarchical ! dependency grammar Cristina Barbero and Leonardo Lesmo and Vincenzo Lombardo Dipartimento di Informatica I Universit~ di Torino - Italy

Paola Merlo Universit6 de Gen~ve - Switzerland I IRCS - University of Pennsylvania

I Abstract level provided by syntactic rules is necessary to avoid In this paper, we propose to introduce syntactic the loss of generalization which would arise if class- classes in a lexicalized dependency formalism. Sub- level information were repeated in all lexical items. i categories of words are organized hierarchically from In parsing, a predictive component is required to a general, abstract level (syntactic categories) to a guarantee the valid prefiz property, namely the ca- word-specific level (single lexical items). The formal- pabifity of detecting as soon as possible whether a ism is parsimonious, and useful for processing. We substring is a valid prefix for the language defined I also sketch a parsing model that uses the hierarchi- by the grammar. Knowledge of syntactic categories, cal mixed-grain representation to make predictions which does not depend on the input, is needed for a on the structure of the input. parser to be predictive. In this paper we address the problem of the in- I 1 Introduction teraction between syntactic and lexical information in dependency grammar. We introduce many inter- Much recent work in linguistics and computational mediate levels between lexical items and syntactic linguistics emphasizes the role of lexical information categories, by organizing the grammar around the i in syntactic representation and processing. notion of subcategorizetion. Intuitively, a subcat- This emphasis given to the lexicon is the result egorization frame for a lexical item L is a specifi- of a gradual process. The original trend in linguis- cation of the number and type of elements that L I tics has been to individuate categories of words hav- requires in order, for ml utterance that contains L, ing related characteristics - the traditional syntactic to be well-formed. For example, within the syntac- categories like verb, noun, adjective, etc. - and to tic category VERB, different verbs require different express the structure of a sentence in terms of con- numbers of nominal dependents for a well-formed I stituents, or phrases, built around these categories. sentence. In Italian (our case study), an intransi- Subsequent considerations lead to a lexicalization of tive verb such as dormirv, "sleep", subcategorizes for grammar. Linguistically, the constraints expressed only one nominal element (the subject), while a tran- on syntactic categories are too general to explain sitive verb such as baciare, "kiss", subcategorizes for I facts about words - e.g. the relation between a verb two nominal elements (the subject and the object) and its nominalization, "destroy the city" and "de- 1. Grammatical relations such as subject and object struction of the city" - or to account uniformly for a are primitive concepts in a dependency paradigm, number of phenomena across languages - e.g. pas- | i.e. they directly define the structure of the sen- sivization. In parsing, the use of individual item tence. Consequently, the dependency paradigm is information reduces the search space of the possi- particularly suitable to define the grammar in terms ble structures of a sentence. From a mathematical of constraints on subcategorization frames. point of view, lexicalized grammars exhibit proper- Our proposal is to use subcategories organized in i ties - like finite ambiguity (Schabes, 1990) - that are of a practical interest (especially in writing real- a hierarchy: the upper level of the hierarchy corre- istic grammars). Dependency grammar is naturally sponds to the syntactic categories, the other levels I suitable for a lexicalization, as the binary relations correspond to subcategories that are more and more representing the structure of a sentence are defined 1We include the subject relation in the subcategorization, with respect to the head (that is a word). or valency, of a verb - cf. (Hudson, 1990) (Mel'cuk, 1988). Pure lexicalized formalisms, however, have also In most constituency theories, on the contrary, the subject is I several disadvantages. Linguistically, the abstract not part of the valency of a verb. I

i 58 I specific as one descends the hierarchy. This repre- 2 A dependency formalism sentation is advantageous because of its compact- The basic idea of dependency is that the syntac- ness, and bemuse the hierarchical mixed-grained or- tic structure of a sentence is described in terms of ganization of the information is useful in processing. binary relations (dependency relations) on pairs of In fact, using the general knowledge at the upper words, a head (or parent), and a dependent (daugh- level of the hierarchy, we can make predictions on ter), respectively; these relations form a tree, the de- the structure of the sentence before encountering the pendency tree. In this section we introduce a formal lexical head. dependency system, which expresses the syntactic Hierarchical formalisms have been proposed in knowledge through dependency rules. The grammar some theories. Pollard and Sag (1987) suggested and the lexicon coindde, since the rules are lexical- a hierarchical organization of lexical information: ized: the head of the rule is a word of a certain cate- as far as subcategorization is concerned, they in- gory, namely the lexical anchor. The formalism is a troduced a "hierarchy of lexical types". A specific shnplified version of (Lombardo and Lesmo, 1998); formalisation of this hierarchy has never reached a we have left out the treatment of long-distance de- wide consensus in the HPSG community, but sev- pendencies to focus on the subcategorization knowl- eral proposals have been developed - see for example edge, which is to be represented in a hierarchy. (Meurers, 1997), that uses head subtypes and lexical principles to express generalizations on the valency A dependency grammar is a five-tuple , where Hudson (1990) adopts a dependency approach and W is a finite set of words of a natural language; uses hierarchies to organize different kinds of lin- C is a finite set of syntactic categories; guistic information, for instance a hierarchy includ- S is a non-empty set of categories (S _C C) that can ing word classes and lexical items. The subcatego- act as head of a sentence; rization constraints, however, are specified for each D is the set of dependency relations, for instance lexical item (for instance STAND -4 STAND-intrans, SUB J, OBJ, XCOMP, P-OB3, PRED; STAND-trans): this is highly redundant and misses H is a set of dependency rules of the form important generalizations. z:X ( ... # ... In LTAG (Joshi and Schabes, 1996), pure syntac- ) tic information is grouped around shared subcatego- 1) z E W, is the head of the rule; rization constraints (tree families). Hierarchical rep- 2) X E C, is its syntactic category; resentations of LTAG have been proposed: (Vijay- 3) an dement is a d-pair (which descri- Shanker and Schabes, 1992), (Becker, 1993), (Evans bes a dependent); the sequence of d-pairs, in- et al., 1995), (Candito, 1996), (Doran et al., 1997). eluding the special symbol # (representing the However, none of these works proposes to use the hi- linear position of the head), is called the d-pair erarchical representation in processing - just Vijay- sequence. We have that Shanker and Schabes (1992) mention, as a possible 3a) rj E D, j E {1,...,i - 1,i + 1 .... ,rn}; future investigation, the definition of parsing strate- 3b) Y~ ~ C,j ~ {1,...,i-l,i+l,...,m}; gies that take advantage of the hierarchical repre- Intuitively, a dependency rule constrains one node sentation. (head) and its dependents in a dependency tree: the The goal of our hierarchical formalism is twofold. d-pair sequence states the order of elements, both On one side, we want to provide a hierarchical orga- the head (# position) and the dependents (d-pairs). nization to a lexicalized dependency formalism: sim- The grammar is lexicalized, because each depen- ilarly to the hierarchical representations of LTAG, dency rule has a lexieal anchor in its head (z:X). the aim is to solve the problems of redundancy and A d-pair identifies a dependent of category lexicon maintenance of pure lexicalized approaches. Yi, connected with the head via a dependency rela- On the other side, we want to explore how a hierar- tion rl. chical formAllgm can be used in processing in order As an example, consider the grammar 2: to get the maximum benefit from it. The paper is organized as follows: in section 2 we G--< describe a lexiealized dependency formalism that is a W : {gli, un, amici, eroe, lo, credevano} simplified version of (Lombardo and Lesmo, 1998). 2We use Italian terms to label grammatical relations - Starting from this formalism, we define in section see table 1. Since subcategorization frames are language- 3 the hierarchy of subcategories. In section 4, we dependent, we prefer to avoid confusions due to different ter- sketch a parsing model that uses the hierarchical minology across languages. For example, the relation Ter- mine - see the caption of figure 4 - actually corresponds to the grammar. In section 5, we describe an application indirect object in English. However l-Objundergoes the dou- of the formalism to the classification of 101 Italian ble accusative transformation into Obj, while Termine does verbs. Section 6 concludes the paper. not.

59 grammar) and sets of subcategories, L : W --~ 2 T- {}. That is, each word can "belong" to one or more subcategories; D is a set of dependency relations (as in section 2); Q is a set of subcategorization frames. Each subcategorization frame is a total mapping q : D -4 Rx 2 T, where R is the set of pairs of natural numbers Figure 1: Dependency tree of the sentence Gg arnici io such that nl _> 0,n2 _> 0 and nl ~ n2; credevano un eroe, "The friends considered him a hero", F is a bijection between subcategories and subcatv- given the grammar G. The word order is indicated by gorization frames, F : T -4 Q; the numbers 1, 2,... associated with the nodes - am/c/, --T is an ordering relation among subcategories. -- ~riend', is a left dependent of the head, as it precedes the head in the linear order of the input string, eroe, In order to define _, In the relation d in the subcategorization frame q. where H includes the following dependency rules: tuitively, Nq(d) is the number of times the depen- I. gli: DETERM (#); dency relation d can be instantiated according to the 2. un: DETERM (#); subcategorization frame q; Vq (d) is the set of subcat- 3. amici: NOUN ( #); egories that can be in relation d with a subcategory 4. eroe: NOUN ( #); having q as a subcategorization frame. 5. lo: PRON (#); Let _ # ); By applying the rules of the grammar, we obtain R, < R,, R2 iff the dependency tree in-figure 1 .forthe sentence Gli rain(R,) > rain(R2) ^ maz(R,) < maz(R2) arnici lo credevano un eroe, '~he friends considered namely, the range RI is inside the range R2. him a hero". Let -, where: hierarchy specifies a correspondence between words T is a finite set of subcategorie.r, and rules. Moreover, we show that the hierarchy L is a mapping between W (the words, defined in the acts as a taxonomy: given that rules(t,) C H is the Sin this paper we focus our attention to verbal subcatego- set of dependency rules whose head is the syntactic rization frames. category t,, we have that

60 Vtz, t~ E T Vdr E H DepSetq = (t, <_T t2 ^ dr ~ rules(h) --~ dr ~ rules(t~)) { {<,ogg, N>, #}, In order to specify the correspondence between sub- { , , #}, categorization frames and dependency rules, we first {<#og 9, N>, , , #}, define {, , #}, { , , , #}, = {ml m = [< d,t > I t e V0(d)] ^ {, , , , #}, minNq(d) < Card(m) < maxNq(d)} { , , #}, Given a subcategorization frame q and a relation d, {, , , #}, { <.oqg, N> , , <~,.W, P> , , #} Depq(d) is the set of all multisets of pairs < d, t >, where t is a subcategory E Vq(d). The multisets If we take all the permutations of the various subsets, come from the fact that the same relation can be we finally obtain the rules. So that if we have instantiated many times (depending on the range). In order to compute the sets of dependency relations L("to aprong") ffi {ttsT} that the subcategorization frame includes, we form F(ttaT) = q the cartesian product of the various Depe(d): we obtain dependency rules of the form in the pre- vious section: Carte = I]aeD Depq(d) to apron9 : ttsr( ~) and we evaluate the union of each member of Carte; each of them is extended by including the special to spron9 : tzsT(# ) #: to sprong : tz3?( #) symbol to aprong : t,av( # ) DepSet, = {m I m = (U.es, sec°.t.s) U {#}} where the union is a mukiset union, preserving du- This procedure has the goal of mapping the subcate- plications. Finally, by picking all the permutations gorization frames onto the dependency rules. In the of each member of DepSet¢, we get the set of rules actual practice, the frames are not multiplied out be- (also called subcategorization patterns): fore processing (for instance, exactly 200 rules would be generated for our very simple example). Process- Rulesq = {rJ r E Permute(m) A m 6 DepSetq} ing issues will be sketched in section 4. An example should make clear how the above defi- 3.2 Ordering among dependents nitions work. Let's assume that The hierarchy, and in particular the subcategorization frames, does not enforce a specific ordering D = {so9g, ogg ,o~ml~} among dependents of the same head. We propose an q = {, {N}>>, extension of the formalism that prevents some per- <099, <<0, z>, {N, C}>>, mutations of the rules from being generated. The , {P}>>} definition of subcategorization frame is modified in (where C is short for CHESUB - subordinating con- the following way: junction - and P for PREP). Then we have: Q is a set of ordered snbcategorization frames. Each of them is a pair consisting of a subcategorization Depq(so#9) = { { } } frame and a set of ordering constraints. Depq(ogg) = {{},{}, {<099 ,C>}} Vq E Q [q :< x20>], where 1t is Dep (co.. ) = {{}, {}, as before and O is a set of pairs where {, }} d,,d2 e DU{#}. Cartq = The pairs in O define a partial order on the rel- { <{<8ogg, zv>},{}, {} >, ative positions of the dependency relations and the <{<8ogg, N>}, {}, >, . head. If both dl and d2 are members of D, the <{ }, {}, {, } >, constraint specifies that the dependent whose gram- <{}, {}, {} >, matical relation is d, (if any) must precede linearly <{<8ogg, N>}, {}, {} >, the dependent whose grammatical relation is d2 (if <{<8og9, Jr>}, {}, { , any). If the first (second) member of the constraint } >, is #, it is specified that the dependent whose gram- <{<8o9g, N>}, {}, {} >, matical relation is d2 (dl respectively), if any, must <{<8099, N>}, {}, {} >, follow (precede) the head. The "if any" clauses say <{ }, {}, {, that in all cases where one of the two elements is } >} optionally present (minimum of the range equal to

61 0), the constraint is assumed to be respected in case I t137 the number of actual instantiations is 0. The ordering relation is transitive, namely: [1,1]j¢" |O,1]P ~ [0,21 ! iN} iN, {Prep) if E O= A E O= then Chesub ) E O= We require that the set of ordering constraints O= associated with any subcategorization frame be con- tl80 I sistent: II, l] IOtO) '0 for at e, e D u {4#}, ¢ Of iml I ! b)/or at e,,e~ e D U {#}, il e Of I men ¢ Of Figure 2: An example of subsumption between two subcategories. Finally, we modify the -~T relation (which defines the hierarchy): identify t~ from tl, and Dif/(tl, t~) is a notation i for specifying the difference between the constraints for each tl, t~ E T: associated with tt, and the ones associated with t2. t~ <_r t2 i~ So, we can say that the constraints associated with ! (OF(t,) _D OF(t=)) ^ t, are determined as the composition of the ones in- VdED her/ted from t2 and the ones specified locally (the (NF(t~)(d) <_~ NF(t,)(d) ^ differentia) for tl. VF(t,)(d) , <#,ogg>}, specifying that the sub- subcategorization frame q shown in the example of ject must precede the verbal head, which, in turn, paragraph 3.1 - and tlso. must precede the direct object. If each p~mutation For the sake of clarity, we show the subcategoriza- I in Rulesq is checked to verify if it satisfies the con- tion frame associated with t137 with a graph. In tlso straints, then only 40 rules are left, corresponding (subsumed by t137), we specify the local constraint to the possible (free) positions of the (0 to 2) com- restrictions: the number restrictions of eGG become plements. [1,1], and those of COMPL become [0, 0]. Moreover, the value restrictions of OGG become {N} (CHESUB I 3.3 Inheritance is ruled out). By inheriting the constraints of t,s7 We briefly mention here a notational convention and restricting them locally, we obtain that tlso re- which is useful to simplify the description of the sub- quires an obligatory nominal subject and an oblig- i categorization frames; this convention is widespread atory nominal object, and cannot have any comple- in almost all taxonomic hierarchies. For details ment. The order constraints - not shown in the fig- about inheritance we remind to the extensive liter- ure - are also inherited in the obvious way. ature on semantic networks, frames and description A more significative example is in figure 4, that we i logics (Nebel, 1990). will describe in section 5. We define: 4 Parsing issues tl

63 ZITI~'~IF.ATZOm IFJUCDZC~'~ tJU~XOH I I NOUN~ NOUN~

SP~C/ SP~ ---.2 sP~/ I DETERNr~-] 1 DETZRM~'~ ~.

$¢A/@C.T/~ PJU~ZC~ZCW O~OI.g27CW I V-TR [credevano] V-TR ['c~dev..o~ V-TR [clredevano] 4

I ,-.~-~--, ~ 3 ,-.-7.~.~ 2 3 _

I Figure 3: Analysis of the sentence Gli amid |o credevano ~ ~ ~ friends con.~dered him a hero". eral literary genres of written Italian. The whole hierarchy has 6 levels: the top level (class The information required by our formalism -- the VERB) represents the general constraints for Italian I grammatical relations associated to the dependents, verbs, the top+l level distinguishes the constraints their number (Nq(d)) and the set of categories for impersonal(V1), intransitive (VERB-INTR) and (Vq(d)) that can realize them -- was partly obtained transitive (VERB-TRANS) verbs, the top+2, top+3, by consulting Italian dictionaries, partly based on etc. levels represent specific classes of verbs (from I native speakers intuitions, and mostly from the anal- V2 to VS0). ysis of the corpus. 5.3 Results All the sentences containing the verbs under anal- The graph in figure 5 shows the distribution of verbs ysis were automatically extracted from the corpus, I and the subcategorization patterns (rules) exhibited by type, namely how the number of verbs covered by the classes grows in relation to the number of classes. by the verbs in those sentences were manually col° We can see that the first (more common) class covers lected. 15 verbs, the first and second more common classes ! We represented the set of subcategorization patterns together covers 26 verbs, etcetera. With the first 9 (rules) as subcategorization frames, by associating classes we cover 63 verbs, giving rise to a reduction with each grammatical relation - according to the of 85.7% compared to having a distinct subcatego- formalism - the related number (Nv(d)) and value rization frame for each verb. With the first 18 classes (Vq(d)) restrictions computed on the corpus. In this l we cover 81 verbs (reduction of 77.7%). The whole test, we have kept the order between the dependents set of verbs requires, however, 50 classes (reduction of a verb free, so there are no ordering constraints. of 50.5%): in fact, we have found many verbs with its I Each class tt is connected to supexclass t2. very idiosyncratic behaviours. Diff(tl, t2), the difference between the constraints Table 2 shows the distribution of verbs by token associated with tl and the ones associated with t.~, (sum of the occurrences, in the corpus, of all the is expressed by specifying, for each relation that is verbs referring to each class), level by level. The I restricted from t2 to tl, the relation itself with the fact that some rare classes occur is interesting if com- new number and value restrictions. pared to the percentage of reduction in the represen- 5.2 Hierarchy tation. There is a compression of 55,7%, while still taking care of very low frequency patterns, where i Figure 4 illustrates a small portion of the resulting compression is almost 0%. hierarchy. This hierarchy is based on the depen- In Table 3, we show, for each level, the number dency relations for a generic Italian verb summa- of subcategorization patterns represented by all the I rized in Table 1 s. classes of that level, namely the sum of the patterns 6Usually the adjuncts are not indicated as part of the sub- of each class at that level. The number of patterns categorization frames of the verbs: they are not obligatorily decreases rapidly by d~,cending the hierarchy. required by the verbs themselves. We have specified them anyway, as the hierarchy represents the grammar - which in- The representation of the syntactic knowledgeconcerning ad- I cludes all the information about the dependents, adjuncts in- juncts is currently a research goal. Most authors tend to cluded. Moreover, by specifying the information about the avoid it in the representation of subcategorlzation frames - adjuncts at the top level, we maintain the clarity of the rep- see (Hudson, 1990) and the "adjoining" operation in LTAG I resentation and the mapping on the formal grammar. (Josh| and Schabes, 1996). I l 64 t GRAMMATICAL SYNTACTIC CATEGORY EXAMPLE + TRANSLATION RELATION (value restriction) subject nominal group, N Paolo area Maria (Soot) "Paolo loves Maria" embedded clause headed by the Mi diverte che tu dica ci~), complementizer che, "that", Cheaub "It amuses me that you say this" preposition di, "off, Prep[di] Non mi interessa di venire, "I am not interested in coming" infinitive verb, Verb[inf] ~qdar¢ t be/to, "Skiing is nice" object nominal group, N Gianni mangia una mela, (Oct) "Gianni eats an apple" embedded clause headed by the Credo J~e_z~l_d/lffizlm~, complementizer ehe, "that", Chesub "I think it is amusing" preposition d/, "of", Prep[di] Aepetto di partite, "I am waiting to leave" predicative complement nominal group, N Gon.~idero Piero ~ Omico, (PRED) "I consider Piero a friend" adjective group, Adj Luigi ~ gentile, "Luigi is prepositional group, Prep Tuo zio ~ senza ritegno, "Your uncle is without reserve" complements prepositional group, Prep Metro il vaso (COMPL), "I put the vase on the table" at most 3 clitic, Clitic Gli ho dato un libro, "I gave him a book" adjunct prepositional group, Prep Procedevo di buon p~so, (AGGIUNTO) "I was walking at a brisk pace" adverb, Ado Luigi corse veloeemente, ,Luigi ran quickly" conjunction, Conjdip Telefonami fuando puoi, "Call me when you can" verb of non-finite mood, Verb[non.fin] Camminavo ~schiettando, "I was walking while whistling"

Table I: The grammatical relations of a generic Italian verb, with their possible realizations and related examples.

For the patterns found in the texts, we observe a de- The hierarchical formalism has shown to be effective crease similar but less marked than the grammatical in representing parsimoniously - that is, without re- patterns. Even the more specific classes describe a dundancy - the syntactic and lexical knowledge in good portion of the patterns in the texts, so confirm- an empirical test on 101 Italian verbs. ing the usefulness of very specific information in the Moreover, we have sketched a left-to-right predictive analysis. parsing model that takes advantage of the hierarchi- Table 2 show this point more clearly. The lower, cal knowledge representation in order to make pre- more specific levels, while having fewer classes, still dictions on the structure of the input sentence. cover many occurrences of verbs in the text. In the next future we will address a massive empirical test of Italian corpora, and the formal specifica- 6 Conclusion tion of the parsing model, together with a complex- ity analysis. The paper has presented a hierarchical organization of a dependency formalism. The hierarchy is defined by the subsumption relation on subcategories, defined as a mapping between subcategories and sub- 7 Acknowledgements categorization frames. Subcategorization frames, in turn, define the number of possible instantiations This research was partly sponsored by the Swiss of a dependency relation and the subcategories that National Science Foundation, with fellowhip 8210- can realize it. 46569 to P. Merlo.

65 VERB

AGGIUNTO lo.,lv llOo,l ",,!,o.,! 1 [O, Lnf] IN, Chesub, {N, Chesub, {N,AdJ, {Prep, {P repo Adv, Verblinf], Prep[dil } Prep} Clltle} ConJdlp, Prep[di]} Verb[no flnl}

VERB-ZNTR ~-T~NS

SOGG OGG COMPL [0.0l V°.01 10,Ol [1,1l # [0.0] • l {N, Chesub, { } {N, Chesub, {N, Chesub, {} 1} 1} 1} Verb [Infl, Verb[inf] } Prep[dl] } l Prep [dl] }

V$O

TERMINE I[0,1] |lO, l {Prep[a], {N} {Adj] {Preplda] } Clitic [dat ] }

Figure 4: A portion of the hierarchy. Subclasses inherit and restrict the constraints at the top of the hierarchy. The top class, VERB, has three daughters. V1 is the class of impersonal verbs, that can only have adjuncts as dependents - the restriction is on the range, [0, 0], of the other relations. For example, we can say Piove o,i tetti della cittd, "It rains on the roofs of the town". The classes VERB*INTR and VERB-TRANS correspond to intransitive and transitive verbs, respectively. VERB-INTR requires an obligatory subject ([1,1]) and it cannot have a direct object ([0, 0]). VERB-TRANS requires an obligatory subject ([1,1}), that can be headed by a nominal element, a conjunction ch¢ or an infinitive verb, and an obligatory object ([1,1]). A subclass of VERB-INTR, V2, is shown: its only restriction is on the relation COMPL, which is specialized on the subrelation TERMINE, "Indirect Object", having a range [0,1], and having Prep[a]and Clitic[dat]as associated categories (preposition lexically realized by a, "to", and dative clitlc). For example, sembrare~ "seem", is a verb belonging to this class: we can say A Luigi Maria aembra beUissima, "To Lui~ Maria seems very beautiful". VS0 is a subclass of VERB-TRANS: it restrics the sets of categories associated to the relations OGG and PRED, and specializes the relation Co~lPl. on the subrelation SEPARAZIONE, "Separation" (realized by the preposition da, "from", Prep[da)). The verb allontanare, "distance", belongs to VS0: Luigi mi aIlontanb da re, "Luigi distanced me from you".

Number of I01 verbs belon?lnq to ~he classes 81

4S,

12.

21,.

15'

|234S tD X| SO

N~rof ClasBas Figure 5: Distribution of verbs by type.

66 ii LEVEL DISTRIBUTIONOF CLASS SIZE 1 2 5 3 423 322 318 308170 169 148 136 99 77 67 545451 47 46 45 21 20 16 14 12 11 10 3 2 4 321 292 229 116 103 897850 41206 5 5 397332 212 111 52 15 6 299 239 2 Table 2: Distribution of class size by level.

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