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<Paper uid="E91-1008">
  <Title>INDEXING AND REFERENTIAL DEPENDENCIES WITHIN BINDING COMPUTATIONAL FRAMEWORK</Title>
  <Section position="1" start_page="0" end_page="0" type="metho">
    <SectionTitle>
INDEXING AND REFERENTIAL DEPENDENCIES WITHIN BINDING
COMPUTATIONAL FRAMEWORK
Fabio Pianesi
</SectionTitle>
    <Paragraph position="0"> Istituto per la Ricerea Scientifica e Tecnologica 38050, Pante' di Povo - Trento - Italy pianesi@irshit</Paragraph>
  </Section>
  <Section position="2" start_page="0" end_page="0" type="metho">
    <SectionTitle>
THEORY: A
ABSTRACT
</SectionTitle>
    <Paragraph position="0"> This work is concerned with the development of instruments for GB parsing. An alternative to the well known indexation system of (Chomsky, 1981) will be proposed and then used to formalize the view of Binding Theory in terms of the generation of constraints on the referential properties of the NPs of a sentence. Finally the problems of verification and satisfiability of BT will be addressed within the proposed framework.</Paragraph>
  </Section>
  <Section position="3" start_page="0" end_page="0" type="metho">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> This work is concerned with the development of instruments for GB parsing (see Barton, (1984); Berwick (1987); Kolb &amp; Tiersch, (1990)); in particular, our attention will be focused on the Binding Theory (henceforth, BT) a module of the theory of Government and Binding (henceforth, GB; see Chomsky (1981; 1986)). It has been pointed out (eg. in Kolb &amp; Tiersch, (1990)) that the lack of a complete and coherent formalization of a linguistic theory like GB can be a major obstacle in addressing the issue of principle-based parsing; this is true of BT too, in particular with respect to the indexing system of Chomsky (1981), the shortcomings of which have often been pointed out in the literature. A formalism for the treatment of the referential relationships among the NPs of a sentence will be presented that is more expressive than indexation and more effective as a computational tool.</Paragraph>
    <Paragraph position="1"> In Section 2 the indexing system and the role it plays within BT will be discussed; in Section 3, an alternative will be developed that overcomes some of the shortcomings of indexing. Such a system will, then, be used to formalize the view of BT as a device that generates (syntactic) constraints on reference. In Section 4, it will be shown how our proposal could be applied to some computational problems, i.e. the problems of verification and satisfiability within BT.</Paragraph>
  </Section>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
2 Preliminaries
</SectionTitle>
    <Paragraph position="0"> Since Chomsky (1981), it has become commonplace to denote the interpretative relations among the NPs of a sentence by means of indices, i.e. integers attached to NPs in such a way that elements bearing the same index are taken to denote the same object(s), while different indices correspond to different denotations; most of the statements of BT have been |aid down in terms of this system (Chomsky, 1981, 1986). In a number of works (see Chomsky (1981), Higginbotham (1983) and Lasnik &amp; Uriagereka (1988)), however, it has been pointed out that the indexing device is not adequate to capture certain referential relations; this is true for the relation between pronouns and split antecedents, i.e. antecedents composed of two or more arguments bearing different thematic roles, l Furthermore, indices blur the distinction between coindexing under c-command and coindexing without c-command, thereby making it difficult to capture the dependence of an element, behaving like a variable, upon its antecedent (see Reinhart, (1983)). 2 The replacement of indices with index sets has been proposed as a way to address the first problem (see Higginbotham, (1983)): an ordinary index is substituted by a singleton; when there are pluralities, e.g. when an NP is coindexed with a split antecedent, it is annotated with the (set) union of the index sets of each component of the plurality; therefore, coindexing amounts to equating index sets. In this view, the ordinary conditions on disjoint reference (Principles B and C of BT) must be extended to avoid not only identical reference but, more generally, reference intersection. It has also been argued (Higginbotham, 1983) that indices should be abandoned and substituted by the non symmetric relation of linking; when the antecedent is split, a plurality of links should be used. This way, however, two different situations are collapsed together: the one in which an item is coindexed with a plurality of elements all of which share the same index, and the case of true split antecedents, where the elements composing the antecedent do not have the same index. Furthermore, the asymmetric behaviour of linking has no clear correlate at the structural level; it will be suggested below that c-command should continue to play a role here.</Paragraph>
    <Paragraph position="1"> Computational works about BT have been mainly concerned with providing lists of possible or impossible antecedents for the NPs of a sentence (see Correa (1988); Ingria &amp; Stallard (1989)); additional procedures select actual antecedents 1 R-expressions can take split antecedents too, at least in certain cases (epithets); however, we will not explicitly address this point here. Anaphors, instead, can never take ~lit antecedents.</Paragraph>
    <Paragraph position="2"> There is a full range of phenomena for which such a distinction seems crucial, eg. weak crossover and sloppy reading of pronouns (Reinhart, 1983); donkey sentences and the so called indirect binding (Ha'de, 1984; Reinhart, 1987). However, only few of them will be addressed here.</Paragraph>
    <Paragraph position="3"> - 39 among the potential ones. Berwick (1989) considers only R-expressions and a device (actually, a Turing machine) assigning the same index to multiple occurences of the same R-expression (names); furthermore, a set of disjoint indices is associated with each item. Finally, Fong (1990) performs a combinatorial analysis of the paradigm of free indexation, as proposed in (Chomsky, 1981); he shows that free indexation gives rise to an exponential number of alternatives and argues for the necessity of interleaving indexing and structure computation. In any case, indexing has been either explicitly or implicitly assumed, so that most of the computational approaches to BT suffer the same shortcomings pointed out above. In particular, given that both split antecedents and the distinction between binding and coreference cannot be easily accounted for, this results in an impoverished input being provided to the semantic (intepretative) routine.</Paragraph>
    <Paragraph position="4"> In the following section a formal system will be discussed that tries to address such problems by explicitly distinguishing between binding and coreference; at the same time, BT will be seen as a theory that states very general constraints (constraint schemata), which are then (at least in part) instantiated according to the structural properties of the sentence at hand. These instantiated constraints are then used to test sets of positive specifications (indexations) which constitute the input to further semantic processing. 3</Paragraph>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
3 The formal apparatus
</SectionTitle>
    <Paragraph position="0"> For a given sentence w, let N={n 1, n 2 ..... nm} be the set of its NPs; furthermore let us indicate with A, P and R the subset of N whose members are anaphors, pronouns and R-expressions, respectively. Sets A, P, R, constitute a partition of set N. Finally, Q denotes the set of quantified expressions and syntactic variables. Split antecedents will be considered as members of the power set of N, P(N); for the sake of uniformity, single NPs will be denoted by members of P(N) with cardinality equal to one, i.e. by singletons.</Paragraph>
    <Paragraph position="1"> Definition 1 A relation s ~(P(N)xP(N))is defined such that (9 ~)es iff C/={m}, ly={n I .....</Paragraph>
    <Paragraph position="2"> np} , p&gt; l and me lg.</Paragraph>
    <Paragraph position="3"> For any C/i~=(n), neN, sets .~(n), B(n) and G{n) will denote the set of elements that c-command n and lie 3Disjoint reference constraints arising from Principles B and C of BT are not carried over to semantic routines but are resolved at an earlier stage. Furthermore, it is assumed that, whatever processing the semanti~ routines perform, their default behaviour consists of assigning non-sharing semantic import to different NPs, unless otherwise stated in the input constraint set.</Paragraph>
    <Paragraph position="4"> inside its binding domain whenever, respectively, n eA, nEP or neR; finally, if n is a pronoun D(n) will denote the set of NPs c-commanding it and lying outside its binding domain. 4 Definition 2 Given a sentence w, a relation b ~ (P(N)xP(N)) is defined, such that (9 ~)eb iff one of the following conditions obtains: (i) ~={n't}, nieA , ~={nj} and nje.~(ni); (ii) ~={ni}, nieP, II/={nj}, and njeD(ni).</Paragraph>
    <Paragraph position="5"> Definition 3 Given a sentence w, a relation d~ ( P(N) x P(N) ) is defined, such that (9 ~)e d iff ~={ni}, II/={nj} and either njeB(ni) or njeC(ni), depending on whether nieP or nieR.</Paragraph>
    <Paragraph position="6"> In the following, b(.)and s(.), the inverse relations, will be used as well.</Paragraph>
    <Paragraph position="7"> Definition 4 Given a sentence w and a phrase structure tree representation for it, Zw, the set of binding constraints for T,v is the set ~R,,={(C/ r ~) I 9, ~veP(N), r is a symbol, re {d, b, b(.) } }, such that (9 r ~)e~R,, iff (9 Ig)er, where r is the corresponding relation. 5 Given sentence w and a phrase structure representation, a binding constraint set states disjoint reference constraints (essentially, the consequencies of Principle B and C of BT) and the range of the binding relation (see below) for each NP. The meaning of the formers is that whenever (a d \]\])e 9?,,, the intersection of the references of ct and 13 is empty. Note that 3,, does not exhaust the range of possible constraints on reference; for instance, those preventing weak crossover violations or circular readings are not included in ~,, but will be discussed later on; furthermore, split antecedents are not mentioned in 9t,, Let us, now, focus the attention on how to represent positive referential relationships. To this p~arpose, two fundamental relations on ~N), coreference and binding (more precisely, the bound variable reading, in the terminology of Reinhart (1983)) are introduced. The former is a tran~sitive, symmetric and reflexive relation, therefore an equivalence relation; the latter is irreflexive, intransitive and non symmetric, it only obtains under c-command and denotes the dependence of an item upon another one for its interpretation. 6 An</Paragraph>
    <Paragraph position="9"> item can be bound by, at most, one other element; on the contrary, an NP can corefer more than once and even with itself. Split antecedents cannot be bound and, finally, it is not possible for an item, ct, to be bound and, at the same time, to corefer; on the other hand, ct can be a binder and, at the same time, corefer. The binding relation will be denoted by the symbol I.</Paragraph>
    <Paragraph position="10"> The differences between binding and coreference are at both the structural and the interpretative level. Binding can only obtain under c-command while this is not a prerequisite for coreference; at the interpretative level, the reference of the binder can be accessed to form the reference of the bindee.</Paragraph>
    <Paragraph position="11"> Instead, coreference corresponds to a sort of extensional identity and simply amounts to equating independent references; of course, items that do not refer (e.g., quantified expressions and anaphors) cannot corefer. 7 Bound items behave similarly, i.e. even a pronoun, when bound, loses the capability of autonomously referring and, therefore, of coreferring. Transitivity has not been assumed for binding, in order to avoid reducing the interpretation of a sequence of elements al .... an, such that each ai is bound by ai+l, upon that of the  last element; consider the following sentence: (1) John and Mary told each other PRO to leave.</Paragraph>
    <Paragraph position="12"> and the two readings: (2) (i) John told Mary that Mary should leave and Mary told John that John should leave.</Paragraph>
    <Paragraph position="13"> (ii)* John told Mary that John should leave and Mary told John that Mary should leave.</Paragraph>
    <Paragraph position="14">  Because of obligatory control, PRO is bound by the reciprocal, which, in its turn, is bound by the matrix's subject. If binding were transitive, we should conclude that the interpretation of PRO is entirely dependent upon that of John and Mary (in this being on a par with the reciprocal) and the relevant reading would be (2.ii). However, (1) has only the first of the two readings in (2) and this requires that PRO inherits reciprocality from each other; therefore, the correct dependencies are between PRO and each other and between the latter and the matrix subject. 8 Note that a sentence like they are largely determined by structural properties. No pragmatic import is assumed for coreference, as is done by Reinhart (1983).</Paragraph>
    <Paragraph position="15"> 7See tla'ik (1984) for a discussion about the distinction between referring and non referring NPs. 8Here, it is assumed that a VP conlaining a reciprocal, e.g. told each other, is true of each element a such that a is in the interpretation of each other and told(a, b) is true. where b is also in the interpretation of each other and a;~b; see  (3) John and Mary told each other that they should leave.</Paragraph>
    <Paragraph position="16"> admits both readings, given that the subject of the  dependent clause can be bound either by the reciprocal or by the matrix subject. In this work, then, binding has a functional nature which may well reflect properties of semantic processing; even in this case, however, the point is that syntax only addresses an abstract property, i.e. functionality. Since coreference is an equivalence, the representation could be simplified by considering a minimal relation corresponding to coreference. The connected parts of the graph of the coreference relation are complete subgraphs; for each of them, A=(V, E), choose an arbitrary vertex, ~t, and consider the graph Amin=(V, {(~ 0~)\] 1~:0~, (1~ a)~ E}). By iterating the procedure and then taking the union of the results, a (directed) graph is obtained that represents the minimal relation corresponding to coreference. 9 We will denote such a minimal relation with the symbol c and call it 'coreference' tout court. The inverses of both I and c, I(.) and c(.) will be used as well.</Paragraph>
    <Paragraph position="17"> At this point, the notion of indexation set can be defined.</Paragraph>
    <Paragraph position="18"> Definition 5 A indexation set for a sentence w is the set ~3 w= { ( ~ r u,/) I q), ~/~ P(N), r is a symbol and re {c, c(.) , l, l(.), s, s(.)} } such that (~ r 9')~$w iff (C/ ~)~r, where r is now interpreted as the corresponding relation.</Paragraph>
    <Paragraph position="19"> Note that split antecedents (relation s) are seen as part of the indexation set of the sentence since they do not have any independent status within syntax; furthermore, this move permits us to only consider a limited number of them every time, instead of the exponential number of possible split antecedents arising by free combinatorics.</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
3.1 Compatibility of an indexation set
</SectionTitle>
      <Paragraph position="0"> with BT An indexation set is composed of positive specifications that interpretative procedures process in order to assign actual references. Before this could happen, however, it must be verified that each of such specifications does not contradict the sentence particular constraints of ~R,v and general BT restrictions. In other words, a way is needed to enforce the overall compatibility of ~,, w.r.t~. BT. A path in ~3 w is a sequence of elements p=(C/o rl ~1) (~1 r2 ~2)... ($m-1 rm ~m), m&gt;-l; if ~O=~m Higginbotham (1981, 1985).</Paragraph>
      <Paragraph position="1"> 9No information is lost in the passage from coreference to its minimal counterpart; the original graph can, in fact, be easily recovered by reintroducing transitivity, symmetricity and reflexivity. Of course, the choice of 0t does not affect the result.</Paragraph>
    </Section>
    <Section position="2" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
-41 -
</SectionTitle>
      <Paragraph position="0"> then p is a circular path. Furthermore, the string wp=rl r2 ... rm is called the path string associated with p. Path strings may be used to define the following regular languages that will be useful to state many of the conditions about index sets in a compact form: Ll=l*(c+c(.)+cc(.)+c(.)c+l+l(.))l(.)*,</Paragraph>
      <Paragraph position="2"> their meaning. The paths from an element, C/~, to another one, ~, with strings in LI encode all the possible ways in which C/~ and ~ can be related by a combination of binding and coreference relations (in such a way, of course, that their definitory properties are respected). In this respect, Ll replaces the traditional notion of coindexation (although we will continue to use this (improper) term to denote the existence of a path with string in Ll).</Paragraph>
      <Paragraph position="3"> Therefore, given a sentence like the following one (where subscripts are only used to single out constituents): (4) His1 mother told John 2 that he3 should leave a possible indexation set may contain the following elements: (3 1 2), (2 c I) and the string lc for the path from 3 to 1, may be taken to substitute the old notion of coindexation. Consider, now, the notion of referential contribution; the basic case is given by the configuration (~ s ~)e5 w (i.e., an element contributing to a split antecedent); by extension, language L2 encodes all the cases in which an element contributes to the reference of another one. For instance, a possible indexation set for the following sentence (5) John1 told Mary 2 that they3 should leave is {(1 s 4), (2 s 4), (3 l 4)}; in this case, 1 and 2 are both contributing to the reference of 4 (the split antecedent) and of 3. On the other hand, language L 3 encodes all the cases in which an element receives a referential contribution from ~. Finally, consider overlapping reference between two items; the basic instance is given by two split antecedents some component of which are either shared or coindexed; the general configuration gives rise to paths with strings in the language L3L2, the concatenation of L3 with L2 .10 An example is the following sentence: 10In the linguistic literature, the term overlapping reference is used for all cases in which the reference of two items is not disjoint; of course, this implies that at least one of them denotes a plurality. However, the present use of this term, and that of referential contribution as well, is restricted to split antecedents, seen as the means, available to syntax, to compose pluralities and does not generalize to all the possible different varieties of plurals, such as those considered in (Lasnik (1976) and Higginbotham (1983)). (6) John I told Mary 2 that they 3 should avoid telling Henry 4 that theY5 had been discovered with the following indexation set: {(1 s 6), (2 s 6), (1 s 7), (4 s 7), (3 l 6), (5 l 7)}. In this case, two split antecedents (6 and 7) are introduced that share the component 1; therefore, overlapping reference obtains between 6 and 7 and between 3 and 5.</Paragraph>
      <Paragraph position="4"> The BT version considered here consistes of Principles A, B and C, as given by Chomsky (1986), weak crossover (see Reinhart (1983)) and some restrictions on circular readings. Now we can state the following: Theorem 1 Conditions for the compatibility of an index set with BT Given a sentence w, a tree representation zw and the b~nding constraint set, ~w, an index set, ~3w, complies with BT iff the following statements  hold: (i) for any pair (~={ni}, v={njt ..... nip}, l_&lt;p, if ((; l Ig)~w then (~ b Vk)e g~ w , l_~k_&lt;p, where ~k={ni~}; i.e. binding relations cannot be arbitrarily introduced in ~3 w, but must be derived from the relation b in 9~ w.</Paragraph>
      <Paragraph position="5"> (ii) for any C/={n} there is no circular path in ~w, from C/, with string in l+; i.e. there are no circular dependencies; (iii) for any C/={n}, no circular path in BW gives  rise to strings in L2; i.e. an element is never coindexed with another one and, at the same time, contributes to its reference; (iv) for any C/={n}, if neA then there is a ~ such that (C/l v/)e~w and I~1=1; i.e. each anaphor is bound in ~w and never takes a split antecedent; (v) for any C/)={n}, if n eQ then there is no element W such that (~ c ~)EBw or (IF c C/)e~ w ; i.eo quantifiers and syntactic variables cannot corefer;, therefore, they can only function as binders;l 1 (vi) if (~ d ~)~9~ w then there are no paths, in ~w. from C/ to V with strings either in Ll or in L2 or in L3L2; i.e. if two elements are in a principle B or principle C configuration then: they cannot be coindexed; no one of them can contribute Co the reference of the other and, finally, their references do not overlap.</Paragraph>
      <Paragraph position="6"> This theorem states the conditions for the compatibility of an indexation set for a sentence w with BT. Note, that certain constraints, expecially those in (vi), make crucial use of the set 9~ w ; other constraints, instead, directly apply to ~w, e.g. that i l(v ) together with (i) enforces the ban against weak crossover in that (v) checks that no quantifier corefers and (i) only admits binding under e-command.</Paragraph>
      <Paragraph position="7"> - 42 preventing weak crossover.</Paragraph>
    </Section>
  </Section>
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