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<Paper uid="C90-3022">
  <Title>A Computational Approach to Binding Theory*</Title>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
3 The Algorithms
</SectionTitle>
    <Paragraph position="0"> With respect to the two principles considered here, i.e.</Paragraph>
    <Paragraph position="1"> Principles A and B, the output of BT can be represented &amp;quot;hi'he definition in the text is Ihe one foand in Giorgi (1987); cfr also Chomsky (1986). Note that, in most cases, the CFC coincides wilh the first Senience or Noun Phrase dominating the ilem in qudstion. However, this is not always the case and the systems defining the binding domain this way often nm into trouble; this point will be fnrther considc~cd in Seclion 5; sue also Giorgi, Plainest, Satta (1989a).</Paragraph>
    <Paragraph position="2"> 61npro-drop languages (sec Chomsky 1981) typically, lhe subieet can be noJl lexica\[, i.e. can be an empty calegory, or can bc expressed postvel%ally, leaving an cxpletive empty category in subject position.</Paragraph>
    <Paragraph position="3"> as a lomml language Lv. More precisely, given a sentence w, let T be the set of all tuples t=&lt;z'w, (x, fit, .... /7,~&gt;, n_&gt;0, where &amp;quot;cw is a parse tree for w, a is either an anaphor or a pronoun and the components fll ... fin represent any set of NPs. v Let us define Le c T to be the set of all tuples such that the following conditions hold: (i) if o: is an anaphor, \]31...fin are all and only the items that can be antecedents for c~, according to</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
Principle A of BT;
</SectionTitle>
      <Paragraph position="0"> (it) if c~ is a pronoun, then ill...fin are all and only the items disjoint from ~x, according to Principle B of BT.</Paragraph>
      <Paragraph position="1"> The algorithms to be presented can be seen as recogniscrs for L s.</Paragraph>
    </Section>
    <Section position="2" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
3.1 Definitions
</SectionTitle>
      <Paragraph position="0"> Let N:(nl ..... nq}, q&gt;_l, be the set of all nodes in rw.</Paragraph>
      <Paragraph position="1"> We will also assume the following functions and predicates: s a function father from N to Nu {_t_}; a function siblings from N to '/'(30; a binary predicate agreement, defined on NxN, such that agreement(n1, n2)=TRUE iff the agreement features of n 1 and n2 are mutually compatible; a unary predicate pt-antecedent, defined in -N, such that pt-antecedent(n)=TRUE iff n is a maximal projection of a head N o (a Noun) within T~.</Paragraph>
      <Paragraph position="2"> Definition 1 A binary predicate domain is defined in NxN in such a way that domain(ha, hal=TRUE if f: (i) n,~ is the least constituent such that either all the O-roles pertaining to a lexical head are realized, or all the grammatical functions pertaining to the same lexical head are realized; (it) ny is the lexical governor of na and filther(n ~)~filther(n,4).9 0 Condition (it) has been explicitely introduced in order to lake care of cases of government across the boundaries (see Section 3.3).</Paragraph>
      <Paragraph position="3"> To account for the interaction of BT with pro-drop (cf. ex.(8) above), we also need the following definitions. Let ch-mark be a procedure defined on Nx&amp;(: whenever ch-mark(n,nm) is invoked, if n is a landing-site of a chain ~deg c within rw, every node nc such that nc belongs to c, gets marked with a distinctive mm'ker, which will be assumed to be the second argument node n m. This marking relation will hold until a new call to the procedure takes place for any node corresponding to the same chain as n, with a different node-marker. We need 7We assume lhat all the principles of the theory have already been applied to tile sentence. Such an asstimption is reasonable, given the modular nature of tile theory; see Chorasky (1981,  functional-.roles, lexical governor and government can be found in Chomsky (1981, 1986). A computaliona\] account of fl~e no,.iov, of \]ocal domain can be found in Giorgi, Pianesi and Satta (1989a), along with some tbrmal properties of the predica'.e domain. 10l(oughly, tim notion of chain can be defined (cf. Chomsky, 1986) as the set of coindexed positions (landing sites) pertaining to the same syntactic object (uhetv only one of sLicl't posili'.,ns is lexically filled).</Paragraph>
      <Paragraph position="4"> 2 121 also a function ch-marker, from 5% to Nvo (+-\], defined such that ch-marker(n)=nm iff n is a landing-site of a chain c within ~,,, and a previous call to the procedure ch-mark has m~ked each node in c with the marker-node n m .</Paragraph>
    </Section>
    <Section position="3" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
3.2 Algorithm Schemata
</SectionTitle>
      <Paragraph position="0"> The two algorithms behave in a very similar way; they take as input a node in N corresponding to an NP in Vw, and analyse some specific relations between the input node and each node in N that c-commands the input node, up to certain specific domain. The c-commanding relation is implicitly encoded in the way in which the algorithms apply the two flmctionsfather and siblings.</Paragraph>
      <Paragraph position="1"> An Algorithm for Principle A Given an input node n which corresponds to an anaphor in l:w, the algorithm outputs a list of nodes corresponding to &amp;quot;actual antecedents&amp;quot; for the anaphor itself. The algorithm looks for a &amp;quot;potential antecedent&amp;quot; of the input anaphor, starting from node n and proceeding from bottom-to-top. As soon as a potential antecedent is found, the algorithm restricts its search to the local domain it has just identified. Note that each potential antecedent must pass the agreement check to be considered an actual antecedent n We also consider some cases of referential circularity; in particular, problems arising in pro-drop constructions. More specifically, a node which belongs to the chain also containing the anaphor, cannot be collected as a potential antecedent. The following circularity check is therefore included: every chain whose landing-sites dominate the input anaphor, up to the domain of interest, is marked by the procedure ch-mark using the input-node as a marker. In this way a node ccomnmnding the input node and corresponding to a landing-site of a chain marked by the latter, cannot be taken as a potential antecedent for the input-node itself (for more discussion, see Section 5). The same mechanism also ensures that, for every possible chain, only one of its landing-sites is ever considered as a potential matezedent.</Paragraph>
      <Paragraph position="2"> Algorithm 1 input-node: A node corresponding to an anaphor in 'r~. Output: A list of nodes in N corresponding to actual antecedents for the input anaphor.</Paragraph>
      <Paragraph position="3"> Method.</Paragraph>
      <Paragraph position="4"> Step 1: Let input-node be the value of the program variable present-node, hfitialize also the program variable local .domain-flag to the value FALSE and invoke the procedure ch-mark(present-node, input- node). Step 2: For each value of the program variable present-sibling in siblings(t)resent-node), if ch-marker(presentsibling)veinput-node and pt-antecedent(presentsibling)=TRUE, perform the following actions. Set the program variable local-domain-flag to TRUE if it is FALSE and invoke the procedure ch-mark(presentl lAccording to Chomsky (1986), the existence of the potential anlecedent for an anaphor is crucial in defining its local domain. Note timt such an item is not necessarily the actual antecedenl, sibling, input-node); furthermore, if agreement(present-sibling, input-node)=TRUE then output present-sibling.</Paragraph>
      <Paragraph position="5"> Step 3: If father(present-node)=L, go to Step 4, otherwise let father(present-node) be assigned as the value of present-node. Invoke the procedure chmark(present-node, input-node). If local-domainflag=FALSE then restart at Step 2. Otherwise there are two possibilities: if domain(present-node, input-node )=FALSE then restart at Step 2; if domain(present-node, input-node)=TRUE go to Step 4.</Paragraph>
      <Paragraph position="6"> Step 4: Stop. ca An Algorithm for Principle B The algorithm starts from an input node that corresponds to a pronoun in rw. The algorithm visits all nodes in Nwhich correspond to elements c-commanding the input pronoun and lie inside the local domain; finally, it outputs a list of disjoint elements. Indee(1 the algorithm is procedurally very similar to the one given for Principle A, with minor changes due to the differences in the definitions of the local domain.</Paragraph>
      <Paragraph position="7"> Algorithm 2 considers each chain only once, as does Algorithm 1. Observe that if a pronoun belongs to a certain chain, it cannot be disjoint from other elements of the same chain. An identity check is then carried out by the algorithm in the following way: the chain, which the input-node belongs to, is mitiatly marked by the procedure ch-mark. Then, every node that c-commands the input-node inside its local domain, corresponding to a landing-site of this marked chain, will not be inserted in the output list of Algorithm 2 (see Section 5). The details are the lbllowing:</Paragraph>
    </Section>
    <Section position="4" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
3.3 Some Formal Results
</SectionTitle>
      <Paragraph position="0"> Some properties of Algorithms 1 and 2 will be stated; see also Giorgi, Pianesi and Satin (198%).</Paragraph>
      <Paragraph position="1"> Theorem 1 The predicate domain(present..node, inputnode) holds true at Step 3 in Algorithms 1 and 2 iff present-node corresponds in rw to the minimal CFC containing both input-node and n~, where n 7 is the lexical governor of input-node.</Paragraph>
      <Paragraph position="3"> present-node is a CFC, as defined in (5). Furthermore presenbnode dominates input-node at Step 3, as it is easy to show. It remains to demonstrate that present-node dominates n.C/. A government relation between n 7 and input-node can only be attained within the following three structural configurations. In the first, government is realize.d under sisterhood; thus, every node that dominates the governee will also dominate the governor.</Paragraph>
      <Paragraph position="4"> In the second configuration the govcrnee is attached higher than its governor, within the maximal projection of the latter; again, every node that dominates the former will also dominate the latter. The third possibility concerns the so called government across boundaries: when a maximal projection ZP (or a Small Clause) is in sisterhood relation with a lexical category X deg, then the latter can govern the specifier position of Ihe former (or the subjex:t position, in the case of a Small Clause). ZP may well be a CFC, in the sense of (5), but it does not contain the governor X deg. Condition (it) in Definition 1 explicitely rules out this case, so the claim is proved.</Paragraph>
      <Paragraph position="5"> 'If' The proof immediately follows from the given analysis of the possible configurations of government between the nodes n 7 and input-node, and from</Paragraph>
      <Paragraph position="7"> Theorem 2 Let &amp;quot;Cw be an X-.bar description for some sentence w such that all the principles of GB hold trtte for it, apart from the BT principles, and let N be the set (~f all nodes ill &amp;quot;~w. When input-node is assigned a value, which corresponds to an anaphor a in &amp;quot;rw, Algorithm 1 computes the whole list of nodes in N that corresponds to the antecedents of a, in the sense of Principle A of BT.</Paragraph>
      <Paragraph position="8"> f:'roof Omitted. La Theorem 3 Let &amp;quot;Cw and N be as in Theorem 2. Given as input a node that corresponds to a pronoun ~ in &amp;quot;cw, Algorithm 2 computes the whole list of nodes in N that must be disjoint from a, in the sense of Principle B of BT.</Paragraph>
      <Paragraph position="9"> Proof Omitted. c) Questions about time complexity are now addressed for Algorithms 1 and 2 (we assume, as the reference model for comi)utation, a RAM).</Paragraph>
      <Paragraph position="11"> Figure 3 Theorem 4 The running times of Algorithms 1 and 2 are given by two functions fAz andfA2, such thatf4 ~, fa2e O(n), where n is the length of the sentence under analysis. ',2 Proof (outline) From elementary considerations about X-bar Theory, ~3 it can be argued that set N has .'dze bounded by an expression of the form cxn+c2, It is easy to show that no node in N is visited more than once by Algorithms l and 2 and that a constant amount of time is spent in visiting each node; then the result follows.</Paragraph>
    </Section>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
4 A running example
</SectionTitle>
    <Paragraph position="0"> Let us see how Algorithm 1 works with the following sentence, giving, as input node, the one corresponding to the anaphor herself: (9) lIP Mary \[r \[vt' \[v' sees herself \[pp in the minvr\]\]\]\]\] 14 At the beginning, the variable present-node is set to the value of input-node, i.e. the anaphor node, while the variable local-domain-flag is set to FALSE. Then Algorithm 1 enters Step 2, where it scans the anaphor's siblings. Once present-sibling is set to the PP node, ptantecedentQ)resent-sibling)=FALSE. At this point, Algorithm 1 exits Step 2 and enters Step 3, where present-node is set to the value of father(present-node), in this case, the V' node. Given that local-domain.flag=FALSE, Algorithm 1 enters Step 2 again. We do not follow the whole computation but directly skip to the point where Algorithm 1 enters Step 2 with present- null node=I'. The only sibling of I' is the subject NP and, setting present-sibling to it, one has that ptantecedent(present-sibling)=TRUE. Algorithm 1 then sets local-domain-flag to TRUE; furthermore, agreement(present-sibling, present-node)=TRUE, so that the value of present-sibling is output. After that, Algorithm 1 enters Step 3 and sets present-node to father(present-node), i.e. to the topmost (IP) node. Now local-domain-flag=TRUE and domain(present-node, input-node)=TRUE (i.e. IP is the local domain for the input anaphor); therefore Algorithm 1 enters Step 4 and then stops.</Paragraph>
  </Section>
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