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<?xml version="1.0" standalone="yes"?> <Paper uid="P90-1014"> <Title>Free Indexation: Combinatorial Analysis and A Compositional Algorithm*</Title> <Section position="3" start_page="0" end_page="105" type="metho"> <SectionTitle> 2 Free Indexation </SectionTitle> <Paragraph position="0"> Consider the ambiguous sentence: (1) John believes Bill will identify him *The author would like to acknowledge Eric S. Ristad, whose interaction helped to motivate much of the analysis in this paper. Also, Robert C. Berwick, Michael B. Kashket, and Tanveer Syeda provided many useful comments on earlier drafts. This work is supported by an IBM Graduate Fellowship.</Paragraph> <Paragraph position="1"> In (1), the pronominal &quot;him&quot; can be interpreted as being coreferential with &quot;John&quot;, or with some other person not named in (1), but not with &quot;Bill&quot;. We can represent these various cases by assigning indices to all noun phrases in a sentence together with the interpretation that two noun phrases are coreferential if and only if they are coindexed, that is, if they have the same index. Hence the following indexings represent the three coreference options for pronominal &quot;him&quot; :1 (2) a. John1 believes Bill2 will identify him1 b. John1 believes Bill2 will identify him3 c. *John1 believes Bills will identify him2 In the principles-and-parameters framework (Chomsky \[3\]), once indices have been assigned, general principles that state constraints on the locality of reference of pronominals and names (e.g.</Paragraph> <Paragraph position="2"> &quot;John&quot; and &quot;Bill&quot;) will conspire to rule out the impossible interpretation (2c) while, at the same time, allow the other two (valid) interpretations.</Paragraph> <Paragraph position="3"> The process of assigning indices to noun phrases is known as &quot;free indexation,&quot; which has the following general form: (4) Assign indices freely to all noun phrases? In such theories, free indexation accounts for the fact that we have coreferential ambiguities in language. Other principles interact so as to limit the 1Note that the indexing mechanism used above is too simplistic a framework to handle binding examples involving inclusion of reference such as: (3) a. We1 think that I1 will win b. We1 think that Is will win c. *We1 like myself 1 d. John told Bill that they should leave Richer schemes that address some of these problems, for example, by representing indices as sets of numbers, have been proposed. See Lasnik \[9\] for a discussion on the limitations of, and alternatives to, simple indexation. Also, Higginbotham \[7\] has argued against coindexation (a symmetric relation), and in favour of directed links between elements (linking theory). In general, there will be twice as many possible 'linkings' as indexings for a given structure. However, note that the asymptotic results of Section 3 obtained for free indexation will also hold for linking theory.</Paragraph> <Paragraph position="4"> number of indexings generated by free indexation to those that are semantically well-formed.</Paragraph> <Paragraph position="5"> In theory, since the indices are drawn from the set of natural numbers, there exists an infinite number of possible indexings for any sentence.</Paragraph> <Paragraph position="6"> However, we are only interested in those indexings that are distinct with respect to semantic interpretation. Since the interpretation of indices is concerned only with the equality (and inequality) of indices, there are only a finite number of semantically different indexings. 3 For example, &quot;John1 likes Mary2&quot; and &quot;John23 likes Mary4&quot; are considered to be equivalent indexings. Note that the definition in (4) implies that &quot;John believes Bill will identify him&quot; has two other indexings (in addition to those in (2)): (5) a. *John1 believes Bill1 will identify him1 b. *John1 believes Bill1 will identify him2 subsets. For example, a set of four elements {w, x, y, z} can be partitioned into two subsets in the following seven ways: {w, z}{y} {w, y, y} y, z){w} The number of partitions obtained thus is usually represented using the notation {~} (Knuth \[8\]). In general, the number of ways of partitioning n elements into m sets is given by the following formula. (See Purdom & Brown \[10\] for a discussion of (6).)</Paragraph> <Paragraph position="8"> In some versions of the theory, indices are only freely assigned to those noun phrases that have not been coindexed through a rule of movement (Move-a). (see Chomsky \[3\] (pg.331)). For example, in &quot;Who1 did John see \[NPt\]l?&quot;, the rule of movement effectively stipulates that &quot;Who&quot; and its trace noun phrase must be coreferential. In particular, this implies that free indexation must not assign different indices to &quot;who&quot; and its trace element. For the purposes of free indexation, we can essentially 'collapse' these two noun phrases, and treat them as if they were only one. Hence, this structure contains only two independent noun phrases. 4</Paragraph> </Section> <Section position="4" start_page="105" end_page="106" type="metho"> <SectionTitle> 3 The Combinatorics of </SectionTitle> <Paragraph position="0"> Free Indexation ........</Paragraph> <Paragraph position="1"> In this section, we show that free indexation generates an exponential number of indexings in the number of independent noun phrases in a phrase structure. We achieve this result by observing that the problem of free indexation can be expressed in terms of a well-known combinatorial partitioning problem.</Paragraph> <Paragraph position="2"> Consider the general problem of partitioning a set of n elements into m non-empty (disjoint) versions of the theory. For example, in Chomsky \[4\] (pg.59), free indexation is restricted to apply to A-positions at the level of S-structure, and to A-positions at the level of logical form.</Paragraph> <Paragraph position="3"> ZIn other words, there are only a finite number of equivalence classes on the relation 'same core\[erence relatlons hold.' This can easily be shown by induction on the number of indexed elements.</Paragraph> <Paragraph position="4"> 4TechnicaJly, &quot;who&quot; and its trace are said to form a chain. Hence, the structure in question contains two distinct chains.</Paragraph> <Paragraph position="5"> for n,m > 0 The number of ways of partitioning n elements into zero sets, {o}, is defined to be zero for n > 0 and one when n = 0. Similarly, {,no}, the number of ways of partitioning zero elements into m sets is zero for m > 0 and one when m = 0.</Paragraph> <Paragraph position="6"> We observe that the problem of free indexation may be expressed as the problem of assigning 1, 2,... ,n distinct indices to n noun phrases where n is the number of noun phrases in a sentence. Now, the general problem of assigning m distinct indices to n noun phrases is isomorphic to the problem of partitioning n elements into m non-empty disjoint subsets. The correspondence here is that each partitioned subset represents a set of noun phrases with the same index. Hence, the number of indexings for a sentence with n noun phrases is:</Paragraph> <Paragraph position="8"> Bell's Exponential Number B.; see Berge \[2\].) The recurrence relation in (6) has the following solution (Abramowitz \[1\]): (8) Using (8), we can obtain a finite summation form for the number of indexings:</Paragraph> <Paragraph position="10"> It can also be shown (Graham \[6\]) that Bn is asymptotically equal to (10):</Paragraph> <Paragraph position="12"> where the quantity mn is given by: (11) 1 mn In mn= n - - 2 That is, (10) is both an upper and lower bound on the number of indexings. More concretely, to provide some idea of how fast the number of possible indexings increases with the number of noun phrases in a phrase structure, the following table exhibits the values of (9) for the first dozen values</Paragraph> </Section> <Section position="5" start_page="106" end_page="108" type="metho"> <SectionTitle> 4 A Compositional Algorithm </SectionTitle> <Paragraph position="0"> In this section, we will define a compositional algorithm for freeindexation that provably enumerates all and only all the possible indexings predicted by the analysis of the previous section.</Paragraph> <Paragraph position="1"> The PO-PARSER is a parser based on a principles-and-parameters framework with a uniquely flexible architecture (\[5\]). In this parser, linguistic principles such as free indexation may be applied either incrementally as bottom-up phrase structure construction proceeds, or as a separate operation after the complete phrase structure for a sentence is recovered. The PO-PARSER was designed primarily as a tool for exploring how to organize linguistic principles for efficient processing. This freedom in principle application allows one to experiment with a wide variety of parser configurations.</Paragraph> <Paragraph position="2"> Perhaps the most obvious algorithm for free indexation is, first, to simply collect all noun phrases occurring in a sentence into a list. Then, it is easy to obtain all the possible indexing combinations by taking each element in the list in turn, and optionally coindexing it with each element following it in the list. This simple scheme produces each possible indexing without any duplicates and works well in the case where free indexing applies after structure building has been completed.</Paragraph> <Paragraph position="3"> The problem with the above scheme is that it is not flexible enough to deal with the case when free indexing is to be interleaved with phrase structure construction. Conceivably, one could repeatedly apply the algorithm to avoid missing possible indexings. However, this is very inefficient, that is, it involves much duplication of effort. Moreover, it may be necessary to introduce extra machinery to keep track of each assignment of indices in order to avoid the problem of producing duplicate indexings. Another alternative is to simply delay the operation until all noun phrases in the sentence have been parsed. (This is basically the same arrangement as in the non-interleaved case.) Unfortunately, this effectively blocks the interleaved application of other principles that are logically dependent on free indexation to assign indices. For example, this means that principles that deal with locality restrictions on the binding of anaphors and pronominals cannot be interleaved with structure building (despite the fact that these particular parser operations can be effectively interleaved).</Paragraph> <Paragraph position="4"> An algorithm for free indexation that is defined compositionally on phrase structures can be effectively interleaved. That is, free indexing should be defined so that the indexings for a phrase is some function of the indexings of its sub-constituents.</Paragraph> <Paragraph position="5"> Then, coindexings can be computed incrementally for all individual phrases as they are built. Of course, a compositional algorithm can also be used in the non-interleaved case.</Paragraph> <Paragraph position="6"> Basically, the algorithm works by maintaining a set of indices at each sub-phrase of a parse tree. 5 Each index set for a phrase represents the range of indices present in that phrase. For example, &quot;Whoi did Johnj see tiT' has the phrase structure and index sets shown in Figure 1.</Paragraph> <Paragraph position="7"> There are two separate tasks to be performed whenever two (or more) phrases combine to form a larger phrase, s First, we must account for the possibility that elements in one phrase could be coindexed (cross-indexed) with elements from the other phrase. This is accomplished by allowing indices from one set to be (optionally) merged with distinct indices from the other set. For example, the phrases &quot;\[NpJohni\]&quot; and &quot;\[vP likes himj\]&quot; have index sets {i} and {j}, respectively. Free indexation must allow for the possibilities that &quot;John&quot; and &quot;him&quot; could be coindexed or maintain distinct indices. Cross-indexing accounts for dices. The actual algorithm keeps track of additional information, such as agreement features like person, number and gender, associated with each index. For example, irrespective of configuration, &quot;Mary&quot; and &quot;him&quot; can never have the same index.</Paragraph> <Paragraph position="8"> \[cP \[NP who/\] \[~- did \[IP \[NP Johnj\] \[vP see \[NP tdl\]\]\] {i,j} {i} {/,j} {i,j} {j} {i} {/} Figure 1 Index sets for &quot;Who did John see?&quot; b. Johni likes himj, i not merged with j Secondly, we must find the index set of the aggregate phrase. This is just the set union of the index sets of its sub-phrases after cross-indexation. In the example, &quot;John likes him&quot;, (12a) and (125) have index sets {i} and {i, j}.</Paragraph> <Paragraph position="9"> More precisely, let Ip be the set of all indices associated with the Binding Theory-relevant elements in phrase P. Assume, without loss of generality, that phrase structures are binary branching. 7 Consider a phrase P = Iv X Y\] with immediate constituents X and Y. Then: 1. Cross Indexing: Let fx represent those elements of Ix which are not also members of Iv, that is, (Ix -Iv). Similarly, let iv be (Iv - Ix). s (a) If either ix or fr are empty sets, then done.</Paragraph> <Paragraph position="10"> (b) Let x and y be members of ix and fy, respectively.</Paragraph> <Paragraph position="11"> (c) Eifher merge indices z and y or do nothing. null (d) Repeat from step (la) with ix_ - {z} in place of ix. Replace Ir with Iv - {y} if and y have been merged.</Paragraph> <Paragraph position="12"> 2. Index Set Propagation: Ip = Ix O Iv.</Paragraph> <Paragraph position="13"> The nondeterminism in step (lc) of crossindexing will generate all and only all (i.e. without duplicates) the possible indexings. We will show this in two parts. First, we will argue that eSome reaPSlers may realize that the algorithm must have an additional step in cases where the larger phrase itself may be indexed, for instance, as in \[NPi\[NP, John's \] mother\]. In such cases, the third step is slCmply to merge the singleton set consisting of the index of the larger phrase with the result of crossindexing in the first step. (For the above example, the extra step is to just merge {i} with {j}.) For expository reasons, we will ignore such cases. Note that no loss of generality is implied since a structure of the form \[NPI \[NPj... ~.. -\]... ~...\] can be can always be handled as \[P1 \[NPi\]\[P2\[NPj... oC/...\].../~...\]\]. rThe algorithm generalizes to n-ary branching using iteration. For example, a ternary branching structure such as \[p X Y Z\] would be handled in the same way as \[p X\[p, Y Z\]\].</Paragraph> <Paragraph position="14"> SNote that ix and iv are defined purely for notational convenience. That is, the algorithm directly operates on the elements of Ix and Iy.</Paragraph> <Paragraph position="16"> the above algorithm cannot generate duplicate indexings: That is, the algorithm only generates distinct indexings with respect to the interpretation of indices. As shown in the previous section, the combinatorics of free-indexlng indicates that there are only B, possible indexings. Next, we will demonstrate that the algorithm generates ex- null actly that number of indexings. If the algorithm satisfies both of these conditions, then we have proved that it generates all the possible indexings exactly once.</Paragraph> <Paragraph position="17"> 1. Consider the definition of cross-indexing, ix represents those indices in X that do not appear in Y. (Similarly for iv.) Also, whenever two indices are merged in step (lb), they are 'removed' from ix and iv before the next iteration. Thus, in each iteration, z and y from step (lb) are 'new' indices that have not been merged with each other in a previous iteration. By induction on tree structures, it is easy to see that two distinct indices cannot be merged with each other more than once.</Paragraph> <Paragraph position="18"> Hence, the algorithm cannot generate duplicate indexings.</Paragraph> <Paragraph position="19"> 2. We now demonstrate why the algorithm generates exactly the correct number of indexings by means of a simple example. Without loss of generality, consider the right-branching phrase scheme shown in Figure 2.</Paragraph> <Paragraph position="20"> Now consider the decision tree shown in Figure 3 for computing the possible indexings of the right-branching tree in a bottom-up fashion. null Each node in the tree represents the index set of the combined phrase depending on whether the noun phrase at the same level is cross- null indexed or not. For example, {i} and {i, j} on the level corresponding to NPj are the two possible index sets for the phrase Pij. The path from the root to an index set contains arcs indicating what choices (either to coindex or to leave free) must have been made in order to build that index set. Next, let us just consider the cardinality of the index sets in the decision tree, and expand the tree one more level (for NP~) as shown in Figure 4.</Paragraph> <Paragraph position="21"> Informally speaking, observe that each decision tree node of cardinality i 'generates' i child nodes of cardinality i plus one child node of cardinality i + 1. Thus, at any given level, if the number of nodes of cardinality m is cm, and the number of nodes of cardinality m- 1 is c,,-1, then at the next level down, there will be mcm + c,n-1 nodes of cardinality m.</Paragraph> <Paragraph position="22"> Let c(n,m) denote the number of nodes at level n with cardinality m. Let the top level of the decision tree be level 1. Then: (13) c(n+l, re+l) = c(n, m)+(m+l)c(n, re+l) Observe that this recurrence relation has the same form as equation (6). Hence the algorithm generates exactly the same number of indexings as demanded by combinatorial analysis.</Paragraph> </Section> class="xml-element"></Paper>