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<Paper uid="P06-1052">
  <Title>Sydney, July 2006. c(c)2006 Association for Computational Linguistics An Improved Redundancy Elimination Algorithm for Underspecified Representations</Title>
  <Section position="4" start_page="409" end_page="411" type="metho">
    <SectionTitle>
2 Dominance graphs
</SectionTitle>
    <Paragraph position="0"> The basic underspecification formalism we assume here is that of (labelled) dominance graphs (Althaus et al., 2003). Dominance graphs are equivalent to leaf-labelled normal dominance constraints (Egg et al., 2001), which have been discussed extensively in previous literature.</Paragraph>
    <Paragraph position="1">  Definition 1. A (compact) dominance graph is a directed graph (V,EunionmultiD) with two kinds of edges, tree edges E and dominance edges D, such that: 1. The graph (V,E) defines a collection of node disjoint trees of height 0 or 1. We call the trees in (V,E) the fragments of the graph.</Paragraph>
    <Paragraph position="2"> 2. If (v,vprime) is a dominance edge in D, then v is  a hole and vprime is a root. A node v is a root if v doesnothaveincomingtreeedges;otherwise, v is a hole.</Paragraph>
    <Paragraph position="3"> A labelled dominance graph over a ranked signature S is a triple G = (V,E unionmultiD,L) such that (V,E unionmultiD) is a dominance graph and L : V squigglerightS is a partial labelling function which assigns a node v a label with arity n iff v is a root with n outgoing tree edges. Nodes without labels (i.e. holes) must have outgoing dominance edges.</Paragraph>
    <Paragraph position="4"> We will write R(F) for the root of the fragment F, and we will typically just say &amp;quot;graph&amp;quot; instead of &amp;quot;labelled dominance graph&amp;quot;.</Paragraph>
    <Paragraph position="5"> An example of a labelled dominance graph is shown to the left of Fig. 1. Tree edges are drawn as solid lines, and dominance edges as dotted lines, directed from top to bottom. This graph can serve as an USR for the sentence &amp;quot;a representative of a company saw a sample&amp;quot; if we demand that the holes are &amp;quot;plugged&amp;quot; by roots while realising the dominance edges as dominance, as in the two configurations (of five) shown to the right. These configurationsaretreesthatencodesemanticrepresen- null tations of the sentence. We will freely read configurations as ground terms over the signature S.</Paragraph>
    <Section position="1" start_page="409" end_page="410" type="sub_section">
      <SectionTitle>
2.1 Hypernormally connected graphs
</SectionTitle>
      <Paragraph position="0"> Throughout this paper, we will only consider hypernormally connected (hnc) dominance graphs.</Paragraph>
      <Paragraph position="1"> Hnc graphs are equivalent to chain-connected dominance constraints (Koller et al., 2003), and areclosely relatedto dominance nets (Niehrenand Thater, 2003). Fuchss et al. (2004) have presented a corpus study that strongly suggests that all dominance graphs that are generated by current large-scale grammars are (or should be) hnc.</Paragraph>
      <Paragraph position="2"> Technically, a graph G is hypernormally connected iff each pair of nodes is connected by a simple hypernormal path in G. A hypernormal path (Althaus et al., 2003) in G is a path in the undirected version Gu of G that does not use two dominance edges that are incident to the same hole.</Paragraph>
      <Paragraph position="3"> Hnc graphs have a number of very useful structural properties on which this paper rests. One which is particularly relevant here is that we can predict in which way different fragments can dominate each other.</Paragraph>
      <Paragraph position="4"> Definition 2. Let G be a hnc dominance graph. A fragment F1 in G is called a possible dominator of another fragment F2 in G iff it has exactly one hole h which is connected to R(F2) by a simple hy-</Paragraph>
      <Paragraph position="6"> pernormal path which doesn't use R(F1). We write ch(F1,F2) for this unique h.</Paragraph>
      <Paragraph position="7"> Lemma 1 (Koller and Thater (2006)). Let F1, F2 be fragments in a hnc dominance graph G. If there isaconfigurationC ofGinwhichR(F1)dominates R(F2), then F1 is a possible dominator of F2, and in particular ch(F1,F2) dominates R(F2) inC.</Paragraph>
      <Paragraph position="8"> By applying this rather abstract result, we can derive a number of interesting facts about the example graph in Fig. 1. The fragments 1, 2, and 3 arepossibledominatorsofallotherfragments(and of each other), while the fragments 4 through 7 aren't possible dominators of anything (they have  noholes);so4through7mustbeleavesinanyconfiguration of the graph. In addition, if fragment 2 dominates fragment 3 in any configuration, then in particularthe right holeof2willdominatetheroot of 3; and so on.</Paragraph>
    </Section>
    <Section position="2" start_page="410" end_page="411" type="sub_section">
      <SectionTitle>
2.2 Dominance charts
</SectionTitle>
      <Paragraph position="0"> Below we will not work with dominance graphs directly. Rather, we will use dominance charts (Koller and Thater, 2005b) as our USRs: they are more explicit USRs, which support a more fine-grained deletion of reading sets than graphs.</Paragraph>
      <Paragraph position="1"> A dominance chart for the graph G is a mapping of weakly connected subgraphs of G to sets of splits (see Fig. 2), which describe possible ways of constructing configurations of the subgraph.</Paragraph>
      <Paragraph position="2"> A subgraph Gprime is assigned one split for each fragment F in Gprime which can be at the root of a configuration of Gprime. If the graph is hnc, removing F from the graph splits Gprime into a set of weakly connected components (wccs), each of which is connected to exactly one hole of F. We also record the wccs, and the hole to which each wcc belongs, in the split. In order to compute all configurations represented by a split, we can first compute recursively the configurations of each component; then we plug each combination of these sub-configurations into the appropriate holes of the root fragment. We define the configurations associated with a subgraph as the union over its splits, and those of the entire chart as the configurations associated with the complete graph.</Paragraph>
      <Paragraph position="3"> Fig. 2 shows the dominance chart corresponding to the graph in Fig. 1. The chart represents exactly the configuration set of the graph, and is minimal in the sense that every subgraph and every split in the chart can be used in constructing some configuration. Such charts can be computed efficiently (Koller and Thater, 2005b) from a dominance graph, and can also be used to compute the configurations of a graph efficiently.</Paragraph>
      <Paragraph position="4"> The example chart expresses that three fragments can be at the root of a configuration of the complete graph: 1, 2, and 3. The entry for the split with root fragment 2 tells us that removing 2 splits the graph into the subgraphs {1,4,5} and {3,6,7} (see Fig. 3). If we configure these two subgraphs recursively, we obtain the configurations shown in the third column of Fig. 3; we can then plug these sub-configurations into the appropriate holes of 2 and obtain a configuration for the entire graph.</Paragraph>
      <Paragraph position="5"> Notice that charts can be exponentially larger than the original graph, but they are still exponentially smaller than the entire set of readings because common subgraphs (such as the graph {2,5,7}intheexample)arerepresentedonlyonce,  and are small in practice (see (Koller and Thater, 2005b) for an analysis). Thus the chart can still serve as an underspecified representation.</Paragraph>
    </Section>
  </Section>
  <Section position="5" start_page="411" end_page="411" type="metho">
    <SectionTitle>
3 Equivalence
</SectionTitle>
    <Paragraph position="0"> Now let's define equivalence of readings more precisely. Equivalence of semantic representations is traditionally defined as the relation between formulas (say, of first-order logic) which have the same interpretation. However, even first-order equivalence is an undecidable problem, and broad-coverage semantic representations such as those computed by the ERG usually have no well-defined model-theoretic semantics and therefore no concept of semantic equivalence.</Paragraph>
    <Paragraph position="1"> On the other hand, we do not need to solve the full semantic equivalence problem, as we only want to compare formulas that are readings of the same sentence, i.e. different configurations of the same USR. Such formulas only differ in the way that the fragments are combined. We can therefore approximateequivalencebyusingarewritesystem that permutes fragments and defining equivalence of configurations as mutual rewritability as usual.</Paragraph>
    <Paragraph position="2"> By way of example, consider again the two configurations shown in Fig. 1. We can obtain the second configuration from the (semantically equivalent) first one by applying the following rewrite rule, which rotates the fragments 1 and 2:</Paragraph>
    <Paragraph position="4"> Thus we take these two configurations to be equivalent with respect to the rewrite rule. (We could also have argued that the second configuration can be rewritten into the first by using the inverted rule.) We formalise this rewriting-based notion of equivalence as follows. The definition uses the abbreviation x[1,k) for the sequence x1,...,xk[?]1, and x(k,n] for xk+1,...,xn.</Paragraph>
    <Paragraph position="5"> Definition 3. A permutation system R is a system of rewrite rules over the signature S of the following form: f1(x[1,i), f2(y[1,k),z,y(k,m]),x(i,n])f2(y[1,k), f1(x[1,i),z,x(i,n]),y(k,m]) The permutability relation P(R) is the binary relation P(R) [?] (SxN)2 which contains exactly the tuples ((f1,i),(f2,k)) and ((f2,k),(f1,i)) for each suchrewriterule.Twotermsareequivalent withrespect to R, s[?]R t, iff there is a sequence of rewrite steps and inverse rewrite steps that rewrite s into t. If G is a graph over S and R a permutation system, then we write SCR(G) for the set of equivalence classes Conf(G)/[?]R, where Conf(G) is the set of configurations of G.</Paragraph>
    <Paragraph position="6"> The rewrite rule (3) above is an instance of this schema, as are the other three permutations of existential quantifiers. These rules approximate classical semantic equivalence of first-order logic, as they rewrite formulas into classically equivalent ones. Indeed, all five configurations of the graph in Fig. 1 are rewriting-equivalent to each other.</Paragraph>
    <Paragraph position="7"> In the case of the semantic representations generated by the ERG, we don't have access to an underlying interpretation. But we can capture linguisticintuitionsabouttheequivalenceofreadings null in permutation rules. For instance, proper names and pronouns (which the ERG analyses as scopebearers, although they can be reduced to constants without scope) can be permuted with anything. Indefinites and definites permute with each other if they occur in each other's scope, but not if they occur in each other's restriction; and so on.</Paragraph>
  </Section>
  <Section position="6" start_page="411" end_page="413" type="metho">
    <SectionTitle>
4 Redundancy elimination
</SectionTitle>
    <Paragraph position="0"> Given a permutation system, we can now try to get ridofreadingsthatareequivalenttootherreadings.</Paragraph>
    <Paragraph position="1"> One way to formalise this is to enumerate exactly one representative of each equivalence class. However, after such a step we would be left with a collection of semantic representations rather than an USR, and could not use the USR for ruling out further readings. Besides, a naive algorithm which  first enumerates all configurations would be prohibitively slow.</Paragraph>
    <Paragraph position="2"> We will instead tackle the following underspecified redundancy elimination problem: Given an USR G, compute an USR Gprime with Conf(Gprime) [?] Conf(G) and SCR(G) = SCR(Gprime). We want Conf(Gprime) to be as small as possible. Ideally, it would contain no two equivalent readings, but in practice we won't always achieve this kind of completeness. Our redundancy elimination algorithm willoperateonadominancechartandsuccessively delete splits and subgraphs from the chart.</Paragraph>
    <Section position="1" start_page="412" end_page="412" type="sub_section">
      <SectionTitle>
4.1 Permutable fragments
</SectionTitle>
      <Paragraph position="0"> Because the algorithm must operate on USRs rather than configurations, it needs a way to predict from the USR alone which fragments can be permuted in configurations. This is not generally possible in unrestricted graphs, but for hnc graphs it is captured by the following criterion.</Paragraph>
      <Paragraph position="1"> Definition 4. Let R be a permutation system. Two fragments F1 and F2 with root labels f1 and f2 in a hnc graph G are called R-permutable iff they are possible dominators of each other and ((f1,ch(F1,F2)),(f2,ch(F2,F1)))[?]P(R).</Paragraph>
      <Paragraph position="2"> For example, in Fig. 1, the fragments 1 and 2 are permutable, and indeed they can be permuted in any configuration in which one is the parent of the other. This is true more generally: Lemma 2 (Koller and Thater (2006)). Let G be a hnc graph, F1 and F2 be R-permutable fragments with root labels f1 and f2, and C1 any configuration of G of the form C(f1(..., f2(...),...)) (where C is the context of the subterm). Then C1 can be R-rewritten into a tree C2 of the form C(f2(..., f1(...),...)) which is also a configuration of G.</Paragraph>
      <Paragraph position="3"> TheproofusesthehnconnectednessofGintwo ways: in order to ensure that C2 is still a configuration of G, and to make sure that F2 is plugged into the correct hole of F1 for a rule application (cf. Lemma 1). Note thatC2 [?]R C1 by definition.</Paragraph>
    </Section>
    <Section position="2" start_page="412" end_page="413" type="sub_section">
      <SectionTitle>
4.2 The redundancy elimination algorithm
</SectionTitle>
      <Paragraph position="0"> Now we can use permutability of fragments to define eliminable splits. Intuitively, a split of a subgraph G is eliminable if each of its configurations is equivalent to a configuration of some other split of G. Removing such a split from the chart will rule out some configurations; but it does not change the set of equivalence classes.</Paragraph>
      <Paragraph position="1"> Definition 5. Let R be a permutation system. A split S=(F,...,hi mapsto-Gi,...) of a graph G is called eliminable in a chartCh if some Gi contains a fragment Fprime such that (a) Ch contains a split Sprime of G with root fragment Fprime, and (b) Fprime is R-permutable with F and all possible dominators of Fprime in Gi.</Paragraph>
      <Paragraph position="2"> In Fig. 1, each of the three splits is eliminable.</Paragraph>
      <Paragraph position="3"> For example, the split with root fragment 1 is eliminable because the fragment 3 permutes both with 2 (which is the only possible dominator of 3 in the same wcc) and with 1 itself.</Paragraph>
      <Paragraph position="5"> Proof. Let C be an arbitrary configuration of S = (F,h1 mapsto- G1,...,hn mapsto- Gn), and let Fprime [?] Gi be the root fragment of the assumed second split Sprime.</Paragraph>
      <Paragraph position="6"> Let F1,...,Fn be those fragments in C that are properly dominated by F and properly dominate Fprime. All of these fragments must be possible dominators of Fprime, and all of them must be in Gi as well, so Fprime is permutable with each of them. Fprime must alsobepermutablewith F.Thismeansthatwecan apply Lemma 2 repeatedly to move Fprime to the root of the configuration, obtaining a configuration of Sprime which is equivalent toC.</Paragraph>
      <Paragraph position="7"> Notice that we didn't require that Ch must be the complete chart of a dominance graph. This means we can remove eliminable splits from a chart repeatedly, i.e. we can apply the following redundancy elimination algorithm:  Prop. 3 shows that the algorithm is a correct algorithm for the underspecified redundancy elimination problem. The particular order in which eliminable splits are removed doesn't affect the correctness of the algorithm, but it may change the number of remaining configurations.</Paragraph>
      <Paragraph position="8"> The algorithm generalises an earlier elimination algorithm (Koller and Thater, 2006) in that the earlier algorithm required the existence of a single split which could be used to establish eliminability of all other splits of the same subgraph.</Paragraph>
      <Paragraph position="9"> We can further optimise this algorithm by keeping track of how often each subgraph is referenced  by the splits in the chart. Once a reference count drops to zero, we can remove the entry for this subgraph and all of its splits from the chart. This doesn't change the set of configurations of the chart, but may further reduce the chart size. The overall runtime for the algorithm is O(n2S), where S is the number of splits in Ch and n is the number of nodes in the graph. This is asymptotically not much slower than the runtime O((n+m)S) it takes to compute the chart in the first place (where m is the number of edges in the graph).</Paragraph>
    </Section>
    <Section position="3" start_page="413" end_page="413" type="sub_section">
      <SectionTitle>
4.3 Examples and discussion
</SectionTitle>
      <Paragraph position="0"> Let's look at a run of the algorithm on the chart in Fig. 2. The algorithm can first delete the eliminablesplitwithroot1fortheentiregraphG.After null this deletion, the splits for G with root fragments 2 and 3 are still eliminable; so we can e.g. delete the split for 3. At this point, only one split is left for G. The last split for a subgraph can never be eliminable, so we are finished with the splits for G. This reduces the reference count of some sub-graphs (e.g. {2,3,5,6,7}) to 0, so we can remove thesesubgraphstoo.Theoutputofthealgorithmis the chart shown below, which represents a single configuration (the one shown in Fig. 3).</Paragraph>
      <Paragraph position="2"> In this case, the algorithm achieves complete reduction, in the sense that the final chart has no two equivalent configurations. It remains complete for all variations of the graph in Fig. 1 in which some or all existential quantifiers are replaces by universal quantifiers. This is an improvement over our earlier algorithm (Koller and Thater, 2006), which computed a chart with four configurations for the graph in which 1 and 2 are existential and 3 is universal, as opposed to the three equivalence classes of this graph's configurations.</Paragraph>
      <Paragraph position="3"> However, the present algorithm still doesn't achieve complete reduction for all USRs. One exampleisshowninFig.4.Thisgraphhassixconfig- null urations in four equivalence classes, but no split of the whole graph is eliminable. The algorithm will deleteasplitforthesubgraph{1,2,4,5,7},butthe final chart will still have five, rather than four, configurations. A complete algorithm would have to recognise that {1,3,4,6,7} and {2,3,5,6,7} have splits (for 1 and 2, respectively) that lead to equivalent configurations and delete one of them. But it is far from obvious how such a non-local decision could be made efficiently, and we leave this for future work.</Paragraph>
    </Section>
  </Section>
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