<?xml version="1.0" standalone="yes"?>
<Paper uid="P95-1013">
  <Title>Compilation of HPSG to TAG*</Title>
  <Section position="4" start_page="93" end_page="93" type="metho">
    <SectionTitle>
&amp;quot;N-L \[SL,SH E\]\]
</SectionTitle>
    <Paragraph position="0"/>
    <Paragraph position="2"> Finally, we assume that rule schemata and principles have been compiled together (automatically or manually) to yield more specific subtypes of the schemata. This does not involve a loss of generalization but simply means a further refinement of the type hierarchy. LP constraints could be compiled out beforehand or during the compilation of TAG structures, since the algorithm is lexicon driven.</Paragraph>
  </Section>
  <Section position="5" start_page="93" end_page="98" type="metho">
    <SectionTitle>
3 Algorithm
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="93" end_page="94" type="sub_section">
      <SectionTitle>
3.1 Basic Idea
</SectionTitle>
      <Paragraph position="0"> While in TAG all arguments related to a particular functor are represented in one elementary tree structure, the 'functional application' in HPSG is distributed over the phrasal schemata, each of which can be viewed as a partial description of a local tree.</Paragraph>
      <Paragraph position="1"> Therefore we have to identify which constituents in aWe choose such a lexicalized approach, because it will allow us to maintain a restriction that every TAG tree resulting from the compilation must be rooted in a non-emtpy lexical item. The approach will account for extraction of complements out of complements, i.e., along paths corresponding to chains of government relations. null As far as we can see, the only limitation arising from the percolation of SLASH only along head-projections is on extraction out of adjuncts, which may be desirable for some languages like English. On the other hand, these constructions would have to be treated by multi-component TAGs, which axe not covered by the intended interpretation of the compilation algorithm anyway.</Paragraph>
      <Paragraph position="2">  a phrasal schema count as functors and arguments.</Paragraph>
      <Paragraph position="3"> In TAG different functor argument relations, such as head-complement, head-modifier etc., are represented in the same format as branches of a trunk projected from a lexical anchor. As mentioned, this anchor is not always equivalent to the HPSG notion of a head; in a tree projected from a modifier, for example, a non-head (ADJUNCT-DTR) counts as a functor. We therefore have to generalize over different types of daughters in HPSG and define a general notion of a functor. We compute the functor-argument structure on the basis of a general selection relation. Following (Kas92) 4, we adopt the notion of a selector daughter (SD), which contains a selector feature (SF) whose value constrains the argument (or non-selector) daughter (non-SD)) For example, in a head-complement structure, the SD is the HEAD-DTR, as it contains the list-valued feature coMPs (the SF) each of whose elements selects a C0m~-DTR, i.e., an element of the CoMPs list is identified with the SYNSE~4 value of a COMP-DTR.</Paragraph>
      <Paragraph position="4"> We assume that a reduction takes place along with selection. Informally, this means that if F is the selector feature for some schema, then the value (or the element(s) in the list-value) of 1: that selects the non-SD(s) is not contained in the F value of the mother node. In case F is list-valued, we-assume that the rest of the elements in the list (those that did not select any daughter) are also contained in the F at the mother node. Thus we say that F has been reduced by the schema in question.</Paragraph>
      <Paragraph position="5"> The compilation algorithm assumes that all HPSG schemata will satisfy the condition of simultaneous selection and reduction, and that each schema reduces at least one SF. For the head-complement- and head-subject-schema, these conditions follow from the Valence Principle, and the SFs are coMPs and SUBJ, respectively. For the headadjunct-schema, the ADJUNCT-DTR is the SD, because it selects the HEAD-DTR by its NOD feature. The NOD feature is reduced, because it is a head feature, whose value is inherited only from the HEAD-DTR and not from the ADJUNCT-DTR. Finally, for the filler-headschema, the HEAD-DTR is the SD, as it selects the FILLER-DTR by its SLASH value, which is bound off, not inherited by the mother, and therefore reduced.</Paragraph>
      <Paragraph position="6"> We now give a general description of the compilation process. Essentially, we begin with a lexical de4The algorithm presented here extends and refines the approach described by (Kas92) by stating more precise criteria for the projection of features, for the termination of the algorithm, and for the determination of those structures which should actually be used as elementary trees.</Paragraph>
      <Paragraph position="7"> 5Note that there might be mutual selection (as in the case of the specifier-head-relations proposed in (PS94)\[44ff\]). If there is mutual selection, we have to stipulate one of the daughters as the SD. The choice made would not effect the correctness of the compilation. scription and project phrases by using the schemata to reduce the selection information specified by the lexical type.</Paragraph>
      <Paragraph position="8"> Basic Algorithm Take a lexical type L and initialize by creating a node with this type. Add a node n dominating this node.</Paragraph>
      <Paragraph position="9"> For any schema S in which specified SFs of n are reduced, try to instantiate S with n corresponding to the SD of S. Add another node m dominating the root node of the instantiated schema. (The domination links are introduced to allow for the possibility of adjoining.) Repeat this step (each time with n as the root node of the tree) until no further reduction is possible.</Paragraph>
      <Paragraph position="10"> We will fill in the details below in the following order: what information to raise across domination links (where adjoining may take place), how to determine auxiliary trees (and foot nodes), and when to terminate the projection.</Paragraph>
      <Paragraph position="11"> We note that the trees produced have a trunk leading from the lexical anchor (node for the given lexical type) to the root. The nodes that are siblings of nodes on the trunk, the selected daughters, are not elaborated further and serve either as foot nodes or substitution nodes.</Paragraph>
    </Section>
    <Section position="2" start_page="94" end_page="95" type="sub_section">
      <SectionTitle>
3.2 Raising Features Across Domination
Links
</SectionTitle>
      <Paragraph position="0"> Quite obviously, we must raise the SFs across domination links, since they determine the applicability of a schema and licence the instantiation of an SD.</Paragraph>
      <Paragraph position="1"> If no SF were raised, we would lose all information about the saturation status of a functor, and the algorithm would terminate after the first iteration.</Paragraph>
      <Paragraph position="2"> There is a danger in raising more than the SFs.</Paragraph>
      <Paragraph position="3"> For example, the head-subject-schema in German would typically constrain a verbal head to be finite.</Paragraph>
      <Paragraph position="4"> Raising HEAD features would block its application to non-finite verbs and we would not produce the trees required for raising-verb adjunction. This is again because heads in HPSG are not equivalent to lexical anchors in TAG, and that other local properties of the top and bottom of a domination link could differ. Therefore HEAD features and other LOCAL features cannot, in general, be raised across domination links, and we assume for now that only the SFs are raised.</Paragraph>
      <Paragraph position="5"> Raising all SFs produces only fully saturated elementary trees and would require the root and foot of any auxiliary tree to share all SFs, in order to be compatible with the SF values across any domination links where adjoining can take place. This is too strong a condition and will not allow the resulting TAG to generate all the trees derivable with the given HPSG (e.g., it would not allow unsaturated VP complements). In SS 3.5 we address this concern by  using a multi-phase compilation. In the first phase, we raise all the SFs.</Paragraph>
    </Section>
    <Section position="3" start_page="95" end_page="95" type="sub_section">
      <SectionTitle>
3.3 Detecting Auxiliary Trees and Foot
Nodes
</SectionTitle>
      <Paragraph position="0"> Traditionally, in TAG, auxiliary trees are said to be minimal recursive structures that have a foot node (at the frontier) labelled identical to the root. As such category labels (S, NP etc.) determine where an auxiliary tree can be adjoined, we can informally think of these labels as providing selection information corresponding to the SFs of HPSG. Factoring of recursion can then be viewed as saying that auxiliary trees define a path (called the spine) from the root to the foot where the nodes at extremities have the same selection information. However, a closer look at TAG shows that this is an oversimplification. If we take into account the adjoining constraints (or the top and bottom feature structures), then it appears that the root and foot share only some selection information.</Paragraph>
      <Paragraph position="1"> Although the encoding of selection information by SFs in HPSG is somewhat different than that traditionally employed in TAG, we also adopt the notion that the extremities of the spine in an auxiliary tree share some part (but not necessarily all) of the selection information. Thus, once we have produced a tree, we examine the root and the nodes in its frontier. A tree is an auxiliary tree if the root and some frontier node (which becomes the foot node) have some non-empty SF value in common. Initial trees are those that have no such frontier nodes.</Paragraph>
      <Paragraph position="2">  In the trees shown, nodes detected as foot nodes are marked with *. Because of the SUBJ and SLASH values, the HEAD-DTR is the foot of T2 below (anchored by an adverb) and COMP-DTR is the foot of T3 (anchored by a raising verb). Note that in the tree T1 anchored by an equi-verb, the foot node is detected because the SLASH value is shared, although the SUBJ is not. As mentioned, we assume that bridge verbs, i.e., verbs which allow extraction out of their complements, share their SLASH value with their clausal complement.</Paragraph>
    </Section>
    <Section position="4" start_page="95" end_page="95" type="sub_section">
      <SectionTitle>
3.4 Termination
</SectionTitle>
      <Paragraph position="0"> Returning to the basic algorithm, we will now consider the issue of termination, i.e., how much do we need to reduce as we project a tree from a lexical item.</Paragraph>
      <Paragraph position="1"> Normally, we expect a SF with a specified value to be reduced fully to an empty list by a series of applications of rule schemata. However, note that the SLASH value is unspecified at the root of the trees T2 and T3. Of course, such nodes would still unify with the SD of the filler-head-schema (which reduces SLASH), but applying this schema could lead to an infinite recursion. Applying a reduction to an unspecified SF is also linguistically unmotivated as it would imply that a functor could be applied to an argument that it never explicitly selected.</Paragraph>
      <Paragraph position="2"> However, simply blocking the reduction of a SF whenever its value is unspecified isn't sufficient. For example, the root of T2 specifies the subs to be a non-empty list. Intuitively, it would not be appropriate to reduce it further, because the lexical anchor (adverb) doesn't semantically license the SUBJ argument itself. It merely constrains the modified head to have an unsaturated SUBS.</Paragraph>
      <Paragraph position="3">  To motivate our termination criterion, consider the adverb tree and the asterisked node (whose SLASH value is shared with SLASH at the root). Being a non-trunk node, it will either be a foot or a substitution node. In either case, it will eventually be unified with some node in another tree. If that other node has a reducible SLASH value, then we know that the reduction takes place in the other tree, because the SLASH value must have been raised across the domination link where adjoining takes place. As the same SLASH (and likewise suB J) value should not be reduced in both trees, we state our termination criteria as follows: Termination Criterion The value of an SF F at the root node of a tree is not reduced further if it is an empty list, or if it is shared with the value of F at some non-trunk node in the frontier.</Paragraph>
      <Paragraph position="4"> Note that because of this termination criterion, the adverb tree projection will stop at this point. As the root shares some selector feature values (SLASH and SUB J) with a frontier node, this node becomes the foot node. As observed above, adjoining this tree will preserve these values across any domination links where it might be adjoined; and if the values stated there are reducible then they will be reduced in the other tree. While auxiliary trees allow arguments selected at the root to be realized elsewhere, it is never the case for initial trees that an argument selected at the root can be realized elsewhere, because by our definition of initial trees the selection of arguments is not passed on to a node in the frontier.</Paragraph>
      <Paragraph position="5"> We also obtain from this criterion a notion of local completeness. A tree is locally complete as soon as all arguments which it licenses and which are not licensed elsewhere are realized. Global completeness is guaranteed because the notion of &amp;quot;elsewhere&amp;quot; is only and always defined for auxiliary trees, which have to adjoin into an initial tree.</Paragraph>
    </Section>
    <Section position="5" start_page="95" end_page="95" type="sub_section">
      <SectionTitle>
3.5 Additional Phases
</SectionTitle>
      <Paragraph position="0"> Above, we noted that the preservation of some SFs along a path (realized as a path from the root to the foot of an auxiliary tree) does not imply that all SFs need to be preserved along that path. Tree T1 provides such an example, where a lexical item, an equi-verb, triggers the reduction of an SF by taking a complement that is unsaturated for SUBJ but never shares this value with one of its own SF values.</Paragraph>
      <Paragraph position="1"> To allow for adjoining of auxiliary trees whose root and foot differ in their SFs, we could produce a number of different trees representing partial projections from each lexical anchor. Each partial projection could be produced by raising some subset of SFs across each domination link, instead of raising all SFs. However, instead of systematically raising all possible subsets of SFs across domination links, we can avoid producing a vast number of these partial projections by using auxiliary trees to provide guidance in determining when we need to raise only a particular subset of the SFs.</Paragraph>
      <Paragraph position="2"> Consider T1 whose root and foot differ in their SFs. From this we can infer that a SUBJ SF should not always be raised across domination links in the trees compiled from this grammar. However, it is only useful to produce a tree in which the susJ value is not raised when the bottom of a domination link has both a one element list as value for SUBJ and an empty COMPS list. Having an empty SUBJ list at the top of the domination link would then allow for adjunction by trees such as T1.</Paragraph>
      <Paragraph position="3"> This leads to the following multi-phase compilation algorithm. In the first phase, all SFs are raised. It is determined which trees are auxiliary trees, and then the relationships between the SFs associated with the root and foot in these auxiliary trees are recorded. The second phase begins with lexical types and considers the application of sequences of rule schemata as before. However, immediately after applying a rule schema, the features at the bottom of a domination link are compared with the foot nodes of auxiliary trees that have differing SFs at foot and root. Whenever the features are compatible with such a foot node, the SFs are raised according to the relationship between the root and foot of the auxiliary tree in question. This process may need to be iterated based on any new auxiliary trees produced in the last phase.</Paragraph>
    </Section>
    <Section position="6" start_page="95" end_page="97" type="sub_section">
      <SectionTitle>
3.6 Example Derivation
</SectionTitle>
      <Paragraph position="0"> In the following we provide a sample derivation for the sentence (I know) what Kim wants to give to Sandy. Most of the relevant HPSG rule schemata and lexical entries necessary to derive this sentence were already given above. For the noun phrases what, Kim and Sandy, and the preposition to no special assumptions are made. We therefore only add the entry for the ditransitive verb give, which we take to subcategorize for a subject and two object complements. null</Paragraph>
    </Section>
    <Section position="7" start_page="97" end_page="98" type="sub_section">
      <SectionTitle>
Ditransitive Verb
</SectionTitle>
      <Paragraph position="0"> L cdegMPS imp\[ \]pp\[ 1) From this lexical entry, we can derive in the first phase a fully saturated initial tree by applying first the lexical slash-termination rule, and then the head-complement-, head-subject and filler-headrule. Substitution at the nodes on the frontier would yield the string what Kim gives to Sandy.</Paragraph>
      <Paragraph position="1">  The derivations for the trees for the matrix verb want and for the infinitival marker to (equivalent to a raising verb) were given above in the examples T1 and T3. Note that the suBJ feature is only reduced in the former, but not in the latter structure.</Paragraph>
      <Paragraph position="2"> In the second phase we derive from the entry for give another initial tree (Ts) into which the auxiliary tree T1 for want can be adjoined at the topmost domination link. We also produce a second tree with similar properties for the infinitive marker to (T6).  By first adjoining the tree T6 at the topmost domination link of T5 we obtain a structure T7 corresponding to the substring what ... to give to Sandy. Adjunction involves the identification of the foot node with the bottom of the domination link and identification of the root with top of the domination link. Since the domination link at the root of the adjoined tree mirrors the properties of the adjunction site in the initial tree, the properties of the domination link are preserved.</Paragraph>
      <Paragraph position="3">  The final derivation step then involves the adjunction of the tree for the equi verb into this tree, again at the topmost domination link. This has the effect of inserting the substring Kim wants into what ... to give to Sandy.</Paragraph>
    </Section>
  </Section>
class="xml-element"></Paper>