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<Paper uid="E95-1002">
  <Title>Principle Based Semantics for HPSG</Title>
  <Section position="3" start_page="10" end_page="12" type="metho">
    <SectionTitle>
3 UDRS Construction in HPSG
</SectionTitle>
    <Paragraph position="0"> In the following we will design a syntax-semantics interface for the construction of UDRSes in HPSG, focussing on the underspecified representation of scope and plural. To overcome the problems discussed in Section 2 we chose to depart from the semantics used in standard HPSG (Pollard/Sag 1994), and instead allow for the construction of (U)DRScs. The structure of the CONTENT attribute as well as the Semantics Principle will be changed substantially, since the construction of (U)DRSes allows for inherently different information structures and processing mechanisms. The former CONTENT attribute is replaced by a complex feature structure UDRS, consisting of three attributes, LS, SUBORD and CONDS.</Paragraph>
    <Paragraph position="2"> CONDS is a set of labelled DRS-conditions, ~i, the form of which is determined by lexical entries. SUBORD contains information about the hierarchical structure of a DRS. It is expressed by means of a subordination relation, &lt;, between labels. If &amp;quot;)'1 and &amp;quot;72 are two DRS-conditions with labels ll and 12 such that ll &lt;_ 12 is contained in SUBORD, then this is equivalent to saying that ~/1 and &amp;quot;)'2 will occur in DRSs/(1 and/(2 such that/(1 is weakly subordinate to/(2, i.e. /(1 is either identical to I(2 or nested within it. SUBORD thus imposes the structure of an upper semi-lattice with one-element, lT, to the set of labels. The attribute LS defines the distinguished labels, which indicate the upper and lower bounds for a DRS-condition within the semilattice.</Paragraph>
    <Paragraph position="3"> The main task in constructing UDRSes consists in appropriately relating the labels of the DRS-conditions that are to be combined. This is performed by the association of DRS-conditions with distinguished labels in the lexical entries on the one hand and by conditions governing the projection of the distinguished labels on the other. The role of the distinguished labels is most transparent with verbs and quantifiers.</Paragraph>
    <Paragraph position="4"> In the lexical entry of a transitive verb, for example, the DRS-condition stated in CONDS is a relation  holding between discourse referents. 6 This condition is associated with an identifying label 1. In addition 1 is identified as the minimal distinguished label of the verbal projection by coindexation with L-MIN.</Paragraph>
    <Paragraph position="5"> rcAsE .... 1 rOASEo ol 1 OAT,HISC&lt; \[D&amp;quot;EFm \]'\[O.EP\[\]\] &gt; r,+\[,-+ l (2) / SUBOrtD {} / \[ uo~s / f \[LABEL Iml \]/ REL hire /H L t L ARo2 \[\] J JJ Generalized quantifiers, as in (3), introduce two new labels which identify the DRS-conditions of their restricter and nuclear scope. The quantificational relation holding between them is stated in terms of the relation attribute, REL. In the lexical entry for every, given in (3), a new discourse referent is introduced, in the restrictor DRS, labelled 111, which is identified with the label of the subcategorized NP.</Paragraph>
    <Paragraph position="6"> The feature SUBORD defines the labels of restrictor and scope to be subordinate to the label 11 which identifies the entire condition. The label 11 is defined as the upper bound, or distinguished maximal label of the quantificational structure, whereas the lower bound, or distinguished minimal label is given by the label of the nuclear scope, 112.</Paragraph>
    <Paragraph position="7">  The entry for the indefinite singular determiner, (4), introduces a new individual type referent. As indefinites do not introduce any hierarchical structure into a DRS the identity statement 11 = 112 for the minimal and maximal labels is defined in SUBORD.</Paragraph>
    <Paragraph position="9"> The construction of UDRSes will be defined in terms of clauses of the Semantics Principle: In (5), clause (I) of the Semantics Principle defines the inheritance of the partial DRSes defined in the CONDS attributes of the daughters to the CONDS value of the phrase. Contrary to the Semantics Principle of (Pollard/Sag 1994) the semantic conditions are always inherited from both daughters (we assume bidegThe reference to discourse referents of the syntactic arguments is only provisionally stated here. For the precise definition see (10) below. The use of SUBCAT (SC) as a head attribute is motivated in (Frank 1994).</Paragraph>
    <Paragraph position="10"> nary branching) and therefore project to the uppermost sentential level. Furthermore, clause (I) applies to head-comp- and head-adj-structures in exactly the same way. 7 Clause (II) of the Semantics Principle defines the inheritance of subordination restrictions: The subordination restrictions of the phrase are defined by the union of the SUBORD values of the daughters. Clause (Ill) of the Semantics Principle states the distinguished labels LS of the phrase to be identical to the distinguished labels of the HEADdaughter. It is therefore guaranteed that in binary branching structures the minimal and maximal labels of the head category are available all along the (extended) head projection, s This prepares clauses (IV) and (V) of the Semantics Principle, which define the binding of discourse markers and locality of quantificational scope, respectively. We will first consider clause (IV) and will come back to clause (V) in the next Section.</Paragraph>
    <Paragraph position="11"> In a (U)DRS, the partial structure of the verb has to be (weakly) subordinate to the scope of all the partial DRSes that introduce the discourse markers corresponding to the verb's arguments. This guarantees that all occurrences of discourse markers are properly bound by some superordinated DRS. The constraint is realized by clause (IV) of the Semantics Principle, the Closed Formula Principle. It guarantees that the label associated with the verb, which is identified with the distinguished minimal label of the sentential projection, is subordinated to the minimal label, or lower bound of each of the verb's arguments.</Paragraph>
    <Paragraph position="12"> Note that with quantified arguments the predicate of the verb must be subordinate to the nuclear scope of the quantifier. As defined in (3), it is in fact the nuclear scope of the quantified structure that will be accessed by the distinguished minimal label of the quantified NP. Thus the Closed Formula Principle (IV) in (5) states that in every (non-functional) head-comp-struc a further subordination restriction is unioned to the phrase's SUBORD value, which subordinates the minimal label of the head -here the minimal label associated with the verb- to the minimal label of its actual complement, which in case of a quantified argument identifies the nuclear scope.</Paragraph>
    <Paragraph position="13">  SFunctional categories inherit the distinguished labels of their complement (see (7)). The distinguished labels therefore project along the extended head projection.</Paragraph>
    <Paragraph position="14">  Note that generalized quantifiers were marked as scope bearing by non-identical values of minimal and maximal labels; and singular indefinite NPs were marked as not scope bearing by identifying minimal and maximal labels. As plural NPs introduce a quantificational condition when they are interpreted distributively but behave like indefinites when interpreted collectively, in a representation of their meaning that is underspecified with respect to the distributive/collective ambiguity plural NPs must be marked as potentially scope bearing. This can be achieved if in the lexicon entry of a plural determiner (6) we do not completely specify the relation between the minimal label 112 and the maximal label l~, but only require that 112 is weakly subordinate to 11. This weak subordination relation will be further restricted to either identity or strict subordination when more information is available from the semantic or pragmatic context that allows the ambiguity to be resolved. By monotonically adding further constraints a collective or quantificational (distributive or generic) reading of the plural NP may then be specified, xl If a distributive reading is chosen, the minimal label 112 will identify the nuclear scope of the quantified structure, and in the case of a collective reading the relation of (weak) subordination between minimal and maximal label will be reduced to identity. We will state this in detail in Section 4.</Paragraph>
    <Paragraph position="15"> F rHEAD rAGB.\] NUM pl\] \] q /degAT CL&amp;quot;BEL &gt;l /</Paragraph>
    <Paragraph position="17"> Together with the structure of the lexical entries illustrated above, the clauses (I) - (IV) of the Semantics Principle given in (5) define the core mechanism for UDRS construction: The Semantics Principle defines the inheritance of the labelled DRS conditions and of the subordination restrictions between these labels, which define the semilattice for the complete UDRS structure. The subordination restrictions are projected from the lexicon or get introdhced monotonical- null comp-structures. For head-subj- and head-adj-structures corresponding clauses have to be stated. For head-fillerstructures we only define inheritance of CONDS, SUB-ORD, and LS from the HEAD-DTR.</Paragraph>
    <Paragraph position="18"> lOThe dots indicate that further subordination restrictions will be unioned to the phrase's SUBORD value by clause (V) of the Semantics Principle, defined below.</Paragraph>
    <Paragraph position="19"> llXVe are not in the position to discuss the factors that determine these constraints here.</Paragraph>
    <Paragraph position="20"> ly, e.g. by the Closed Formula Principle to ensure the correct binding of discourse referents. Further subordination restrictions will be added - monotonically - by the remaining clauses of the Semantics Principle, to be introduced in the next Section.</Paragraph>
  </Section>
  <Section position="4" start_page="12" end_page="15" type="metho">
    <SectionTitle>
4 Quantifier Scope and Plural
</SectionTitle>
    <Paragraph position="0"> Disambiguation Quantificational Scope Since the conditions on quantificational scope for generalized quantifiers and distributive readings of plural NPs are dependent on syntactic structure, the Semantics Principle will be supplemented by further clauses governing the interface between syntactic constraints and semantic representation. Note that genuine quantifiers as well as distributive readings of plural NPs differ in their scope potential from indefinite NPs and collectivcly interpreted plural NPs. Whereas the latter may take arbitrarily wide scope, the scope of the former is clause bounded, i.e. they are allowed to take scope only over elements that appear in their local domain.</Paragraph>
    <Paragraph position="1"> We implement this restriction by requiring that the maximal label of a generalized quantifier be subordinate to the distinguished label that identifies the upper bound of the local domain. For plural NPs, a similar constraint must be stated in case a distributive reading is chosen which specifies the plural NP as scope bearing.</Paragraph>
    <Paragraph position="2"> The distinction between scope bearing and not scope bearing NPs was defined by strict subordination and identity of the distinguished labels, respectively. In case a distributive reading is chosen by the clauses for plural disambiguation, to be stated below, the relation of weak subordination in (6), is strengthened to strict subordination. Yet, plural disambiguation may take place rather late in subsequent discourse, while the syntactic constraints for quantificational scope can only be determined locally. The Quantitier Scope Principle (V) will therefore introduce conditionalized subordination restrictions to define the clause-boundedness of both generalized quantifiers and distributively quantified plural NPs. ~2 For finite sentences the local domain for quantified verb arguments comes down to the local IP projection (Frey 1993). In a functional HPSG grammar (see (Frank 1994)) this local domain corresponds to the functional projection of the finite VP. The distinguished maximal label lmax which identifies the upper bound of the local domain for quantified vcrb arguments will therefore be instantiated by the complementizer heading a finite sentence, as in (7).</Paragraph>
    <Paragraph position="3"> X2The scoping principles described in (Frank/lleyle 1994) further account for the scope restrictions of generalized quantifiers and distributive plural NPs.</Paragraph>
    <Paragraph position="4">  Due to the projection of the distinguished labels by clause (III) of the Semantics Principle and the definition of functional categories, the upper bound for the local domain of quantifier scope, lma~, is available throughout the extended projection, where clause (V) of the Semantics Principle, the Quantifier Scope Principle, applies. In (8), the Quantifier Scope Principle (V) states that if the complement is a generalized quantifier (type quant) or a potentially scope bearing plural NP (type plura 0 the SUBORD value of the phrase will contain a further conditionalized subordination constraint, which states that - if the argument is, or will be characterized as a scope bearing argument by strict subordination of its minimal and maximal label - the complement's maximal label lq~,a,u is subordinate to the label lmax which identifies the upper bound of the local domain.</Paragraph>
    <Paragraph position="5">  We argued that for an underspecified representation of plural NPs as regards the collective/distributive ambiguity, their meaning has to be represented by potentially scope bearing partial DRSs. This was achieved by stating the minimal label of the plural NP to be weakly subordinated to its maximal label in (6). Yet, in order to allow for an underspecified representation of the example given in (9), the lexical entry of the verb, stated in (2), has to be refined as indicated in (10).</Paragraph>
    <Paragraph position="7"> Note that as long as it is not determined whether a distributive or collective reading will be chosen for the plural NP, the discourse referent which occupies the corresponding argument place of the verb cannot be identified with the group referent introduced by the plural NP the lawyers. Instead, the mapping between NP meanings and the corresponding argument slots of the verb will be defined by a function dre\]_res, which returns the value of the appropriate discourse referent once a particular plural interpretation is chosen for (9).</Paragraph>
    <Paragraph position="8"> But as long as the plural ambiguity is unresolved the function dre\]_res will be undefined. Thus, if context does not provide us with further, disambiguating information, (11) will be the final, underspecified representation for (9). Here, the function dref_res is undefined for the (underspeeified) plural subject NP.</Paragraph>
    <Paragraph position="10"> Note that the requirement for an underspecified representation of the discourse referent to fill the argument place of the verb cannot be implemented by use of a type hierarchy or similar devices which come to mind straightforwardly. For it is not appropriate for the issue of underspecified representations to compute the set of disjunctive readings, which would ensue automatically if we took such an approach. Instead, the function dre/_res will be implemented by using delaying techniques. The conditions which determine the delayed evaluation of the function dre/_res are defined in its second argument Cond. As long as the variable Cond is not instantiated, the evaluation of ~dref_res will be blocked, i.e. delayed. 13 The three clauses of the function dref_res in (12) and (13) distinguish between not scope bearing, scope bearing and potentially scope bearing elements.</Paragraph>
    <Paragraph position="11"> co.os .. \[\] .. L {\[ }J (12) ,, l The first clause of (12), which takes as its first argument the UDRS value of a verb argument, as defined in (10), is only appropriate for non-quantificational singular NPs (4). The SUBORD value pertaining to the argument is constrained to contain a condition which identifies its minimal and maximal labels: 11 = In. The second clause applies if the semantic structure of the argument contains a subordination restriction which characterizes the NP as scope bearing. This is the case for generalized quantifiers (3). The values of the minimal and maximal labels are lain the CUF system (Doerre/Dorna 1993) delay statements are defined by the predicate wait. The delayed function can only be evaluated when all specified argument positions are instantiated. The delay statement for dref_res is wait(dref_res(udrs, subord_info)), where subord_info is the type of a member of SUBOILD.</Paragraph>
    <Paragraph position="12">  characterized as non-identical by strong subordination: 11 &gt; 112.</Paragraph>
    <Paragraph position="13"> If a clause is applied successfully, by coindexation of the differentiating subordination restrictions with the second argument of dre\]_res, the latter gets properly instantiated and the function is relieved from its delayed status. It returns the discourse referent which in the argument's UDRS is associated with the maximal label for not scope bearing NPs, and with the label of the restrictor 111 for scope bearing NPs. For plural NPs, which are represented as potentially scope bearing by a weak subordination constraint as shown in (6), the clauses in (12) will fail: the required subordination conditions will not be contained in the SUBORD value of the verb argument. 14 Underspecified as well as disambiguated plural NPs, characterized by a weak subordination constraint in the local UDRS, are captured by the third clause of dre/_res in (13).</Paragraph>
    <Paragraph position="14"> (13) ~rer-~es \] s L~_~,~ .c .....</Paragraph>
    <Paragraph position="15"> LSUBOaD{.. l\[~ &gt; \[i~..</Paragraph>
    <Paragraph position="16"> In (13) the value of dre/_res is undefined (T) and the variable Cond, which is subject to the delay conditions on dref_res, is not instantiated by coindexation with a subordination restriction in the local SUBORD value. The function therefore is delayed, until further disambiguating constraints are available which resolve the plural ambiguity and determine the discourse referent to fill the argument slot of the verb. This is what we aimed at for the special concerns of plural underspecification.</Paragraph>
    <Paragraph position="17"> If, however, a particular reading of a plural NP is determined by the lexical meaning of the verb, as it is the case for gather, an appropriate definition of dref_res in the lexical entry of the verb ensures the correct plural interpretation.</Paragraph>
    <Paragraph position="18"> Plural Disamblguation In most cases, however, disambiguating information for the interpretation of plurals comes from various sources of semantic or pragmatic knowledge. Usually it is provided by subsequent discourse. We therefore define a mechanism for plural disambiguation which may apply at any stage of the derivation, to add disambiguating DRS conditions and subordination constraints to the underspecified representation whenever enough information is available to determine a particular plural interpretation. To this end we extend the Semantics 14This will be so even if - by the function pl_dis to be introduced below - further disambiguating constraints for, e.g., a collective or distributive reading are introduced at a later stage of the derivation: dref_res is defined on the UDRS value of a verb argument in the lexical entry of the verb. The value of thfs local UDRS, and with it the SUBORD attribute, remains unaffected by the introduction of additional subordination restrict.ions by clauses of the Semantics Principle.</Paragraph>
    <Paragraph position="19"> Principle to include a function pidis (plural disambiguation), which applies to a phrase's UDRS value, to render a new value of the same type, which specifies a collective or distributive reading for a plural discourse referent contained in the underspecified representation. The individual clauses of pLdis will have to state constraints for determining the respective plural readings, to be satisfied by the preceding context, represented in UDRS. Ideally, these constraints have access to inference modules, including semantic and pragmatic knowledge. We first state the function pidis for the different readings and then incorporate the function into the Semantics Principle.</Paragraph>
    <Paragraph position="20"> If in clause (14) of pLdis the constraints that determine a collective reading of the plural NP with label 11 are satisfied, the relation of weak subordination between the minimal and maximal label of the plural NP is strenghtened to the identity relation. In tile output value the restriction 11 = In gets unioned to the original SUBORD value. Note that the function pidis is fully monotonic in that its result is a UDRS which is obtained by only adding information to the input values SUBORD and CONDS by union.</Paragraph>
    <Paragraph position="21"> Whenever disambiguation of a plural NP takes place, the function dref_res must be relieved from its delayed status in order to instantiate the corresponding argument slot of the verb. We will access the delayed goal dref_res by reference to the plural NP's maximal and minimal labels 11 and 112, instantiate its second argument by the identity constraint 11 = 112, and define its value by the DREF value X associated with 11. The resulting UDRS for a collective interpretation of (9) is given in (15).</Paragraph>
    <Paragraph position="22">  Disambiguation to a distributive reading is obtained in (16) by adding a quantificational distribution condition to the original value of CONDS. The restrictor In introduces an individual discourse referent x together with the distribution condition x 6 X and the nuclear scope is identified by the minimal label 112. Moreover, (strong) subordination of restrictor and scope is defined in SUBORD. Again, the delayed function dref_res is defined to return the discourse referent x which is to fill the argument slot of the  verb and is un-delayed by instantiation of its second argument.</Paragraph>
    <Paragraph position="23"> LS \[\] (\[SUBORD\[~\] {..\[h'\]&gt;\[~ ..} \]~ pl-dls LA - ~'=&amp;quot;' := t L oNo \[\] {... ...}j) }} \[s..,.o.-&lt;o im &gt; Elm &gt; (16) I ( FLAB~-L\[~ I FLABI~L il\[~klJ -\] Conditions: constraints for a distributive reading (of X) ~ &lt;,.,:,.o&lt;,_:o:,: &lt;,,..,_..:( \Ve now complete the Semantics Principle by the Principle for Plural Disambiguation (VI). In (17), the function pl_dis applies in a coordination structure coord-struc, which recursively, combines pairs of (sequences of) sentences and a sentence. The function pl_dis applies to the phrase's UDRS value, which is defined by application of the basic clauses (I) and (II) of UDRS construction. Depending on the context represented in UDRS, and supplemented by general semantic and/or pragmatic knowledge, pl_dis monotonically redefines the phrase's UDRS value if disambiguating constraints for a specific plural reading can be determined. If the constraints for plural disambiguation (14) and (16) are not satisfied, the trivial clause of pl_dis applies, which returns the</Paragraph>
  </Section>
  <Section position="5" start_page="15" end_page="15" type="metho">
    <SectionTitle>
\[CONDS \[\] JJ \[CONDS \[\] JJ
5 Conclusion and Perspectives
</SectionTitle>
    <Paragraph position="0"> A constraint based semantic formalism for HPSG has been presented to replace the standard HPSG semantics. The new formalism comes closer to a principle based construction of semantic structure and, therefore, is more in the spirit of HPSG philosophy than its standard approach. Furthermore the new formalism overcomes a number of shortcomings of the standard approach in a natural way.</Paragraph>
    <Paragraph position="1"> In particular, we presented an HPSG grammar for English that defines a syntax-semantics interface for the construction of U(nderspecified) D(iscourse) R(epresentation) S(tructure)s. The construction is guided by general principles, which clearly identify the interaction between the modules, i.e. the &amp;quot;interface&amp;quot; between syntax and semantics. In the fragment we defined underspecificied representations for quantificational structures and plural NPs. The principles governing the interaction of syntax and semantics specify scoping relations for quantifiers and quantificational readings of plural NPs.</Paragraph>
    <Paragraph position="2"> In addition to the syntax/semantics interface the Semantics Principle developed in this paper defines a clear interface to contextual and pragmatic knowledge. This interface allows reasoning modules to interact with semantics construction. The approach taken here can, therefore, be generalized to disambiguation problems other than the collective/distributive ambiguity as well as to anaphora resolution. A further issue to which the present account is directly related is incremental interpretation.</Paragraph>
  </Section>
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