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<Paper uid="C92-1035">
  <Title>CATEGORIAL SEMANTICS FOR LFG</Title>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
3 Sentence Interpretation
</SectionTitle>
    <Paragraph position="0"> A sentence such as (1) has the interpretation given in (2): 1  (1) John crashed.</Paragraph>
    <Paragraph position="1"> (2) crash (john)  This interpretation is the outcome of a derivation according to a set of rules to be described below. Some of the rules must be licensed by particular f-structure configurations, while some are unrestricted in their apphcahihty. Example  1 has the following hstructure: (3) \[PRED ,crash (SUBJ) , 1</Paragraph>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
\[SkrBJ \[PRED 'John'\]\]
</SectionTitle>
    <Paragraph position="0"> Annotated phrase structure rules hke the following are assumed: 2</Paragraph>
    <Paragraph position="2"> Notice that these phrase structure rules encode only syntactic information. No semantic information or constraints are required.</Paragraph>
    <Paragraph position="3"> The lexical entries involved in the derivation of sentence (1) are:</Paragraph>
    <Paragraph position="5"> The notation f~, stands for the interpretation of an f-structure f, often referred to as the semantic projection of f (Kaplan, 1987; Halvorsen and Kaplan, 1988). The interpretation for any f-structure f is a sequent:  between c-structure and f-structure and the notation commonly used to represent that relation.</Paragraph>
    <Paragraph position="6"> (4) G:\[o~-M\]  The sequent '\[a ~ M\]' is a pair consisting of a set of assumptions a, somewhat analogous to a 'quantifier store' (Cooper, 1983), and a matrix term M in which free variables introduced by the asstutlptions in a may occur (Pereira, 1990; Pereira, 1991; Dalrymple et al., 1991). In the following, I will speak of such expressions as introducing the meaning M under the assumptions in a.</Paragraph>
    <Paragraph position="7"> I assume a fixed order of application of the meaning of a verb to its semantic arguments, with the order determined by the syntax (though this assmnption is not crucial to the analysis). Arguments are applied in the following order: s  (1) Obliques (2) o,~2 (3) osJ (4) sunJ  The PILED of the f-structure of an active verb such as own will, then, be associated via the a mapping with the following interpretation: (5) Ay.Ax.own(x,y) Notice that the verb is required to combine with the object first, and then the subject, in accordance with the argument ordering given above. \]:'or a passive verb, the ordering will be reversed. For the passive verb (be) owned, the order will be: (6) x~.~v.ow,t(~,v) Here, the verb combines first with the oblique by-phrase, then with the subject.</Paragraph>
    <Paragraph position="8"> The rule for interpreting art f-structure for a clause headed by an intransitive verb is: 4 (7) Clause with intransitive verb: 3This order of application was also proposed by Dowry (1982), and is reminiscent of the obliqueness ordering for arguments in HPSG (Pollard and Sag, 1987). 4This rule should apply when f has a PRED and a sUB J, but no other governable grammatical functions; it should not apply if the verb is transitive and there is a slJl~J and an oB3, although f is unifiable with tile f-structure of a transitive verb as well as an intransitive one. There are several ways of ensuring the needed result: the valence of tire verb can be reflected in its semantic type; f-structures can be typed, with this rule applying only to intransitive f-structures (Zajac and Emele, 1990); or the PROD and its arguments can be separately specified, with the argumarts of the PRED specified as a list which can be mntched against, as in recent work by John Maxwell, Ron Kaplan, and others.</Paragraph>
    <Paragraph position="10"> The derivation of the meaning f~ of an f-structure f with a PRED and SUBJ proceeds by applying the meaning of the PILED to the meaning of the suBJ. The associated assumption set is the union of the assmnptions from the PRED and the SUna. The f-structure for sentence 1 hcenses the following derivation and provides the expected meaning (under a null assumption set):  (8) \['p RED f2:,craah (SUBJ) , J ks.., 'John'\]\]</Paragraph>
    <Paragraph position="12"/>
  </Section>
  <Section position="6" start_page="0" end_page="0" type="metho">
    <SectionTitle>
4 Quantification
</SectionTitle>
    <Paragraph position="0"> Sentence 9 contains a quantified noun phrase and has the meaning represented in (10):  (9) Every car crashed.</Paragraph>
    <Paragraph position="1"> (10) every(Ay.car(y), Az.craMz(x)) This sentence has the f-structure shown in (11), constructed on the basis of the lexical entries below: null (11) \[Pa~D 'c~ash &lt;sv.J) ' \]</Paragraph>
    <Paragraph position="3"> The type of the quantifier every is the familiar generalized quantifier type (e -+ t) ~ (e ~ t) ---* t: quantifiers are functions from properties to properties, yielding a truth value. The following schematic rule is necessary to interpret quantified noun phrases: (12) Quantified noun phrase, preliminary version (handles unmodified nominals only):</Paragraph>
    <Paragraph position="5"> The notation 'a, A' represents the set a plus the singleton A. By this rule, a quant assumption is added to the assumption set for the noun phrase.</Paragraph>
    <Paragraph position="6"> The quant assmnption acts like an element in a Cooper store, keeping together the information associated with the quantified noun phrase, ms in the quant asstmlption is the meaning of the specifier (here, every); z is the variable introduced by the quantifier as the meaning of the quantified norm phrase; and mp is the meaning of the PRED, which will form the first argument of the generalized quantifier every when quantifier discharge takes place. The derivation of the meaning of sentence 9 according to the rnles given thus far proceeds as follows:</Paragraph>
    <Paragraph position="8"> By rule 12, 'Quantified noun phrase': (In), = \[{quant (every, z, Ay.car(y))} W z\] By rule 7, 'Clause with intransitive verb':</Paragraph>
    <Paragraph position="10"> According to these rules, the meaning for f-structure fl is a meaning under an assumption about the variable x. The meaning of fl without assumptions is obtained by discharging the (sole) quantifier assumption in the assumption set. The quantifier discharge rule relates a se- null quent and a syntactic licensing environment to a new sequent: Acres DE COLING-92, NANT~. 23-28 ^O~&amp;quot; 1992 2 1 4 PROC. OF COLING-92, NANTES, AUG. 23-28, 1992 (14) Quantifier discharge:  disch(f, \[a, quaut (ms, x, mR) ~- SCOPE :t\]) = \[a ~ ms&amp;quot; (rap, Ax.SCO P E)\] Conditions on f: none By this discharge rule, the quant assumption is removed from the assumption set, the variable x introduced by the quantifier assmnption is abstracted out of the scope SCOPE (required to be of type t), and the quantifier is applied to its scope. The syntactic licensing environment is the f-structure f. in this rule, f is lmeonstrained; there are no conditions on f. This means that the quantifier discharge rule has art unrestricted syntactic licensing condition. A quantifier may scope over any syntactic constituent, as long as it is of the correct semantic type. 5 To interpret sentence 9, diseh can now be applied to the sequent (fl)~ associated with the f-structure \]1: disch (PS, \[{quant(every, ~, )~y.car(y))} b crash(z:)\]) = \[0 \]- every (Xy.car (y), Xx.crash(x))\] The result is the meaning of fl with all assumptions discharged. I will assume that what is generally referred to as the 'meaning' of an utterance is the meazfing obtained when all assumptions have been discharged.</Paragraph>
    <Paragraph position="11"> In general, assumptions may be discharged after any application of a functor to an argument, as long as the syntactic enviromnent for assumption discharge has been met. Thus, a predicate apply can be defined: (15) apply(f, \[a~ ~ Fun\], \[aA ~ Arg\]) d9 discharge(I, \[aF UaA '- Fun (Avg)\]) apply operates on sequents in a syntactic licensing environment f. discharge(f, S) is the result of applying any number of discharge (disch) rules licensed by the syntactic configuration f to S. (Note that apply is not a function, since the result of apply depends on the Immber and the choice of assumptions to be discharged.) By this function application rule for sequents, then, the meaning of the fimctor is applied to the meaning of the argunlent; the union of the functor assumptions and the argmnent assumptions is taken; and some number of discharge rules may be applied. This definition of apply will be used ~Here \[ will not discuss conditions on preferred scopes for quantifiers (such as the tendency for the quantifier each to outscope other quantifiers, or for quantifiers to scope inside the clause in which they appear).</Paragraph>
  </Section>
  <Section position="7" start_page="0" end_page="0" type="metho">
    <SectionTitle>
ACTES DE COLING-92, NANTES, 23-28 ho~r 1992 2 1 5
</SectionTitle>
    <Paragraph position="0"> in the tbllowing to apply predicates to their arguments and to permit subsequent assumption discharge.</Paragraph>
    <Paragraph position="1"> Given this new definition of apply, interpretation rule 7 for clauses headed by intransitive verbs can be restated: (16) Clause with intransitive verb: The interpretation for an f-structure f, representing an umnodified clause with an intransitive verb, is obtained by applying the pREI) P to the SuBJ S in the syntactic heensing enviroltment f. In general, f~, will constitute an assignment of f to a sequent that satisfies the constraints given by the lexical entries and the rules of interpretation. null It should be noted that rule 16 is incomplete in providing interpretations only for sentences not involving adverbial modification; an analysis of adverbials, though straightforward in this framework, will not be provided here.</Paragraph>
  </Section>
  <Section position="8" start_page="0" end_page="0" type="metho">
    <SectionTitle>
5 Nominal modification
</SectionTitle>
    <Paragraph position="0"> Rule 12 for tile interpretation of quantified norm plLrases is incomplete, since it apphes only to unmodified nominals. Consider sentence (17), its fstructure, displayed in Figure 1, and its meaning,</Paragraph>
    <Paragraph position="2"> Syntactically, a relative clause contains a fronted constituent (a TOPIC; see Bresnan and Mchombo (1987)) which is related to a gapped position in tim sentence. This fronted constituent contains a relative pronoun or that. Tile relative pronoun nlay be deeply embedded in the fronted PROC. ol, COLING-92, NANTES, AUG. 23-28, 1992 constituent, as in the ease of pied piping. Semantically, the interpretation of a relative clause is the property obtained when the position filled by the relative pronoun is abstracted out. For example, here are some relative clauses with a rough representation of their memfings: (19) a. (the man) that I saw: Az.saw(1,~) b. (the man) whose brother's ear I drove:</Paragraph>
    <Paragraph position="4"> I assume that relative pronouns such as that or whose introduce a variable under a tel assumption which is abstracted out in the course of the derivation. The interpretation of a relative clause is obtained by a rule allowing the discharge of the rel assumption associated with the relative pronoun (and possibly other assumptions as well): (20) Relative clause interpretation: \[TOPIC TOP\] f : \[REL R J ~ &amp;quot;f~ = discharge(f,R~) The tel discharge rule applies only under syntactically licensed conditions:</Paragraph>
    <Paragraph position="6"> The relative pronoun must appear in the fronted TOPIC constituent. This is indicated in the second condition by the regular expression TOPIC OF*; this expression involves functional uncertainty (Kaplan and Maxwell, 1988) and requires that the relative pronoun relp must appear at the end of a path that is a member of the language described by the regular exprestmn. Here, the path expression does not constrain where the relative pronoun may be found within the fronted constituent (OF* is a sequence of zero or more grammatical functions); a more complete syntactic analysis of relative clauses would constrain the path appropriately. The result of the application of rule 21 is that the variable introduced by the relative pronoun is abstracted out.</Paragraph>
    <Paragraph position="7"> The value of the MOPS attribute is a set offstructures, interpreted according to the following r ule: (22) The semantic value of a set off-structures is the set of corresponding sequents. If F is a set of f-structures: F~-- {f~rfc F} The rule for the interpretation of quantified noun phrases with nominal modification is given in Figure 2. According to this rule, the derivation of the meaning of a quantified noun phrase proceeds by introducing a variable (x in Figure 2) under a quant assumption, consisting of the meaning of the specifier of the noun phrase, the variable, and the quantifier restriction ILEST.</Paragraph>
    <Paragraph position="8"> Recall the definition of apply given in 15: (23) apply(f, jaR ~- Fun\], \[aA F- Arg\]) d~=y discharge(f, \[aF U aA ~- Fun (Arg)}) This definition will now be extended so apply can take a set ofsequents as its argument. The result is a set of sequents: (24) apply (f, Set, Arg) dff {s \[ Fun E Set A s = apply (f, Fun, Arg)} The function conj is defined as the conjunction of a set of sequents; the matrices of the sequents are conjoined and the union of the assumptions is taken: (25) conj (S) de=/\[ U a F- A M\] IaP-M\]eS \[a~-M\]ES nEST, then, is the conjunction of the result of applying the PRED meaning and the meaning of each of the modifiers in MOPS to the variable z. Finally, a rule for the interpretation of a clause containing a transitive verb is also needed: (26) Clause with transitive verb: ?deg:l</Paragraph>
    <Paragraph position="10"> The interpretation of sentence 17 can now be derived; the derivation is sketched in Figure 3.</Paragraph>
  </Section>
  <Section position="9" start_page="0" end_page="0" type="metho">
    <SectionTitle>
ACRES DE COLING-92. NANTES. 23-28 AO~' 1992 2 1 7 PROC, OF COLING-92. NANTES, AUG. 23-28, 1992
6 Conclusion
</SectionTitle>
    <Paragraph position="0"> The small fragment of English presented above is easily extensible to handle other semantic phenomena, such as sentential modification. Constraining semantic derivations with respect to f-structures is preferable to the standard approach of using phrase structure trees, since f-structures need not be specifically tailored to solving the interpretation problem, but are motivated on independent grounds. The categorial semantics rules presented above provide an interpretation for f-structures directly, without the need for constructing an intermediate level of 'logical form'.</Paragraph>
  </Section>
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