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<?xml version="1.0" standalone="yes"?> <Paper uid="E95-1005"> <Title>The Semantics of Resource Sharing in Lexical-Functional Grammar</Title> <Section position="4" start_page="31" end_page="32" type="metho"> <SectionTitle> X/Y:F CON J:& X/Y:G =~ X/Y:Ax.(Fx&Gx) </SectionTitle> <Paragraph position="0"> The contraction from )~x.Fx and Ax.Gx to )~x.(Fx&Gx) in this rule allows for the single argument to be utilized twice.</Paragraph> <Paragraph position="1"> As noted by Hudson (1976), however, not all examples of RNR involve coordinate structures: (4) Citizens who support, paraded against politicians who oppose, two trade bills.</Paragraph> <Paragraph position="2"> Obviously, such cases fall outside of the purview of the coordination schema. An analysis for this sentence is avi~ilable in the CCG framework by the addition of the xsubstitute combinator (Steedman, p.c.), as defined in Steedman (1987).</Paragraph> <Paragraph position="4"> The use of this combinator assimilates cases of noncoordinate RNR to cases involving parasitic gaps.</Paragraph> <Paragraph position="5"> While this approach has some drawbacks, 1 we do not offer a competing analysis of the syntax of sentences like (4) here. Rather, we seek an analysis of RNR (and of resource sharing in general) that is uniform in the semantics; such a treatment isn't available in CCG because of its tight integration between syntax and semantics.</Paragraph> <Section position="1" start_page="31" end_page="32" type="sub_section"> <SectionTitle> 2.2 Partee and Rooth </SectionTitle> <Paragraph position="0"> Perhaps the most influential and widely-adopted semantic treatment of coordination is the approach of Partee and Rooth (1983). They propose a generalized conjunction scheme in which conjuncts of the same type can be combined...ks is the case with Steedman's operators, contraction inherent in the schema allows for a single shared argument to be distributed as an argument of each conjunct. Type-lifting is allowed to produce like types when necessary; the combination of the co-ordination scheme and type-lifting can have the effect of 'copying' an argument of higher type, such as a quantifier in the case of coordinated intensional verbs. They propose a 'processing strategy' requiring that expressions are interpreted a! the lowest possible type, with type-raising taking place only where necessary.</Paragraph> <Paragraph position="1"> To illustrate. Partee and Rooth assume that extensional verbs such as find are entered in the lexicon with basic type (e, (e, t)}, whereas intensional verbs like want, which require a quantifier as an argument, have type (((e, t}, t), (e, t}) (ignoring intensionality). Two extensional verbs such as find and support are coordinated at their basic types: (6) find and support (type (e, (e, t}}): )W.)~x.\[f ind( x, y) A support(x, y)\] Two intensional verbs such as want and seek are also coordinated at their basic (higher) types: (7) want and seek (type (((e, t), t}, (e, t))): )~P.)~x.\[want(x, 79) A seek(z, 79)\] The argument to this expression is a quantified NP. When an intensional and an extensional verb are coordinated, the extensional verb must be 1We find two problems with the approach as it stands. First, the intuition that one gap is 'parasitic' upon the other in cases like (4) is not strong, whereas the CCG analysis suggests an asymmetry between the two gaps. Second, the combinator appears to cause overgeneration. While it allows sentence (4), it also allows sentence (b), where two trade bills is analyzed as the object of both verbs: (b) *Politicians who oppose, paraded against, two trade bills.</Paragraph> <Paragraph position="2"> type-raised to promote it to the type of the intensional verb: (8) want and find (type <<(e,t>,t),<e,t>>):</Paragraph> <Paragraph position="4"> Again, this leads to the desired result. However, an unwelcome consequence of this approach, which appears to have gone unnoticed in the literature, arises in cases in which more than two verbs are conjoined. If an intensional verb is co-ordinated with more than one extensional verb, a copy of the quantifier will be distributed to each verb in the coordinate structure. For instance, in (9), two extensional verbs and an intensional verb are coordinated.</Paragraph> <Paragraph position="5"> (9) want, find, and support: AP.Ax.\[ want(x, 7 0) A ~P(Ay.find(x, y)) A 7)(Ay.support(x, y)) \] Application of this expression to a quantifier results in two quantifiers being scoped separately over the extensional verbs. This is the wrong result; in a sentence such as Hillary wanted, found, and supported two candidates, the desired result is where one quantifier scopes over both extensional verbs (that is, Hillary found and supported the same two candidates), just as in the case where all the verbs are extensional. Further, there does not seem to be an obvious way to modify the Partee and Rooth proposal so as to produce the correct result, the problem being that the ability to copy quantifiers inherent in their schema is too unrestricted. null A second problem with the account is that, as with Steedman's coordination schema, Partee and Rooth's type-raising strategy only applies to coordinate structures. However, the need to type-raise extends to cases not involving coordination, as in sentence (10).</Paragraph> <Paragraph position="6"> (10) Citizens who seek, paraded against politicians who have, a decent health insurance policy.</Paragraph> <Paragraph position="7"> We will present an analysis that preserves the intuition underlying Partee and Rooth's processing strategy, but that predicts and generates the correct reading for cases such as (9). Furthermore, the account applies equally to examples not involving coordination, as is the case in sentence (10).</Paragraph> </Section> </Section> <Section position="5" start_page="32" end_page="34" type="metho"> <SectionTitle> 3 LFG and Linear Logic </SectionTitle> <Paragraph position="0"> LFG assumes two syntactic levels of representation: constituent structure (c-structure) 2 encodes phrasal dominance and precedence relations, and functional structure (f-structure) encodes syntactic predicate-argument structure. The f-structure for sentence (11) is given in (12): Semantic information is expressed in (1) a meaning language and (2) a language for assembling meanings, or glue language. The meaning language could be that of any appropriate logic: for present purposes, higher-order logic will sufrice. Expressions of the meaning language (such as Bill) appear on the right side of the meaning relation ---~.</Paragraph> <Paragraph position="1"> The glue language is the tensor fragment of linear logic (Girard, 1987). The semantic contribution of each lexical entry, which we will refer to as a meaning constructor, is a linear-logic formula consisting of instructions in the glue language for combining the meanings of the lexical entry's syntactic arguments to obtain the meaning of the f-structure headed by the entry. For instance, the meaning constructor for the verb supported is a glue language formula paraphrasable as: &quot;If my SUBJ means X and ((r)) my OBJ means Y, then ( ---o ) my sentence means supported(X, Y)&quot;.</Paragraph> <Paragraph position="2"> In the system described in Dalrymple et al. (1993a), the ~ relation associates expressions in the meaning language with f-structures. As a result, each f-structure contributed a single meaning constructor as a resource to be used in a derivation. Because linear logic does not have any form of logical contraction (as is inherent in 2For discussion of c-structure and its relation to f-structure, see, for example, Kaplan and Bresnan (1982).</Paragraph> <Paragraph position="3"> the approaches discussed earlier), cases where resources are shared appear to be problematic in this framework. Intuitively. however, the need for the multiple use of an f-structure meaning results not from the appearance of a particular lexical item (e.g., a conjunction) or a particular syntactic construction (e.g., parasitic gap constructions), but instead results from multiple paths to it from within the f-structure that contains it, where structure sharing is motivated on syntactic grounds. We therefore revise the earlier framework to model what we will term occurrences of f-structures as resources explicitly in the logic. F-structures can mathematically be regarded as (finite) functions from a set of attributes to a set of atomic values, semantic forms and (recursively) f-structures. We will identify an occurrence of an f-structure with a path (from the root) to that occurrence; sets of occurrences of an f-structure can therefore be identified with path sets in the f-structure. We take, then, the domain of the a projection to be path sets in the root f-structure. Only those path sets S are considered which satisfy the property that the extensions of each path in S are identical. Therefore the f-structure reached by each of these paths is identical. Hence from a path set S, we can read off an f-structure S I. In the examples discussed in Dalrymple et al. (1993a) there is a one-to-one correspondence between the set of path sets S and the set of f-structures S I picked out by such path sets, so the two methods yield the same predictions for those cases.</Paragraph> <Paragraph position="4"> Relations between path sets are represented explicitly as resources in the logic by R-relations. R-relations are represented as three-place predicates of the form R(F, P, G) which indicate that (the path set) G appears at the end of a path P (of length 1) extending (the path set) F. That is, the f-structure Gf appears at the end of the singleton path P in the f-structure Fy. For example, the f-structure given in (12) results in two R-relations: (i) R(f, SUB J, 9) (ii) R(f, OBJ, h) Because f and g represent path sets entering an f-structure that they label, R-relation (i) indicates that the set of paths (f sun J) (which denotes the set of paths f concatenated with SUB J) is a subset of the set of paths denoted by g. An axiom for interpretation provides the links between meanings of path sets related by R-relations.</Paragraph> <Paragraph position="5"> Axiom I: !(VF, G,P,X. Go-'-*X --o !(R(F,P,G) --o (F P)o.-.~X)) According to this axiom, if a set of paths G has meaning X. then for each R-relation R(F, P,G) that has been introduced, a resource (F P)C/---*.\&quot; can be produced. The linear logic operator '!' allows the conclusion (R(F, P,G) --o (F P)~,.-.~X) to be used as many times as necessary: once for each R-relation R(F, P, G) introduced by the f-structure.</Paragraph> <Paragraph position="6"> We show how a deduction can be performed to derive a meaning for example (11) using the meaning constructors in (13), R-relations (i) and (ii). and Axiom I. Instantiating the lexical entries for Bill, NAFTA, and supported according to the labels on the f-structure in (12), we obtain the fol- null for Bill yields: (14) !VF, P. R(F, P, g) ---o (F P)o...., Bill This formula states that if a path set is R-related to the (path set corresponding to the) f-structure for Bill, then it receives Bill as its meaning. From R-relation (i) and formula (14), we derive (15).</Paragraph> <Paragraph position="7"> giving the meaning of the subject of f.</Paragraph> <Paragraph position="8"> (15) (f suBJ)a&quot;~Bill The meaning constructor for supported combines with (15) to derive the formula for bill-supported shown in (16).</Paragraph> <Paragraph position="9"> (16) V\]&quot;. (fOBJ) &quot;-~r -o f~ ~ supported(Bill, Y) Similarly, using the meaning of NAFTA, R-relation (ii), and Axiom I, we can derive the meaning shown in (17): (17) (f OBJ)o'..*NAFTA and combine it with (16) to derive (18): (18) fo'--* supported( Bill, NAFTA) At each step, universal instantiation and modus ponens are used. A second derivation is also possible, in which supported and NAFTA are combined first and the result is then combined with Bill.</Paragraph> <Paragraph position="10"> The use of linear logic provides a flexible mechanism for deducing meanings of sentences based on their f-structure representations. Accounts of various linguistic phenomena have been developed within the framework on which our extension is based, including quantifiers and anaphora (Dalrymple et al., 1994a), intensional verbs (Dalrympie et al., 1994b), and complex predicates (Dalrymple et al., !993b). The logic fits well with the 'resource-sensitivity' of natural language semantics: there is a one-to-one correspondence between f-structure relationships and meanings; the multiple use of resources arises from multiple paths to them in the f-structure. In the next section, we show how this system applies to several cases of right-node raising.</Paragraph> </Section> <Section position="6" start_page="34" end_page="35" type="metho"> <SectionTitle> 4 Examples </SectionTitle> <Paragraph position="0"/> <Section position="1" start_page="34" end_page="35" type="sub_section"> <SectionTitle> 4.1 RNR with Coordination </SectionTitle> <Paragraph position="0"> First we consider the derivation of the basic case of right-node raising (RN R) illustrated in sentence (i), repeated in (19).</Paragraph> <Paragraph position="1"> supported: gX, Y. (fl soaa)o--~X (r) (k oaa)~-,* Y -o f,o..., supported(X, Y) opposed: VX, Y. (f2 SUBJ)~-~X (r) (f~ osJL~ v -o f2a-,-~opposed(X, y) and: VX, Y. (f CONJ)a&quot;~X Here, we treat and as a binary relation. This suffices for this example, but in general we wiil have to allow for cases where more than two constituents are conjoined. Therefore, a second meaning constructor and2 is also contributed by the appearance of and, prefixed with the linear logic operator '!'. so that it may be used as many times as necessary (and possibly&quot; not at all, as is the case in this example).</Paragraph> <Paragraph position="2"> The R-relations resulting from the feature-value relationships manifest in the f-structure in (20) are: 3 (i) R(f, CONJ. ft) (ii) R(f, CONJ, f2) (iii) R(fl, SUB J, 9) (iv) R(fl, oaa, h) (v) R(f2, SUB J, i) (vi) (A, oBJ, h) There are several equivalent derivation orders: here we step through one. 4 Using the meanings for Bill. supported, Hillary, and opposed, R-relations (iii) and (v), and Axiom I, we can derive meanings for Bill supported and Hillary opposed in the fashion described in Section 3: bill-supported: VY. (ft OBJ}e&quot;'~Y ---o fla &quot;-&quot; supported(Bill, Y ) hillary-opposed:gZ. (f20BJ} o&quot;~ Z .--o f2~, ~ opposed( IIillary, Z) We combine the antecedents and consequents of the foregoing formulae to yield: bill-supported (r) hillary-opposed: VY, Z. (fl (r)B J) ----~Y (r) (f2 oaJ)a&quot;-&quot;Z ---o fla-,-+ supported(Bill, Y) (r) f2a ~ opposed( Hillary, Z) Consuming the meaning of and and R-relations (i) and (ii), and using Axiom I, we derive: bill-suppor ted-and-hillary-opposedl: vY, z. (k osaL ~ r (r) (A oaaL-,-, z</Paragraph> <Paragraph position="4"> aWe treat the CONJ features as unordered, as they are in the f-structure set.</Paragraph> <Paragraph position="5"> Finally, consuming the contribution of NAFT\4, by Ulfiversal instantiation and modus ponens we obtain a meaning for the whole sentence: fo'--*and( supported( Bill, N :tFTA ), opposed( Hillary, NAFTA) ) At this stage, all accountable resources have been consumed, and the deduction is complete.</Paragraph> </Section> <Section position="2" start_page="35" end_page="35" type="sub_section"> <SectionTitle> 4.2 RNR with Coordination and Quantified NPs </SectionTitle> <Paragraph position="0"> We now consider sentence (21), where a quantified NP is shared.</Paragraph> <Paragraph position="1"> (21) Bill supported, and Hillary opposed, two trade bills.</Paragraph> <Paragraph position="2"> Partee and Rooth (1983) observe, and we agree, that the quantifier in such cases only scopes once, resulting in the reading where Bill supported and Hillary opposed the same two bills. 5 Our analysis predicts this fact in the same way as Partee and Rooth's analysis does.</Paragraph> <Paragraph position="3"> The meanings contributed by the lexieal items and f-structure dependencies are the same as in the previous example, except for that of the object NP. Following Dalrymple et al. (1994a), the meaning derived using the contributions from an f-structure h for two trade bills is: two-trade-bills: VH, S. (Vz. h~-.~x --o H~S(~)) -o g..~two(z, tradebill(z), S(z)) The derivation is just as before, up until the final step, where we have derived the formula labeled bill-supported-and-hillary-opposed2. This formula matches the antecedent of the quantified NP meaning, so by universal instantiation and modus ponens we derive: f a &quot;-* two( z, tradebill( z ), and(supported(Bill, z ),</Paragraph> <Paragraph position="5"> With this derivation, there is only one quantifier meaning which scopes over the meaning of the coordinated material. A result where the quantifier meaning appears twice, scoping over each conjunct separately, is not available with the rules we have given thus far; we return to this point in Section 5.</Paragraph> <Paragraph position="6"> The analysis readily extends to cases of noncoordinate RNR such as example (4), repeated as example (22).</Paragraph> <Paragraph position="7"> SWe therefore disagree with Hendricks (1993), who claims that such sentences readily allow a reading involving four trade bills.</Paragraph> <Paragraph position="8"> (22) Citizens who support, paraded against politicians who oppose, two trade bills.</Paragraph> <Paragraph position="9"> In our analysis, the f-structure for two trade bills is resource-shared as ttle object of the two verbs, just as it is in the coordinated case.</Paragraph> <Paragraph position="10"> Space limitations preclude our going through the derivation; however, it is straightforward given the semantic contributions of the lexical items and R-relations. The fact that there is no coordination involved has no bearing on the result, since the s,.mantles of resource-sharing is distinct from that of coordination in our analysis. As previously noted. this separation is not possible in CCG because of the tight integration between syntax and semantics. In LFG, the syntax/semantics interface is more loosely coupled, affording the flexibility to handle coordinated and non-coordinated cases of RNR uniformly in the semantics. This also allows for our semantics of coordination not to r,'quire schemas nor entities of polymorphic type: our meaning of and is type t x t --+ t.</Paragraph> </Section> </Section> <Section position="7" start_page="35" end_page="37" type="metho"> <SectionTitle> 5 Intensional Verbs </SectionTitle> <Paragraph position="0"> We now return to consider cases involving intensional verbs. The preferred reading for sentence (23), in which only one quantifier scopes over the two extensional predicates, is shown below: The meaning constructors for the lexical items are given in Figure 1. Recall that a second meaning VX, Y. (fl SUBJ)~ ~'&quot; X (Vs,p. (VX. (fl susJ)~'--*X -o s--~p(X)) --o s-,~ Y(p)) ---o flz&quot;'* wanted(X, &quot;}&quot;) VX, Y. (f2 sUBJ)~-~.Y= '.9 (f20BJ)a&quot;&quot;~ Y --,o f2~-.-~found(X, Y) VX, Y. (f3 SUBJ),,---+X (r) (f30BJ)o&quot;--~Y --<, f3o-.--supported(X, Y) VX, Y. (f CONJ)a&quot;,~X @ (f CONJ)o.&quot;,.-~ Y ---o fo.-.-+and(X, Y) !(VX, Y. (f CONJ)cy&quot;c*X @ fa-.~ Y --o fo-.~and(X, Y)) constructor and2 is introduced by and in order to handle cases where there are more than two conjuncts; this contribution will be used once in the derivation of the meaning for sentence (23). The following R-relations result from the f-structural relationships: (i) R(f, CONJ. fl) (ii) R(f, CON J, f2) (iii) R(f, CONJ, f3) (iV) ~(fl, SUBJ, g) (v) R(f2, SUB J, g) (vi) /~(f3, SUB J, g) (vii) R(I1, OBJ, h) (viii) R(f2, OBJ, h) (ix) R(f3, OBJ, h) Following the analysis given in Dalrymple et al. (1994b), the lexical entry for want takes a quantified NP as an argument. This requires that the quantified NP meaning be duplicated, since otherwise no readings result. We provide a special rule for duplicating quantified NPs when necessary: o \[vtL s. (vx. F~x --o H~S(x)) --o H-,~,Q(S)\] \]) In the interest of space, again we only show a few steps of the derivation. Combining the meanings for Hillary, found, supported, and and, Axiom I, and R-relations (ii), (iii), (v), (vi), (viii), and (ix), we can derive: ha..~ x ---o fC/,-,-*and(found( Hillary, x), supported( Hillary, x ) ) ) We duplicate the meaning of two candidates using QNP Duplication, and combine one copy with the foregoing formula to yield: f o ..... t wo( z, candidate(z), and(found( Hillary, z), supported( Hillary, z ) ) ) We then combine the other meaning of two candidates with the meanings of Hillary and wanted. and using Axiom I and R-relations (i), (iv), and (vii) we obtain:</Paragraph> <Paragraph position="2"> Finally, using and2 with the two foregoing formulae, we deduce the desired result:</Paragraph> <Paragraph position="4"> We can now specify a Partee and Rooth style processing strategy, which is to prefer readings which require the least use of QNP duplication. This strategy predicts the readings generated for the examples in Section 4. It also predicts the desired reading for sentence (23), since that reading requires two quantifiers. While the reading generated by Partee and Rooth is derivable, it requires three quantifiers and thus uses QNP duplication twice, which is less preferred than the reading requiring two quantifiers which uses QNP duplication once. Also, it allows some flexibility in cases where pragmatics strongly suggests that quantitiers are copied and distributed for multiple extensional verbs; unlike the Partee and Rooth account, this would apply equally to the case where there are also intensional verbs and the case where there are not. Finally, our account readily applies to cases of intensional verbs without coordination as in example (10), since it applies more generally to cases of resource sharing.</Paragraph> </Section> class="xml-element"></Paper>