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<Paper uid="P01-1019">
  <Title>An Algebra for Semantic Construction in Constraint-based Grammars</Title>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
3 Interpretation
</SectionTitle>
    <Paragraph position="0"> The SEPs and EQs can be interpreted with respect to a first order model &lt;E,A,F&gt; where:  1. E is a set of events 2. A is a set of individuals 3. F is an interpretation function, which as- null signs tuples of appropriate kinds to the predicates of the language.</Paragraph>
    <Paragraph position="1"> The truth definition of the SEPs and EQs (which we group together under the term SMRS, for simple MRS) is as follows:  1. For all events and individuals v, [v]&lt;M,g&gt; = g(v).</Paragraph>
    <Paragraph position="2"> 2. For all n-predicates Pn,</Paragraph>
    <Paragraph position="4"> Thus, with respect to a modelM, an SMRS can be viewed as denoting an element of P(G), where G is the set of variable assignment functions (i.e., elements ofGassign the variablese,...andx,...</Paragraph>
    <Paragraph position="5"> their denotations): [smrs]M = {g : g is a variable assignment function and M |=g smrs} We now consider the semantics of the algebra. This must define the semantics of the operationop in terms of a function f which is defined entirely in terms of the denotations of op's arguments. In other words, [op(a1,a2)] = f([a1],[a2]) for some function f. Intuitively, where the SMRS of the SEMENT a1 denotes G1 and the SMRS of the SEMENT a2 denotes G2, we want the semantic value of the SMRS of op(a1,a2) to denote the following: G1 [?]G2 [?] [hook(a1) = hole(a2)] But this cannot be constructed purely as a function of G1 and G2.</Paragraph>
    <Paragraph position="6"> The solution is to add hooks and holes to the denotations of SEMENTS (cf. Zeevat, 1989). We define the denotation of a SEMENT to be an element of I x I x P(G), where I = E [?] A, as follows: Definition 4 Denotations of SEMENTs If a negationslash= [?] is a SEMENT, [[a]]M = &lt;[i],[iprime],G&gt; where:  1. [i] = hook(a) 2. [iprime] = hole(a) 3. G = {g : M |=g smrs(a)}</Paragraph>
    <Paragraph position="8"> So, the meanings of SEMENTs are ordered threetuples, consisting of the hook and hole elements (from I) and a set of variable assignment functions that satisfy the SMRS.</Paragraph>
    <Paragraph position="9"> We can now define the following operation f over these denotations to create an algebra: Definition 5 Semantics of the Semantic Construction Algebra &lt;I xI xP(G),f&gt; is an algebra, where:</Paragraph>
    <Paragraph position="11"> f([[a1]],[[a2]])).</Paragraph>
    <Paragraph position="12"> This follows from the definitions of [[]], op and f.</Paragraph>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
4 Labelling holes
</SectionTitle>
    <Paragraph position="0"> We now start considering the elaborations necessary for real grammars. As we suggested earlier, it is necessary to have multiple labelled holes.</Paragraph>
    <Paragraph position="1"> There will be a fixed inventory of labels for any grammar framework, although there may be some differences between variants.3 In HPSG, complements are represented using a list, but in general there will be a fixed upper limit for the number of complements so we can label holes COMP1,  between SPR and SUBJ that is often made in other HPSGs. the ERG is: SUBJ, SPR, SPEC, COMP1, COMP2, COMP3 and MOD (see Pollard and Sag, 1994).</Paragraph>
    <Paragraph position="2"> To illustrate the way the formalization goes with multiple slots, consider opsubj: Definition 6 The definition of opsubj opsubj(a1,a2) is the following: Ifa1 = [?] ora2 =</Paragraph>
    <Paragraph position="4"> where Tr stands for transitive closure.</Paragraph>
    <Paragraph position="5"> There will be similar operations opcomp1, opcomp2 etc for each labelled hole. These operations can be proved to form an algebra &lt;S,opsubj,opcomp1,...&gt; in a similar way to the unlabelled case shown in Theorem 1. A little more work is needed to prove that opl is closed on S. In particular, with respect to clause 2 of the above definition, it is necessary to prove that opl(a1,a2) = [?] or for all labels lprime, |holelprime(opl(a1,a2)) |[?] 1, but it is straightforward to see this is the case.</Paragraph>
    <Paragraph position="6"> These operations can be extended in a straight-forward way to handle simple constituent coordination of the kind that is currently dealt with in the ERG (e.g., Kim sleeps and talks and Kim and Sandy sleep); such cases involve daughters with non-empty holes of the same label, and the semantic operation equates these holes in the mother SEMENT.</Paragraph>
  </Section>
  <Section position="6" start_page="0" end_page="0" type="metho">
    <SectionTitle>
5 Scopal relationships
</SectionTitle>
    <Paragraph position="0"> The algebra with labelled holes is sufficient to deal with simple grammars, such as that in Sag and Wasow (1999), but to deal with scope, more is needed. It is now usual in constraint based grammars to allow for underspecification of quantifier scope by giving labels to pieces of semantic information and stating constraints between the labels. In MRS, labels called handles are associated with each EP. Scopal relationships are represented by EPs with handle-taking arguments.</Paragraph>
    <Paragraph position="1"> If all handle arguments are filled by handles labelling EPs, the structure is fully scoped, but in general the relationship is not directly specified in a logical form but is constrained by the grammar via additional conditions (handle constraints or hcons).4 A variety of different types of condition are possible, and the algebra developed here is neutral between them, so we will simply use relh to stand for such a constraint, intending it to be neutral between, for instance, =q (qeq: equality modulo quantifiers) relationships used in MRS and the more usual [?] relationships from UDRT (Reyle, 1993). The conditions in hcons are accumulated by append.</Paragraph>
    <Paragraph position="2"> To accommodate scoping in the algebra, we will make hooks and holes pairs of indices and handles. The handle in the hook corresponds to the LTOP feature in MRS. The new vocabulary is:  1. The absurdity symbol [?].</Paragraph>
    <Paragraph position="3"> 2. handles h1,h2,...</Paragraph>
    <Paragraph position="4"> 3. indices i1,i2,..., as before 4. n-predicates which take handles and indices as arguments 5. relh and =.</Paragraph>
    <Paragraph position="5"> The revised definition of an EP is as in MRS: Definition 7 Elementary Predications (EPs) An EP contains exactly four components: 1. a handle, which is the label of the EP 2. a relation 3. a list of zero or more ordinary variable arguments of the relation (i.e., indices) 4. a list of zero or more handles corresponding to scopal arguments of the relation.</Paragraph>
    <Paragraph position="6">  sentences can be related to first order models of the fully scoped forms (i.e., to models of WFFs without labels) via supervaluation (e.g., Reyle, 1993). This corresponds to stipulating that an underspecified logical form u entails a base, fully specified form ph only if all possible ways of resolving the underspecification in u entails ph. For reasons of space, we do not give details here, but note that this is entirely consistent with treating semantics in terms of a description of a logical formula. The relationship between the SEMENTS of non-sentential constituents and a more 'standard' formal language such as l-calculus will be explored in future work. This is written h:r(a1,...,an,sa1,...,sam). For instance, h:every(x,h1,h2) is an EP.5 We revise the definition of semantic entities to add the hcons conditions and to make hooks and holes pairs of handles and indices.</Paragraph>
    <Paragraph position="7">  H-Cons Conditions: Where h1 and h2 are handles, h1relhh2 is an H-Cons condition.</Paragraph>
    <Paragraph position="8"> Definition 8 The Set S of Semantic Entities s [?] S if and only if s = [?] or s = &lt;s1,s2,s3,s4,s5&gt; such that: * s1 = {[h,i]} is a hook; * s2 = [?] or {[hprime,iprime]} is a hole; * s3 is a bag of EP conditions * s4 is a bag of HCONS conditions * s5 is a set of equalities between variables.</Paragraph>
    <Paragraph position="9">  SEMENTs are: [h1,i1]{holes}[eps][hcons]{eqs}.</Paragraph>
    <Paragraph position="10"> We will not repeat the full composition definition, since it is unchanged from that in SS2 apart from the addition of the append operation on hcons and a slight complication of eq to deal with the handle/index pairs:</Paragraph>
    <Paragraph position="12"> where Tr stands for transitive closure as before and hdle and ind access the handle and index of a pair. We can extend this to include (several) labelled holes and operations, as before. And these revised operations still form an algebra.</Paragraph>
    <Paragraph position="13"> The truth definition for SEMENTS is analogous to before. We add to the model a set of labels L (handles denote these via g) and a well-founded partial order [?] on L (this helps interpret the hcons; cf. Fernando (1997)). A SEMENT then denotes an element of Hx...HxP(G), where the Hs (= LxI) are the new hook and holes.</Paragraph>
    <Paragraph position="14"> Note that the language S is first order, and we do not use l-abstraction over higher order elements.6 For example, in the standard Montagovian view, a quantifier such as every  we could make use of l-abstraction as a representation device, for instance for dealing with adjectives such as former, cf., Moore (1989).</Paragraph>
    <Paragraph position="15"> is represented by the higher-order expression lPlQ[?]x(P(x),Q(x)). In our framework, however, every is the following (using qeq conditions, as in the LinGO ERG): [hf,x]{[]subj,[]comp1,[hprime,x]spec,...} [he : every(x,hr,hs)][hr =q hprime]{} and dog is: [hd,y]{[]subj,[]comp1,[]spec,...}[hd : dog(y)][]{} So these composes via opspec to yield every dog: [hf,x]{[]subj,[]comp1,[]spec,...} [he : every(x,hr,hs),hd : dog(y)] [hr =q hprime]{hprime = hd,x = y} This SEMENT is semantically equivalent to: [hf,x]{[]subj,[]comp1,[]spec,...} [he : every(x,hr,hs),hd : dog(x)][hr =q hd]{} A slight complication is that the determiner is also syntactically selected by the Nprime via the SPR slot (following Pollard and Sag (1994)). However, from the standpoint of the compositional semantics, the determiner is the semantic head, and it is only its SPEC hole which is involved: the Nprime must be treated as having an empty SPR hole. In the ERG, the distinction between intersective and scopal modification arises because of distinctions in representation at the lexical level. The repetition of variables in the SEMENT of a lexical sign (corresponding to TFS coindexation) and the choice of type on those variables determines the type of modification.</Paragraph>
    <Paragraph position="16"> Intersective modification: white dog: dog: [hd,y]{[]subj,[]comp1,...,[]mod}</Paragraph>
    <Paragraph position="18"> Scopal Modification: probably walks: walks: [hw,eprime]{[hprime,x]subj,[]comp1,...,[]mod} [hw : walks(eprime,x)][]{} probably: [hp,e]{[]subj,[]comp1,...,[h,e]mod} [hp : probably(hs)][hs =q h]{} probably [hp,e]{[hprime,x]subj,[]comp1,...,[]mod} walks: [hp:probably(hs),hw:walks(eprime,x)] (opmod) [hs =q h]{hw = h,e = eprime}</Paragraph>
  </Section>
  <Section position="7" start_page="0" end_page="0" type="metho">
    <SectionTitle>
6 Control and external arguments
</SectionTitle>
    <Paragraph position="0"> We need to make one further extension to allow for control, which we do by adding an extra slot to the hooks and holes corresponding to the external argument (e.g., the external argument of a verb always corresponds to its subject position). We illustrate this by showing two uses of expect; note the third slot in the hooks and holes for the external argument of each entity. In both cases, xprimee is both the external argument of expect and its subject's index, but in the first structure xprimee is also the external argument of the complement, thus giving the control effect.</Paragraph>
    <Paragraph position="1"> expect 1 (as in Kim expected to sleep) [he,ee,xprimee]{[hs,xprimee,xprimes]subj,[hc,ec,xprimee]comp1,...} [he : expect(ee,xprimee,hprimee)][hprimee =q hc]{} expect 2 (Kim expected that Sandy would sleep) [he,ee,xprimee]{[hs,xprimee,xprimes]subj,[hc,ec,xprimec]comp1,...} [h : expect(ee,xprimee,hprimee)][hprimee =q hc]{} Although these uses require different lexical entries, the semantic predicate expect used in the two examples is the same, in contrast to Montagovian approaches, which either relate two distinct predicates via meaning postulates, or require an additional semantic combinator. The HPSG account does not involve such additional machinery, but its formal underpinnings have been unclear: in this algebra, it can be seen that the desired result arises as a consequence of the restrictions on variable assignments imposed by the equalities.</Paragraph>
    <Paragraph position="2"> This completes our sketch of the algebra necessary to encode semantic composition in the ERG.</Paragraph>
    <Paragraph position="3"> We have constrained accessibility by enumerating the possible labels for holes and by stipulating the contents of the hooks. We believe that the handle, index, external argument triple constitutes all the semantic information that a sign should make accessible to a functor. The fact that only these pieces of information are visible means, for instance, that it is impossible to define a verb that controls the object of its complement.7 Although obviously changes to the syntactic valence features would necessitate modification of the hole labels, we think it unlikely that we will need to increase the inventory further. In combination with 7Readers familiar with MRS will notice that the KEY feature used for semantic selection violates these accessibility conditions, but in the current framework, KEY can be replaced by KEYPRED which points to the predicate alone. the principles defined in Copestake et al (1999) for qeq conditions, the algebra presented here results in a much more tightly specified approach to semantic composition than that in Pollard and Sag (1994).</Paragraph>
  </Section>
  <Section position="8" start_page="0" end_page="0" type="metho">
    <SectionTitle>
7 Comparison
</SectionTitle>
    <Paragraph position="0"> Compared with l-calculus, the approach to composition adopted in constraint-based grammars and formalized here has considerable advantages in terms of simplicity. The standard Montague grammar approach requires that arguments be presented in a fixed order, and that they be strictly typed, which leads to unnecessary multiplication of predicates which then have to be interrelated by meaning postulates (e.g., the two uses of expect mentioned earlier). Type raising also adds to the complexity. As standardly presented, l-calculus does not constrain grammars to be monotonic, and does not control accessibility, since the variable of the functor that is l-abstracted over may be arbitrarily deeply embedded inside a lexpression. null None of the previous work on unification-based approaches to semantics has considered constraints on composition in the way we have presented. In fact, Nerbonne (1995) explicitly advocates nonmonotonicity. Moore (1989) is also concerned with formalizing existing practice in unification grammars (see also Alshawi, 1992), though he assumes Prolog-style unification, rather than TFSs. Moore attempts to formalize his approach in the logic of unification, but it is not clear this is entirely successful. He has to divorce the interpretation of the expressions from the notion of truth with respect to the model, which is much like treating the semantics as a description of a logic formula. Our strategy for formalization is closest to that adopted in Unification Categorial Grammar (Zeevat et al, 1987), but rather than composing actual logical forms we compose partial descriptions to handle semantic underspecification.</Paragraph>
  </Section>
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