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<Paper uid="P02-1041">
  <Title>Coupling CCG and Hybrid Logic Dependency Semantics</Title>
  <Section position="4" start_page="0" end_page="11" type="metho">
    <SectionTitle>
2 Indexed semantic representations
</SectionTitle>
    <Paragraph position="0"> Traditionally, categorial grammar has captured meaning using a (simply typed) l-calculus, building semantic structure in parallel to the categorial inference (Morrill, 1994; Moortgat, 1997; Steedman, 2000b). For example, a (simplified) CCG lexical entry for a verb such as wrote isgivenin(1).</Paragraph>
    <Paragraph position="1">  (1) wrote CO B4D7D2D2B5BPD2 : lxBMlyBMwriteB4yBNxB5 Rules of combination are defined to operate on both categories and l-terms simultaneously. For example, the rules allow the following derivation for Ed wrote books.</Paragraph>
    <Paragraph position="2"> (2) Ed wrote books</Paragraph>
    <Paragraph position="4"> Derivations like (2) give rise to the usual sort of predicate-argument structure whereby the order in which the arguments appear (and are bound by the l's) is essentially constitutive of their meaning.</Paragraph>
    <Paragraph position="5"> Thus, the first argument could be taken to correspond to the writer, whereas the second argument corresponds to what is being written.</Paragraph>
    <Paragraph position="6"> Computational Linguistics (ACL), Philadelphia, July 2002, pp. 319-326. Proceedings of the 40th Annual Meeting of the Association for One deficiency of l-calculus meaning representations is that they usually have to be type-raised to the worst case to fully model quantification, and this can reverberate and increase the complexity of syntactic categories since a verb like wrote will need to be able to take arguments with the types of generalized quantifiers. The approach we advocate in this paper does not suffer from this problem.</Paragraph>
    <Paragraph position="7"> For CCG, the use of the l-terms is simply a convenient device to bind arguments when presenting derivations (Steedman, 2000b). In implementations, a more common strategy is to compute semantic representations via unification, a tactic explicitly employed in Unification Categorial Grammar (UCG) (Zeevat, 1988). Using a unification paradigm in which atomic categories are bundles of syntactic and semantic information, we can use an entry such as  (3) for wrote in place of (1). In the unification setting, (3) permits a derivation analogous to (2).</Paragraph>
    <Paragraph position="8"> (3) wrote CO B4D7 : writeB4yBNxB5D2D2:yB5BPD2:x  For creating predicate-argument structures of this kind, strategies using either l-terms or unification to bind arguments are essentially notational variants. However, UCG goes beyond simple predicate-argument structures to instead use a semantic representation language called Indexed Language (InL). The idea of using indexes stems from Davidson (event variables), and are a commonly used mechanism in unification-based frameworks and theories for discourse representation. InL attaches one to every formula representing its discourse referent. This results in a representation such as (4) for the sentence Ed came to the party.</Paragraph>
    <Paragraph position="9"> (4) CJeCLCJpartyB4xB5BNpastB4eB5BNtoB4eBNxB5BNcomeB4eBNEdB5CL InL thus flattens logical forms to some extent, using the indexes to spread a given entity or event through multiple predications. The use of indexes is crucial for UCG's account of modifiers, and as we will see later, we exploit such referents to achieve similar ends when coupling HLDS and CCG.</Paragraph>
    <Paragraph position="10"> Minimal Recursion Semantics (MRS) (Copestake et al., 1999; Copestake et al., 2001) is a framework for computational semantics that is designed to simplify the work of algorithms which produce or use semantic representations. MRS provides the means to represent interpretations with a flat, underspecified semantics using terms of the predicate calculus and generalized quantifiers. Flattening is achieved by using an indexation scheme involving labels that tag particular groups of elementary predications (EPs) and handles (here, h  erence those EPs. Underspecification is achieved by using unresolved handles as the arguments for scope-bearing elements and declaring constraints (with the BP q operator) on how those handles can be resolved. Different scopes can be reconstructed by equating unresolved handles with the labels of the other EPs obeying the BP q constraints. For example, (5) would be given as the representation for every dog chases some white cat.</Paragraph>
    <Paragraph position="11"> (5) CWh</Paragraph>
  </Section>
  <Section position="5" start_page="11" end_page="11" type="metho">
    <SectionTitle>
CVCX
</SectionTitle>
    <Paragraph position="0"> Copestake et al. argue that these flat representations facilitate a number of computational tasks, including machine translation and generation, without sacrificing linguistic expressivity. Also, flatness permits semantic equivalences to be checked more easily than in structures with deeper embedding, and underspecification simplifies the work of the parser since it does not have to compute every possible reading for scope-bearing elements.</Paragraph>
  </Section>
  <Section position="6" start_page="11" end_page="11" type="metho">
    <SectionTitle>
3 Hybrid Logic Dependency Semantics
</SectionTitle>
    <Paragraph position="0"> Kruijff (2001) proposes an alternative way to representing linguistically realized meaning: namely, as terms of hybrid modal logic (Blackburn, 2000) explicitly encoding the dependency relations between heads and dependents, spatio-temporal structure, contextual reference, and information structure. We call this unified perspective combining many levels of meaning Hybrid Logic Dependency Semantics (HLDS). We begin by discussing how hybrid logic extends modal logic, then look at the representation of linguistic meaning via hybrid logic terms.</Paragraph>
    <Section position="1" start_page="11" end_page="11" type="sub_section">
      <SectionTitle>
3.1 Hybrid Logic
</SectionTitle>
      <Paragraph position="0"> Though modal logic provides a powerful tool for encoding relational structures and their properties, it contains a surprising inherent asymmetry: states (&amp;quot;worlds&amp;quot;) are at the heart of the model theory for modal logic, but there are no means to directly reference specific states using the object language.</Paragraph>
      <Paragraph position="1"> This inability to state where exactly a proposition holds makes modal logic an inadequate representation framework for practical applications like knowledge representation (Areces, 2000) or temporal reasoning (Blackburn, 1994). Because of this, computational work in knowledge representation has usually involved re-engineering first-order logic to suit the task, e.g., the use of metapredicates such as Hold of Kowalski and Allen. Unfortunately, such logics are often undecidable.</Paragraph>
      <Paragraph position="2"> Hybrid logic extends standard modal logic while retaining decidability and favorable complexity (Areces, 2000) (cf. (Areces et al., 1999) for a complexity roadmap). The strategy is to add nominals, a new sort of basic formula with which we can explicitly name states in the object language. Next to propositions, nominals are first-class citizens of the object language: formulas can be formed using both sorts, standard boolean operators, and the satisfac-</Paragraph>
      <Paragraph position="4"> p states that the formula p holds at the state named by i.  (There are more powerful quantifiers ranging over nominals, such as AZ, but we do not consider them here.) With nominals we obtain the possibility to explicitly refer to the state at which a proposition holds. As Blackburn (1994) argues, this is essential for capturing our intuitions about temporal reference. A standard modal temporal logic with the modalities BY and C8 (future and past, respectively) cannot correctly represent an utterance such as Ed finished the book because it is unable to refer to the specific time at which the event occurred. The addition of nominals makes this possible, as shown in (6), where the nominal i represents the Reichenbachian event time. (6) CWC8CXB4iCMEd-finish-bookB5 Furthermore, many temporal properties can be defined in terms of pure formulas which use nominals and contain no propositional variables. For example, the following term defines the fact that the relations for BY and C8 are mutually converse:  A few notes on our conventions: pBNqBNr are variables over any hybrid logic formula; iBN jBNk are variables over nominals; d</Paragraph>
      <Paragraph position="6"> denote nominals (for dependent and head, respectively).</Paragraph>
      <Paragraph position="8"> It is also possible to encode a variety of other representations in terms of hybrid logics. For example, nominals correspond to tags in attribute-value matrices (AVMs), so the hybrid logic formula in (8) corresponds to the AVM in (9).</Paragraph>
      <Paragraph position="9">  A crucial aspect of hybrid logic is that nominals are at the heart of a sorting strategy. Different sorts of nominals can be introduced to build up a rich sortal ontology without losing the perspicuity of a propositional setting. Additionally, we can reason about sorts because nominals are part and parcel of the object language. We can extend the language of hybrid logic with CUCBD3D6D8:NominalCV to facilitate the explicit statement of what sort a nominal is in the language and carry this modification into one of the existing tableaux methods for hybrid logic to reason effectively with this information. This makes it possible to capture the rich ontologies of lexical databases like WordNet in a clear and concise fashion which would be onerous to represent in first-order logic.</Paragraph>
    </Section>
    <Section position="2" start_page="11" end_page="11" type="sub_section">
      <SectionTitle>
3.2 Encoding linguistic meaning
</SectionTitle>
      <Paragraph position="0"> Hybrid logic enables us to logically capture two essential aspects of meaning in a clean and compact way, namely ontological richness and the possibility to refer. Logically, we can represent an expression's linguistically realized meaning as a conjunction of modalized terms, anchored by the nominal that identifies the head's proposition:</Paragraph>
      <Paragraph position="2"> CX, and with each dependent we associate a nominal d</Paragraph>
      <Paragraph position="4"> the state where a dependent expressed as a proposition dep i should be evaluated and is a d</Paragraph>
      <Paragraph position="6"> of h, the nominal identifying the head. As an example, the sentence Ed wrote a long book in London receives the represention in (11).</Paragraph>
      <Paragraph position="7">  The modal relations ACT,PAT,LOC, and GR stand for the dependency relations Actor, Patient, Locative,andGeneral Relationship, respectively. See Kruijff (2001) for the model-theoretic interpretation of expressions such as (11).</Paragraph>
      <Paragraph position="8"> Contextual reference can be modeled as a statement that from the current state (anaphor) there should be an accessible antecedent state at which particular conditions hold. Thus, assuming an accessibility relation XS, we can model the meaning of the pronoun he as in (12).</Paragraph>
      <Paragraph position="10"> During discourse interpretation, this statement is evaluated against the discourse model. The pronoun is resolvable only if a state where male holds is XSaccessible in the discourse model. Different accessibility relations can be modeled, e.g. to distinguish a local context (for resolving reflexive anaphors like himself ) from a global context (Kruijff, 2001).</Paragraph>
      <Paragraph position="11"> Finally, the rich temporal ontology underlying models of tense and aspect such as Moens and Steedman (1988) can be captured using the sorting strategy. Earlier work like Blackburn and Lascarides (1992) already explored such ideas. HLDS employs hybrid logic to integrate Moens and Steedman's notion of the event nucleus directly into meaning representations. The event nucleus is a tripartite structure reflecting the underlying semantics of a type of event. The event is related to a preparation (an activity bringing the event about) and a consequent (a state ensuing to the event), which we encode as the modal relations PREP and CONS, respectively. Different kinds of states and events are modeled as different sorts of nominals, shown in (13) using the notation introduced above.</Paragraph>
      <Paragraph position="13"> To tie (13) in with a representation like (11), we equate the nominal of the head with one of the nominals in the event nucleus (E)a and state its temporal relation (e.g. CWPCX). Given the event nucleus in (13), the representation in (11) becomes (14), where the event is thus located at a specific time in the past.</Paragraph>
      <Paragraph position="14">  Hybrid logic's flexibility makes it amenable to representing a wide variety of semantic phenomena in a propositional setting, and it can furthermore be used to formulate a discourse theory (Kruijff and Kruijff-Korbayov'a, 2001).</Paragraph>
    </Section>
    <Section position="3" start_page="11" end_page="11" type="sub_section">
      <SectionTitle>
3.3 Comparison to MRS
</SectionTitle>
      <Paragraph position="0"> Here we consider the properties of HLDS with respect to the four main criteria laid out by Copestake et al. (1999) which a computational semantics framework must meet: expressive adequacy, grammatical compatibility, computational tractability, and underspecifiability.</Paragraph>
      <Paragraph position="1"> Expressive adequacy refers to a framework's ability to correctly express linguistic meaning. HLDS was designed not only with this in mind, but as its central tenet. In addition to providing the means to represent the usual predicate-valency relations, it explicitly marks the named dependency relations between predicates and their arguments and modifiers. These different dependency relations are not just labels: they all have unique semantic imports which project new relations in the context of different heads. HLDS also tackles the representation of tense and aspect, contextual reference, and information structure, as well as their interaction with discourse. null The criterion of grammatical compatibility requires that a framework be linkable to other kinds of grammatical information. Kruijff (2001) shows that HLDS can be coupled to a rich grammatical framework, and in DC4 we demonstrate that it can be tied to CCG, a much lower power formalism than that assumed by Kruijff. It should furthermore be straight-forward to use our approach to hook HLDS up to other unification-based frameworks.</Paragraph>
      <Paragraph position="2"> The definition of computational tractability states that it must be possible to check semantic equivalence of different formulas straightforwardly. Like MRS, HLDS provides the means to view linguistic meaning in a flattened format and thereby ease the checking of equivalence. For example, (15) describes the same relational structure as (11).</Paragraph>
      <Paragraph position="3">  This example clarifies how the use of nominals is related to the indexes of UCG's InL and the labels of MRS. However, there is an important difference: nominals are full citizens of the object language with semantic import and are not simply a device for spreading meaning across several elementary predications. They simultaneously represent tags on sub-parts of a logical form and discourse referents on which relations are predicated. Because it is possible to view an HLDS term as a flat conjunction of the heads and dependents inside it, the benefits described by Copestake et al. with respect to MRS's flatness thus hold for HLDS as well.</Paragraph>
      <Paragraph position="4"> Computational tractability also requires that it is straightforward to express relationships between representations. This can be done in the object language of HLDS as hybrid logic implicational statements which can be used with proof methods to discover deeper relationships. Kruijff's model connecting linguistic meaning to a discourse context is one example of this.</Paragraph>
      <Paragraph position="5"> Underspecifiability means that semantic representations should provide means to leave some semantic distinctions unresolved whilst allowing partial terms to be flexibly and monotonically resolved. (5) shows how MRS leaves quantifier scope underspecified, and such formulas can be transparently encoded in HLDS. Consider (16), where the relations RESTR and BODY represent the restriction and body arguments of the generalized quantifiers, respectively.  MRS-style underspecification is thus replicated by declaring new nominals and modeling BP q as a modal relation between nominals. When constructing the fully-scoped structures generated by an underspecified one, the BP q constraints must be obeyed according to the qeq condition of Copestake etal. Because HLDS is couched directly in terms of hybrid logic, we can concisely declare the qeq condition as the following implication:</Paragraph>
      <Paragraph position="7"> Alternatively, it would in principle be possible to adopt a truly modal solution to the representation of quantifiers. Following Alechina (1995), (generalized) quantification can be modeled as modal operators. The complexity of generalized quantification is then pushed into the model theory instead of forcing the representation to carry the burden.</Paragraph>
    </Section>
  </Section>
  <Section position="7" start_page="11" end_page="11" type="metho">
    <SectionTitle>
4 CCG Coupled to HLDS
</SectionTitle>
    <Paragraph position="0"> In Dependency Grammar Logic (DGL), Kruijff (2001) couples HLDS to a resource-sensitive categorial proof theory (CTL) (Moortgat, 1997). Though DGL demonstrates a procedure for building HLDS terms from linguistic expressions, there are several problems we can overcome by switching to CCG. First, parsing with CCG grammars for substantial fragments is generally more efficient than with CTL grammars with similar coverage. Also, a wide-coverage statistical parser which produces syntactic dependency structures for English is available for CCG (Clark et al., 2002). Second, syntactic features (modeled by unary modalities) in CTL have no intuitive semantic reflection, whereas CCG can relate syntactic and semantic features perspicuously using unification.</Paragraph>
    <Paragraph position="1"> Finally, CCG has a detailed syntactic account of the realization of information structure in English.</Paragraph>
    <Paragraph position="2"> To link syntax and semantics in derivations, every logical form in DGL expresses a nominal identifying its head in the format @</Paragraph>
    <Paragraph position="4"> p. This handles dependents in a linguistically motivated way through a linking theory: given the form of a dependent, its (possible) role is established, after which its meaning states that it seeks a head that can take such a role. However, to subsequently bind that dependent into the verb's argument slot requires logical axioms about the nature of various dependents. This not only requires extra reduction steps to arrive at the desired logical form, but could also lead to problems depending on the underlying theory of roles.</Paragraph>
    <Paragraph position="5"> We present an alternative approach to binding dependents, which overcomes these problems without abandoning the linguistic motivation. Because we work in a lexicalist setting, we can compile the effects of the linguistic linking theory directly into category assignments.</Paragraph>
    <Paragraph position="6"> The first difference in our proposal is that arguments express only their own nominal, not the nominal of a head as well. For example, proper nouns receive categories such as (18).</Paragraph>
    <Paragraph position="7">  This entry highlights our relaxation of the strict connection between syntactic and semantic types traditionally assumed in categorial grammars, a move in line with the MRS approach.</Paragraph>
    <Paragraph position="8"> In contrast with DGL, the semantic portion of a syntactic argument in our system does not declare the role it is to take and does not identify the head it is to be part of. Instead it identifies only its own referent. Without using additional inference steps, this is transmuted via unification into a form similar to DGL's in the result category. (19) is an example of the kind of head category needed.</Paragraph>
    <Paragraph position="9">  To produce HLDS terms that are fully compatible with the way that Kruijff and Kruijff-Korbayov'a (2001) model discourse, we need to mark the informativity of dependents as contextually bound (CB) and contextually nonbound (NB). In DGL, these appear as modalities in logical forms that are used to create a topic-focus articulation that is merged with the discourse context. For example, the sentence he wrote a book would receive the following (simpli null (Moortgat, 1997) to instantiate the values of informativity. In unification-based approaches such as CCG, the transferal of feature information into semantic representations is standard practice. We thus employ the feature inf and mark informativity in logical forms with values resolved syntactically.  resentations comes with adjuncts. With HLDS, we consider the prepositional verbal modifier in the sentence Ed sleeps in the bed as an optional Locative dependent of sleeps. To implement this, we follow DGL in identifying the discourse referent of the head with that of the adjunct. However, unlike DGL, this is compiled into the category for the adjunct.</Paragraph>
    <Paragraph position="10">  bed This approach thus allows adjuncts to insert their semantic import into the meaning of the head, making use of nominals in a manner similar to the use of indexes in Unification Categorial Grammar.</Paragraph>
  </Section>
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