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<?xml version="1.0" standalone="yes"?> <Paper uid="C04-1028"> <Title>Generalizing Dimensionality in Combinatory Categorial Grammar</Title> <Section position="3" start_page="0" end_page="0" type="metho"> <SectionTitle> 2 Combinatory Categorial Grammar </SectionTitle> <Paragraph position="0"> In this section, we give an overview of syntactic combination and semantic construction in CCG. We use CCG's multi-modal extension (Baldridge and Kruijff, 2003), which enriches the inventory of slash types. This formalization renders constraints on rules unnecessary and supports a universal set of rules for all grammars.</Paragraph> <Section position="1" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 2.1 Categories and combination </SectionTitle> <Paragraph position="0"> Nearly all syntactic behavior in CCG is encoded in categories. They may be atoms, like np, or functions which specify the direction in which they seek their arguments, like (s\np)/np. The latter is the category for English transitive verbs; it first seeks its object to its right and then its subject to its left.</Paragraph> <Paragraph position="1"> Categories combine through a small set of universal combinatory rules. The simplest are application rules which allow a function category to consume its argument either on its right (>) or on its left (<):</Paragraph> <Paragraph position="3"> Four further rules allow functions to compose with other functions:</Paragraph> <Paragraph position="5"> The modalities star, diamondmath and x on the slashes enforce different kinds of combinatorial potential on categories. For a category to serve as input to a rule, it must contain a slash which is compatible with that specified by the rule. The modalities work as follows. star is the most restricted modality, allowing combination only by the application rules (> and <). diamondmath allows combination with the application rules and the order-preserving composition rules (>B and <B). x allows limited permutation via the crossed composition rules (>Bx and <Bx) as well as the application rules. Additionally, a permissive modality * allows combination by all rules in the system. However, we suppress the * modality on slashes to avoid clutter. An undecorated slash may thus combine by all rules.</Paragraph> <Paragraph position="6"> There are two further rules of type-raising that turn an argument category into a function over functions that seek that argument:</Paragraph> <Paragraph position="8"> The variable modality i on the output categories constrains both slashes to have the same modality.</Paragraph> <Paragraph position="9"> These rules support the following incremental derivation for Marcel proved completeness: This derivation does not display the effect of using modalities in CCG; see Baldridge (2002) and Baldridge and Kruijff (2003) for detailed linguistic justification for this modalized formulation of CCG.</Paragraph> </Section> <Section position="2" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 2.2 Hybrid Logic Dependency Semantics </SectionTitle> <Paragraph position="0"> Many different kinds of semantic representations and ways of building them with CCG exist. We use Hybrid Logic Dependency Semantics (HLDS) (Kruijff, 2001), a framework that utilizes hybrid logic (Blackburn, 2000) to realize a dependency-based perspective on meaning.</Paragraph> <Paragraph position="1"> Hybrid logic provides a language for representing relational structures that overcomes standard modal logic's inability to directly reference states in a model. This is achieved via nominals, a kind of basic formula which explicitly names states. Like propositions, nominals are first-class citizens of the object language, so formulas can be formed using propositions, nominals, standard boolean operators, and the satisfaction operator &quot;@&quot;. A formula @i(p[?]<F> (j [?]q)) indicates that the formulas p and <F> (j [?] q) hold at the state named by i and that the state j is reachable via the modal relation F.</Paragraph> <Paragraph position="2"> In HLDS, hybrid logic is used as a language for describing semantic interpretations as follows.</Paragraph> <Paragraph position="3"> Each semantic head is associated with a nominal that identifies its discourse referent and heads are connected to their dependents via dependency relations, which are modeled as modal relations. As an example, the sentence Marcel proved completeness receives the representation in (2).</Paragraph> <Paragraph position="4"> (2) @e(prove [?]<TENSE> past [?]<ACT> (m[?]Marcel)[?]<PAT> (c[?]comp.)) In this example, e is a nominal that labels the predications and relations for the head prove, and m and c label those for Marcel and completeness, respectively. The relations ACT and PAT represent the dependency roles Actor and Patient, respectively.</Paragraph> <Paragraph position="5"> By using the @ operator, hierarchical terms such as (2) can be flattened to an equivalent conjunction of fixed-size elementary predications (EPs): (3) @eprove [?] @e<TENSE> past [?] @e<ACT> m [?] @e<PAT> c[?] @mMarcel [?] @ccomp.</Paragraph> </Section> <Section position="3" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 2.3 Semantic Construction </SectionTitle> <Paragraph position="0"> Baldridge and Kruijff (2002) show how HLDS representations can be built via CCG derivations.</Paragraph> <Paragraph position="1"> White (2004) improves HLDS construction by operating on flattened representations such as (3) and using a simple semantic index feature in the syntax.</Paragraph> <Paragraph position="2"> We adopt this latter approach, described below.</Paragraph> <Paragraph position="3"> EPs are paired with syntactic categories in the lexicon as shown in (4)-(6) below. Each atomic category has an index feature, shown as a subscript, which makes a nominal available for capturing syn- null tactically induced dependencies.</Paragraph> <Paragraph position="4"> (4) prove turnstileleft (se\npx)/npy : @eprove [?] @e<TENSE> past [?] @e<ACT> x[?] @e<PAT> y (5) Marcel turnstileleft npm : @mMarcel (6) completeness turnstileleft npc : @ccompleteness Applications of the combinatory rules co-index the appropriate nominals via unification on the categories. EPs are then conjoined to form the resulting interpretation. For example, in derivation (1), (5) type-raises and composes with (4) to yield (7). The index x is syntactically unified with m, and this resolution is reflected in the new conjoined logical form. (7) can then apply to (6) to yield (8), which has the same conjunction of predications as (3). (7) Marcel proved turnstileleft se/npy : @eprove [?] @e<TENSE> past [?] @e<ACT> m[?] @e<PAT> y [?] @mMarcel (8) Marcel proved completeness turnstileleft se : @eprove [?] @e<TENSE> past [?] @e<ACT> m [?]@e<PAT> c[?]@mMarcel [?]@ccompleteness Since the EPs are always conjoined by the combinatory rules, semantic construction is guaranteed to be monotonic. No semantic information can be dropped during the course of a derivation. This provides a clean way of establishing semantic dependencies as informed by the syntactic derivation. In the next section, we extend this paradigm for use with any number of representational levels.</Paragraph> </Section> </Section> <Section position="4" start_page="0" end_page="0" type="metho"> <SectionTitle> 3 Generalized dimensionality </SectionTitle> <Paragraph position="0"> To support a more modular and perspicuous encoding of multiple levels of analysis, we generalize the notion of sign commonly used in CCG. The approach is inspired on the one hand by earlier work by Steedman (2000a) and Hoffman (1995), and on the other by the signs found in constraint-based approaches to grammar. The principle idea is to extend White's (2004) approach to semantic construction (see SS2.3). There, categories and the meaning they help express are connected through coindexation. Here, we allow for information in any (finite) number of levels to be related in this way.</Paragraph> <Paragraph position="1"> A sign is an n-tuple of terms that represent information at n distinct dimensions. Each dimension represents a level of linguistic information such as prosody, meaning, or syntactic category. As a representation, we assume that we have for each dimension a language that defines well-formed representations, and a set of operations which can create new representations from a set of given representations.1 For example, we have by definition a dimension for syntactic categories. The language for this dimension is defined by the rules for category construction: given a set of atomic categories A, C is a category iff (i) C [?] A or (ii) C is of the form A\mB or A/mB with A,B categories and m [?] {star,diamondmathx,*}. The set of combinatory rules defines the possible operations on categories.</Paragraph> <Paragraph position="2"> This syntactic category dimension drives the grammatical analysis, thus guiding the composition of signs. When two categories are combined via a rule, the appropriate indices are unified. It is through this unification of indices that information can be passed between signs. At a given dimension, the co-indexed information coming from the two signs we combine must be unifiable.</Paragraph> <Paragraph position="3"> With these signs, dimensions interact in a more limited way than in HPSG or LFG. Constraints (resolved through unification) may only be applied if they are invoked through co-indexation on categories. This provides a bound on the number of indices and the number of unifications to be made.</Paragraph> <Paragraph position="4"> As such, full recursion and complex unification as in attribute-value matrices with re-entrancy is avoided.</Paragraph> <Paragraph position="5"> The approach incorporates various ideas from constraint-based approaches, but remains based on a derivational perspective on grammatical analysis and derivational control, unlike e.g Categorial Unification Grammar. Furthermore, the ability for dimensions to interact through shared indices brings several advantages: (1) &quot;parallel derivations&quot; (Hoffman, 1995) are unnecessary; (2) non-isomorphic, functional structures across different dimensions can be employed; and (3) there is no longer a need to load all the necessary information into syntactic categories (as with Kruijff (2001)).</Paragraph> <Paragraph position="6"> 1In the context of this paper we assume operations are multiplicative. Also, note that dimensions may differ in what languages and operations they use.</Paragraph> </Section> <Section position="5" start_page="0" end_page="0" type="metho"> <SectionTitle> 4 Examples </SectionTitle> <Paragraph position="0"> In this section, we illustrate our approach on several examples involving information structure. We use signs that include the following dimensions.</Paragraph> <Paragraph position="1"> Phonemic representation: word sequences, composition of sequences is through concatenation Prosody: sequences of tunes from the inventory of (Pierrehumbert and Hirschberg, 1990), composition through concatenation Syntactic category: well-formed categories, combinatory rules (see SS2) Information structure: hybrid logic formulas of the form @d[in]r, with r a discourse referent that has informativity in (theme th, or rheme r) relative to the current point in the discourse d (Kruijff, 2003). Predicate-argument structure: hybrid logic formulas of the form as discussed in SS2.3.</Paragraph> <Paragraph position="2"> Example (9) illustrates a sign with these dimensions. The word-form Marcel bears an H* accent, and acts as a type-raised category that seeks a verb missing its subject. The H* accent indicates that the discourse referent m introduces new information at the current point in the discourse d: i.e. the meaning @mmarcel should end up as part of the rheme (r) of the utterance, @d[r]m.</Paragraph> <Paragraph position="4"> If a sign does not specify any information at a particular dimension, this is indicated by latticetop (or an empty line if no confusion can arise).</Paragraph> <Section position="1" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 4.1 Topicalization </SectionTitle> <Paragraph position="0"> We start with a simple example of topicalization in English. In topicalized constructions, a thematic object is fronted before the subject. Given the question Did Marcel prove soundness and completeness?, (10) is a possible response using topicalization: (10) Completeness, Marcel proved, and sound null ness, he conjectured.</Paragraph> <Paragraph position="1"> We can capture the syntactic and information structure effects of such sentences by assigning the following kind of sign to (topicalized) noun phrases: This category enables the derivation in Figure 1. The type-raised subject composes with the verb, and the result is consumed by the topicalizing category. The information structure specification stated in the sign in (11) is passed through to the final sign. The topicalization of the object in (10) only indicates the informativity of the discourse referent realized by the object. It does not yield any indications about the informativity of other constituents; hence the informativity for the predicate and the Actor is left unspecified. In English, the informativity of these discourse referents can be indicated directly with the use of prosody, to which we now turn.</Paragraph> </Section> <Section position="2" start_page="0" end_page="0" type="sub_section"> <SectionTitle> 4.2 Prosody & information structure </SectionTitle> <Paragraph position="0"> Steedman (2000a) presents a detailed, CCG-based account of how prosody is used in English as a means to realize information structure. In the model, pitch accents and boundary tones have an effect on both the syntactic category of the expression they mark, and the meaning of that expression.</Paragraph> <Paragraph position="1"> Steedman distinguishes pitch accents as markers of either the theme (th) or of the rheme (r): L+H* and L*+H are th-markers; H*, L*, H*+L and H+L* are r-markers. Since pitch accents mark individual words, not (necessarily) larger phrases, Steedman uses the th/r-marking to spread informativity over the domain and the range of function categories.</Paragraph> <Paragraph position="2"> Identical markings on different parts of a function category not only act as features, but also as occurrences of a singular variable. The value of the marking on the domain can thus get passed down (&quot;projected&quot;) to markings on categories in the range. Constituents bearing no tune have an e-marking, which can be unified with either e, th or r. Phrases with such markings are &quot;incomplete&quot; until they combine with a boundary tone. Boundary tones have the effect of mapping phrasal tones into intonational phrase boundaries. To make these boundaries explicit and enforce such &quot;complete&quot; prosodic phrases to only combine with other complete prosodic phrases, Steedman introduces two further types of marking - i and ph - on categories.</Paragraph> <Paragraph position="3"> The ph markings only unify with other ph or i markings on categories, not with e, th or r. These markings are only introduced to provide derivational control and are not reflected in the underlying meaning (which only reflects e, th or r).</Paragraph> <Paragraph position="4"> Figure 2 recasts the above as an abstract specification of which different types of prosodic constituents can, or cannot, be combined.2 Steedman's 2There is one exception we should note: two intermediate phrases can combine if a second one has a downstepped accent. We deal with this exception at the end of the section.</Paragraph> <Paragraph position="5"> trol in prosody First consider the top half of Figure 2. If a constituent is marked with either a th- or r-tune, the atomic result category of the (possibly complex) category is marked with ip. Prosodically unmarked constituents are marked as up. The lexical entries in (12) illustrates this idea.3 (12) MARCEL proved COMPLETENESS H* L+H* sip/(sup\np) (sup\np)/np sip$\(sup$/np) This can proceed in two ways. Either the marked MARCEL and the unmarked proved combine to produce an intermediate phrase (13), or proved and the marked COMPLETENESS combine (14).</Paragraph> <Paragraph position="6"> For the remainder of this paper, we will suppress up marking and write sup simply as s.</Paragraph> <Paragraph position="7"> Examples (13) and (14) show that prosodically marked and unmarked phrases can combine. However, both of these partial derivations produce categories that cannot be combined further. For example, in (14), sip/(s\np) cannot combine with sip\np to yield a larger intermediate phrase. This properly captures the top half of Figure 2.</Paragraph> <Paragraph position="8"> To obtain a complete analysis for (12), boundary tones are needed to complete the intermediate phrases tones. For example, consider (15) (based on example (70) in Steedman (2000a)): (15) MARCEL proved COMPLETENESS H* L L+H* LH% To capture the bottom-half of Figure 2, the boundary tones L and LH% need categories which create complete phrases out of those for MARCEL and proved COMPLETENESS, and thereafter allow them to combine. Figure 3 shows the appropriate categories and complete analysis.</Paragraph> <Paragraph position="9"> We noted earlier that downstepped phrasal tunes form an exception to the rule that intermediate phrases cannot combine. To enable this, we not only should mark the result category with ip (tune), but also any leftward argument(s) should have ip (downstep). Thus, the effect of (lexically) combining a downstep tune with an unmarked category is specified by the following template: add marking xip$\yip to an unmarked category of the form x$\y.</Paragraph> <Paragraph position="10"> The derivation in Figure 5 illustrates this idea on example (64) from (Steedman, 2000a).</Paragraph> <Paragraph position="11"> To relate prosody to information structure, we extend the strategy used for constructing logical forms described in SS2.3, in which a simple index feature on atomic categories makes a nominal (discourse referent) available. We represent information structure as a formula @d[i]r at a dimension separate from the syntactic category. The nominal r stands for the discourse referent, which has informativity i with respect to the current point in the discourse d (Kruijff, 2003). Following Steedman, we distinguish two levels of informativity, namely th (theme) and r (rheme).</Paragraph> <Paragraph position="12"> We start with a minimal assignment of informativity: a theme-tune on a constituent sets the informativity of the discourse referent r realized by the constituent to th and a rheme-tune sets it to r. This is a minimal assignment in the sense that we do not project informativity; instead, we only set informativity for those discourse referents whose realization shows explicit clues as to their information status. The derivation in Figure 4 illustrates this idea and shows the construction of both logical form and information structure.</Paragraph> <Paragraph position="13"> Indices can also impose constraints on the informativity of arguments. For example, in the downstep example (Figure 5), the discourse referents corresponding to ANNA and SAYS are both part of the theme. We specify this with the constituent that has received the downstepped tune. The referent of the subject of SAYS (indexed x) must be in the theme along with the referent s for SAYS. This is satisfied in the derivation: a unifies with x, and we can unify the statements about a's informativity coming from ANNA (@d[th]a) and SAYS (@d[th]x with x replaced by a in the >B step).</Paragraph> </Section> </Section> class="xml-element"></Paper>