A Projection Architecture for Dependency Grammar
and How it Compares to LFG

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Proceedings of the LFG98 Conference

The University of Queensland, Brisbane

Miriam Butt and Tracy Holloway King (Editors)

1998

CSLI Publications

http://www-csli.stanford.edu/publications/

Abstract:

This paper explores commonalities and differences between $\dachs$ , a variant of Dependency Grammar, and Lexical-Functional Grammar. $\dachs$ is based on traditional linguistic insights, but on modern mathematical tools, aiming to integrate different knowledge systems (from syntax and semantics) via their coupling to an abstract syntactic primitive, the dependency relation. These knowledge systems correspond rather closely to projections in LFG. We will investigate commonalities arising from the usage of the projection approach in both theories, and point out differences due to the incompatible linguistic premises. The main difference to LFG lies in the motivation and status of the dimensions, and the information coded there. We will argue that LFG confounds different information in one projection, preventing it to achieve a good separation of alternatives and calling the motivation of the projection into question.

1 Introduction

Dependency Grammar (DG) was introduced into modern linguistics by [Tesniére1959]. Since then, a number of quite different architectures have been proposed based on the dependency relation [Hudson1993]. There are, e.g., rule-based vs. lexicalized approaches, non-derivational vs. stratificational approaches with various levels, and proposals with varying degrees of procedural specifications. Unfortunately from a theoretical as well as computational point of view, all of them exhibit some empirical and/or formal deficits.

On the other hand, the notion of valence and dependency has been incorporated into all major syntactic theories based on phrase structure, under the names of functional coherence and completeness, $\theta$ -grid [Haegeman1994], subcategorization list [Pollard & Sag1994], extended domain of locality [Joshi1995], etc. These theories have in recent years developed a strong formal base, often based on model theory; and comparisons among them as well as implementations of parsers have benefitted from this mathematical work. In addition, the trend to lexicalization has resulted in a number of approaches replacing complex rules systems by few combinatory operations, solely relying on lexical information to express combinatory restrictions.

One aim of this paper, then, is to compare some ideas of LFG to an approach based on the dependency relation as syntactic primitive, working out commonalities (mainly found on the formal side), and identifying differences. We will propose a specific theoretic architecture for DG which postulates an abstract syntactic primitive, the dependency relation, and which conceives other properties such as morphosyntax, ordering, and semantics as consequences or, analytically, indicators of this dependency relation. These premises will result in a completely lexicalized architecture which allows to factor ambiguities from the syntactic representation. To make things more precise, we will give a state-of-the-art formal framework for DG, based on (multi-)modal logic.

The paper is structured as follows: First, we will briefly review current DG practice, and take a look into model-theoretic approaches to syntax. Then, we make some short remarks about LFG (Section 2.1) and German (Section 2.2), which motivated this work. In Sec. 3 we to introduce $\dachs$ (Dependency Approach to Coupling Heterogeneous knowledge Sources), discussing the descriptional dimensions and their linking. Finally, a number of parallels and differences to LFG will be investigated in Sec. 4. Our conclusion will be that the projection architecture of LFG does not fully exploit the advantages of the projection idea.

1.1 Dependency Theory

A very brief characterization of DG is that it recognizes only lexical, not phrasal nodes, which are linked by directed, typed, binary relations to form a dependency tree [Tesniére1959,Hudson1993]. Several formulations assume a rule base determining dominance and precedence, e.g. Slot Grammar [McCord1990], while most of the DG variants lexicalize at least the dominance information in valency frames. A number of them are stratified, i.e., assume several representations linked by correspondence rules. The most prominent in this class is surely Meaning-Text-Theory [Mel'cuk1988], which assumes seven strata of representation ranging from semantic representation through unordered dependency trees to morpheme sequences. Some of the rule types have not yet been specified [Mel'cuk & Pertsov1987, , p.187f]. Another stratified DG is Functional Generative Description [Sgall et al.1986], which assumes a semantic level and an underlying syntactic representation [Petkevic1995], which is mapped via ordering rules [Kruijff1997] to surface representation. Lexicalized, non-derivational accounts such as Dependency Unification Grammar or Lexicase are very similar to modern phrase-based theories. Dependency Unification Grammar [Hellwig1993] tries to provide dependency trees suitable for a psychologically inspired inference model [Hellwig1980], but is based on an operational semantics of its data types only. Lexicase [Starosta1988] employs $\xbar$ -inspired dependency trees and formally very simple word descriptions, namely fully specified feature sets. This has the disadvantage, however, that many lexical ambiguities are required to capture the many different environments a word may occur in. Word Grammar [Hudson1990] comes closest to our goals, being not only lexicalized, non-derivational, and with syntactic and semantic substructures, but additionally defining a propositional description language. There are, however, formal inconsistencies in its word order description and the inheritance mechanism as described in Hudson1990 [Neuhaus & Bröker1997].

1.2 Model-Theoretic Syntax

Logical approaches to syntax may take the form of proof theories or model theories [Rogers & Cornell1997]. Proof theories view a grammar as a set of axioms and deduction rules, and the question of grammaticality is equivalent to the question of provability within the calculus. Categorial Grammars are a prominent example in this class. On the other hand, model theory emphasizes a priori structures, so-called models, which are described by logical formulae. Here, the question of grammaticality is the question whether a model exists which satisfies the descriptions of all words of the utterance. Model-theoretic approaches may try to define new models (e.g., logically equivalent, but simpler to process) for existing theories [Stabler1992], or they may try to formalize models as they already exist in the linguistic literature [Kasper & Rounds1990,Carpenter1992,Blackburn et al.1993,Kracht1995,Rogers1996].

Our approach falls into the last category. Following [Blackburn1994], we will use Kripke models to represent syntactic structures, and define a multi-modal logic [Fitting1984] for describing them. Basing the formalization on modal logic has several welcome consequences. First, the distinction between (logical) meta language and (graphical) object language allows to compare the expressivity with other frameworks, such as unification grammars [Blackburn1994,Kracht1995]. For example, non-functional modalities correspond to set-valued features in unification approaches, both formally and in actual usage (e.g., for adjuncts). Second, the formal precision allows to derive mathematical results on computational complexity of recognition and generative capacity [Neuhaus & Bröker1997], which contrasts previous results on rather impoverished DG conceptions [Gaifman1965,Lombardo & Lesmo1996]. Third, the linguistic descriptions in terms of, e.g., word class or dependency are directly translatable into modal propositions or operators, respectively.

2 LFG and German Word Order

2.1 The LFG Architecture

Given the audience of this talk, and my knowledge of LFG, I will not attempt a summary or overview of LFG theory. I will rather point out which properties of LFG triggered the comparison presented here.

Some profound differences between LFG and $\dachs$ make them unlikely condidates for comparison: LFG employs a (usually large) explicit rule base, while $\dachs$ is completely lexicalized. LFG uses phrasal categories in these rules, which have no status at all in $\dachs$ . On the other hand, there are several shared assumptions, such as the idea that well-formedness corresponds to satisfiability of descriptions, and the surface-orientation that eliminates underlying strata and derivations.

The most prominent commonality, however, is the projection idea: LFG defines a number of levels of representation which are formally different and specialized to different types of information. These levels are linked via structural correspondences, which map elements of one level to elements of another level. The correspondence is not defined explicitly but rather emerges from the joint statement of restrictions on these levels in rules and lexical entries [Kaplan1995].

There have been different proposals as to the number, the content, and the linking of these levels in LFG (compare Halvorsen1995,Butt1996), and here lies the difference to $\dachs$ that will concern us most.

2.2 German Word Order

Again, given the audience, I will only remind you of very general properties of word order in German, and take note of treatments in LFG, as far as I have observed them.

Traditional descriptions of German word order make use of the notion of a $\key {positional field}$ , i.e., a topological position defined with reference to the finite and infinite verb parts (which constitute the so-called $\key {Satzklammer}$ ).

One confirmation for the concept of a topological field (contrasted with the notion of constituent) comes from the fact that the topological field does not have a categorial implication. That is to say, whereas a constituent always is of a certain category, a topological field is (more or less) neutral to the category of its element(s). For example, one finds NPs, APs, VPs, PPs, $\overline{\mbox{S}}$ s, and even separable verb prefixes in the German Vorfeld. Independent of any particular explanation of topicalization or extraposition, it seems that the major category does not play an important role in it.

Now the notion of a topological field cannot be easily defined within LFG, given its architecture. Unlike $\xbar$ -theory, which defines categorially unspecified positions, the context-free backbone of LFG requires one to explicitly specify the categories allowed. This requires a huge number of such rules which list all the different possibilities. Other means independent of the context-free rules are then used to restrict the selection among the rules, most notably the conditions of functional completeness and functional coherence.

The first example of this reliance on restrictions on f-structure that came to my attention is [Netter1986], who proposes for the VP a rule with a maximal set of complements, each being optional (the proposal did not take alternative orderings within Mittelfeld into account). Other proposals amount to stating rules of the form S -> XP NP, where XP is expanded into (nearly) all categories. Completeness and coherence will ultimately weed out the invalid analyses, but the category in these rules surely is part of the problem, not the solution.

Moving order constraints from c-structure to f-structure, as is proposed by Zaenen+Kaplan1995, does not solve the problem either. [:231]Zaenen+Kaplan1995 acknowledge that c-structure in this setting becomes underdetermined in the sense that there are no clear criteria for distinguishing between alternative analyses on c-structure. We will see below how one could define one level exclusively concerned with ordering facts, such that the level is well-defined and the theory becomes more modular.

3 $\dachs$

This section will present the basic design considerations underlying $\dachs$ , and then move to a description of the individual dimensions of description.

$\dachs$ developed in a project in text understanding and knowledge extraction [Hahn et al.1994]. In this setting, processing efficiency and incremental conceptual interpretation were of paramount importance; and we did not relegate them (solely) to the parsing strategy, but took precautions already in the grammar design. Both issues are addressed by the idea of coupling knowledge systems: Lexical ambiguities can be reduced by coupling several specialized knowledge systems, resulting in a smaller search space for parsing, while incremental conceptual interpretation is achieved through the coupling of syntax to a conceptual representation.

Summarizing these requirements and the discussion in Sec. 1.1, we require the following of our dependency grammar:

to retain the traditional semantic motivation of dependencies (this makes them less arbitrary and facilitates the conceptual interpretation of syntactic structure),
to be surface-oriented and non-derivational (this avoids rather arbitrary abstract underlying strata and facilitates incremental processing),
to be strictly lexicalized (this eases grammar specification and may make processing more efficient),
to define dimensions of descriptions which allow isolating alternatives within the dimensions (this reduces lexical ambiguites and improves processing efficiency),
not to assign priorities to individual dimensions as in many stratificational approaches (again, this enhances incrementality),
to be declarative and formally precise (this is the basis for theoretical investigations into the grammar's properties, for comparison with other frameworks, and for computer implementations).

To fulfill these requirements, $\dachs$ postulates an abstract syntactic primitive, the dependency relation, which is linked to different descriptional dimensions, such as morphosyntax, order, and semantics. We conceive of them as consequences (or, analytically: indicators) of the basic dependency relation. As we will argue, the information represented in each dimension should be restricted to one type; otherwise, the dimension will require complex and redundant specifications.

The following sections informally introduce the dependency tree as the central dimension, and the word class and the word order domain as dimensions mapped off the dependency tree. A more precise account based on model theory is given in [Bröker1998], where a modal logic is used to describe the dependency structures. We will also ignore here the feature annotations and the conceptual interpretation of the syntax tree; see [Bröker1998,Bröker1997] for a complete account.

3.1 Dependency Tree

The dependency tree is the backbone of the syntactic representation. As introduced in Sec. 1.1, it consists of a set of word nodes, linked by typed, binary, directed relations. The dependency relations together form a rooted tree over the set of words. We do not require it to be projective, because we assume semantically motivated dependencies, and word order (e.g., various topicalization possibilities) will not allow these to be projective. The dependency tree for the example sentence $\os {Den Mann hat der Junge gesehen}$ ( $\os {the man$_{ACC}$\space -- has -- the boy$_{NOM}$\space -- seen}$ ) is shown in Fig. 1.

**Figure 1:** Example Dependency Tree
$\begin{figure} \centering \epsfig {file=deptree.eps,width=0.4\columnwidth} \end{figure}$

Since we are not concerned with specific analyses here, but rather with the formal architecture, we will not be specific about the dependency relation types, but rather assume a set $\D = \{subj, obj, vpart, \ldots\}$ of linguistically motivated types. A dependency relation of type $d \in \D$ will be written as R_d, and we will abbreviate the union $\bigcup_{d \in \D } R_d$ as $R_ \D$ . If $\W$ is the set of words, totally ordered by the precedence relation $\pre$ , we define a dependency tree as follows.

Definition 1 (Dependency Tree (preliminary)): A dependency tree is a tuple $\tuple {\W, w_r, R_\D}$ where $R_ \D$ forms a tree over $\W$ rooted in w_r.

3.2 Word Class

A rather trivial descriptional dimension is the word class: Each word (within some syntactic analysis) belongs to exactly one word class, which for our present purposes is atomic, i.e., unanalyzable. We will encode word class assignment by a function $V_ \C$ mapping words to word classes, but it should be clear that this mapping could also be presented as a `real' projection, consisting of a set of objects (the word classes) onto which the respective words are mapped. The point to note, however, is that categorial restrictions are a priori independent of any other restriction: One might require a certain category of a subordinated word without requiring anything else of it. We call the set of word classes $\C$ , and define:

Definition 2 (Dependency Tree): A dependency tree is a tuple $\tuple {\W, w_r, R_\D, V_\C}$ where $R_ \D$ forms a tree over $\W$ rooted in w_r and $V_ \C : \W \mapsto \C$ maps each word to a word class.

In a similar way, non-atomic properties of words, such as morphosyntactic features, may be described. Our approach to feature structures is very similar to the one proposed by Blackburn-SLT.

3.3 Order Domain Structure

We have abandoned projectivity of the dependency tree to retain the semantic motivation of dependencies. To formulate order restrictions, we now introduce word order domains. The word order domain structure is a hierarchy of word order domains, which in turn are sets of words. We link the dependency tree to the domain structure and require projectivity not of the dependency tree, but of the domain structure. The flexibility of the linking allows to represent word order variation including discontinuities in the domain structure alone, keeping the dependency tree constant.

More precisely, a word order domain is a set of words, whose cardinality may be restricted to at most one element, at least one element, or - by conjunction - to exactly one element. Each word is associated with a sequence of order domains, one of which must contain the word itself, and each of these domains may require that its elements have certain morphosyntactic features. Order domains are partially ordered based on set inclusion: If an order domain d contains word w (which is not associated with d), every word w' contained in a domain d' associated with w is also contained in d; therefor, $d' \subset d$ for each d' associated with w. This partial ordering induces a tree on order domains, which we call the order domain structure .

**Figure 2:** Dependency Tree (left) and Order Domain Structure (right) for $\os {Den Mann hat der Junge gesehen}$
$\begin{figure} \centering \epsfig {file=depstruc.eps,width=0.5\columnwidth} \hfill \epsfig {file=domtree.eps,width=0.35\columnwidth} \end{figure}$

Again, take the example of German $\os {Den Mann hat der Junge gesehen}$ . The left of Fig.2 shows the word order domains by dashed circles. The finite verb, $\os {hat}$ , defines a sequence of domains, $\tuple {d_1, d_2, d_3}$ , which roughly correspond to the topological fields in the German main clause. The nouns and the participle each define a single order domain. Set inclusion gives rise to the domain structure on the right of Fig.2, where the individual words are attached by dashed lines to their including domains.

Surface order is derived from an order domain structure by propagating precedence relations from domains to their elements, i.e., $\os {Mann}$ precedes (any element of) d₂, $\os {hat}$ follows (any element of) d₁, etc. Order within a domain, e.g., of $\os {hat}$ and d₆, or d₅ and d₆, is based on precedence predicates. There are two different types, one ordering a word w.r.t. any other element of the domain it is associated with (e.g., $\os {hat}$ w.r.t. d₆), and another ordering two modifiers, referring to the dependency relations they occupy (d₅ and d₆, referring to subj and vpart). A verb like $\os {hat}$ introduces two precedence predicates, requiring other words to follow itself and the participle to follow subject and object, resp.:

$\os {hat}$ $\IMPL (<_{*} \AND\ \tuple {vpart} \gt _{\{subject, object\}})$

Informally, the first conjunct is satisfied by any domain in which no word precedes $\os {hat}$ , and the second conjunct is satisfied by any domain in which no subject or object follows a participle. The domain structure in Fig.2 satisfies these restrictions since nothing follows the participle, and because $\os {den Mann}$ is not an element of d₂, which contains $\os {hat}$ . This is an important interaction of order domains and precedence predicates: Order domains define scopes for precedence predicates. In this way, we take into account that dependency trees are flatter than PS-based ones.

Order domains easily extend to discontinuous dependencies. Consider the non-projective tree in Fig.2. Assuming that the finite verb governs the participle, no projective dependency between the object $\os {den Mann}$ and the participle $\os {gesehen}$ can be established. We allow non-projectivity by loosening the linking between dependency tree and domain structure: A modifier (e.g., $\os {Mann}$ ) may not only be inserted into a domain associated with its direct head ( $\os {gesehen}$ ), but also into a domain of a transitive head ( $\os {hat}$ ), which we will call the positional head .

The possibility of inserting a word into a domain of some transitive head raises the questions of how to require continuity (as needed in most cases), and how to limit the distance between the governor and the modifier. Both questions are solved with reference to the dependency relation. From a descriptive viewpoint, the syntactic construction is often cited to determine the possibility and scope of discontinuities [Bhatt1990,Matthews1981]. In PS-based accounts, the construction is represented by phrasal categories, and extraction is limited by bounding nodes (e.g., Haegeman1994,Becker1991). In dependency-based accounts, the construction is represented by the dependency relation, which is typed or labelled to indicate constructional distinctions which are configurationally defined in PSG. Given this correspondence, it is natural to employ dependencies in the description of discontinuities as follows: For each modifier, a set of dependency types is defined which may link the direct head and the positional head of the modifier ( $\os {gesehen}$ and $\os {hat}$ , respectively). If this set is empty, both heads are identical and a continuous attachment results. The impossibility of extraction from, e.g., a finite verb phrase may follow from the fact that the dependency embedding finite verbs, propo, does not appear on any path between a direct and a positional head.

Formally, we define order domains and domain structures as follows.

Definition 3 (Order Domain): An order domain is a continuous subset of $\W$ , i.e., for any two words contained in the order domain, all words in between are also contained in the order domain.

Definition 4 (Order Domain Structure): An order domain structure $\M$ is a set of order domains which satisfy the following restrictions: First, set inclusion defines a hierarchy over $\M$ :

$\begin{displaymath} \forall m,m' \in M: m \subseteq m' \OR m' \subseteq m \OR m \cap m' = \emptyset.\end{displaymath}$

Second, the top element of this hierarchy is equal to $\W$ , i.e., $\M$ contains all words.
These definitions ensure that the domain structure defines a partial order over the words, which can be extended to a total ordering by adding precedence restrictions on elements within one domain. It is thus similar to projective context-free trees, albeit without any categorial information. This will be crucial later on.

3.4 Dependency Structures

We still need to link the dependency tree to the dimensions of feature structure and domain structure. This is achieved by the following definition.

Definition 5 (Dependency Structure): A dependency structure T is a tuple $\tuple {\W, w_r, R_\D, V_\C, \P, R_\F, V_\A, V_\P, \M, V_\M}$ where $\tuple {\W, w_r, R_\D, V_\C}$ is a dependency tree, $\tuple {\P, R_\F, V_\A}$ is a feature structure, and $\M$ is an order domain structure over $\W$ . $V_\P: \W \mapsto \P$ maps each word to a point in the feature structure, $V_ \M : \W \mapsto \M ^n$ maps each word to a sequence of order domains.

Besides other restrictions not relevant here, we require of a dependency structure four more conditions: (1) Each word $w \in \W$ is contained in exactly one of the domains from $V_ \M (w)$ , (2) all domains in $V_ \M (w)$ are pairwise disjoint, (3) each word (except w_r) is contained in at least two domains, one of which is associated with a (transitive) head, and (4) the (partial) ordering of domains (as described by $V_ \M$ ) is consistent with the precedence of the words contained in the domains.

4 Comparison with LFG

We now turn to comparing the $\dachs$ approach to LFG. We have already noted several shared basic assumptions in Section 2.1, so we will pick out three differences here; Lexicalization, projections and their linking, and word order.

4.1 Lexicalization

A recent tendency in linguistics is the attempt to move more and more linguistic information into the lexicon, thereby eliminating rule systems. Apart from linguistic reasons to do so (which may be debatable), there are some very practical consequences of lexicalization. Large rule systems have proven unwieldy over and over, resulting in unforeseen interactions and questions of where to put (and, equally important, find) certain descriptions. The development of lexica structured by inheritance makes it possible to completely lexicalize a grammar without introducing redundancy. Linguistic generalizations are expressed by class formation, but eventually all grammatical information is located at individual words.

Here we see a clear difference between $\dachs$ and LFG. $\dachs$ is - similar to many other DGs - strictly lexicalized, i.e., there are no rules, but only one combination operation which constructs a larger dependency structure from two smaller ones. Although a lot of information can be moved to lexical items, LFG retains a large set of rules for c-structure. We think this is a disadvantage at least when it comes to writing large grammars.

Lexicalization may also reduce the processing cost, as [Schabes et al.1988] argue. This can be the case only if lexicalization itself does not introduce ambiguities. Unfortunately, this is quite often the case because the non-determinism implicit in rule selection has to be made explicit in lexical entries. Due to this reason, L-TAG (and also some variants of CG) suffers from an increase in lexical ambiguity of factor 10 [Joshi & Srinivas1994]! In our view, this results from the combined description of several information types (in this case, category, dominance, and precedence) on one level, represented by the elementary tree. $\dachs$ does not suffer from this increas of lexical ambiguity, because the levels each encode one type of information, and alternatives on one level need not (although sometimes they must) be multiplied into other levels.

4.2 Projection Architecture

Besides the general similarity in using several linked dimensions or projections, there are striking similarities in their informational content. For example, the dependency tree quite closely corresponds to LFG's f-structure: Both are unordered hierarchies representing subcategorization whose relations (dependencies vs. grammatical functions) are even similar. The feature structure which we did not discuss corresponds to the m(orphological)-structure proposed by [Butt et al.1996] for LFG, since both encode morphosyntactic information. Less similarity must be noted for the conceptual structure of $\dachs$ and s-structure of LFG [Halvorsen & Kaplan1995], which have a slightly different motivation. As we will discuss below, there are also some similarities between the domain structure and the c-structure, in that both encode order restrictions.

The major difference between the $\dachs$ and LFG architectures in this area is the linking of levels and their status. $\dachs$ identifies one level, the dependency tree, as the fundamental one and maps other levels off it, whereas LFG assumes all levels to be orthogonal and of equal importance. Correspondingly, the linking relations are different: In $\dachs$ , one might draw the level dependencies as on the left of Fig. 3, while [Kaplan1995] gives the diagram on the right as (one possible) architecture of LFG (with the m-structure of [Butt et al.1996] added; [Halvorsen & Kaplan1995] maps s-structure directly off of c-structure).

**Figure 3:** Projection Architecture in $\dachs$ (left) and LFG (right)
$\begin{figure} \centering \epsfig {file=dachs-dims.eps,width=0.4\textwidth} \hfill \epsfig {file=lfg-proj.eps,width=0.5\textwidth} \end{figure}$

4.3 Description of Word Order

The main point we want to make concerns c-structure. It usually is described to represent more language-specific information than f-structure. C-structure encodes two types of information; categories and precedence. For languages such as German, which exhibit a quite free word order, it is questionable whether confounding these two information types is linguistically motivated and results in readable specifications.

We have already sketched in Section 2.2 the problems resulting from the combination of categorial and linear restrictions in one level. In contrast, $\dachs$ defines category-independent order domains which explicitly recognizes topological fields. Restrictions on the cardinality of the fields may be directly specified without reference to the field's elements. A major achievement in our view is that word order variation is not represented by rules (as in LFG) or by lexical ambiguity (as in L-TAG [Joshi & Srinivas1994] and some versions of CG [Hepple1994]), but rather by disjunctive descriptions of a separate dimension. This not only allows a concise description of the linguistic notions behind precedence and topological fields (because they are stated in a language specially suited for this specification), but also eliminates alternatives for parsing, reducing the search space. In a way, this recapitulates the ambiguity-reducing effect of feature annotations for order restrictions.

5 Conclusion

This paper has explored an architecture based on the dependency relation which exhibits a number of similarities to LFG. As in LFG, we view restrictions on morphosyntactic features, word order, and conceptual interpretation as largely independent, similar to the modularity assumption for LFG projections. In contrast to LFG, we see them as consequences (or, for analytical purposes such as parsing, as indicators) of an abstract syntactic primitive. Consequently, this abstract syntactic representation is given special status, which shows up in, e.g., the star-shaped projections originating on this fundamental level, as opposed to the more linear projection architecture of LFG.

We have argued that the information content on each projection or dimension should be restricted to one type, and that confounding categorial and precedence information in c-structure has undesirable consequences. These consequences materialize in complex rule systems (where large sets of categories must be enumerated in certain positions) or - in lexicalized theories - in lexical ambiguities (representing word order variation). We think that the projection idea allows a better separation of alternatives than in LFG and have defined - as a separate projection - word order domains, which have no categorial implications. One feature of word order domains is that they factor ordering alternatives from the syntactic tree, much like feature annotations do for morphological alternatives. The traditional description in terms of semantically motivated dependencies and topological fields has been backed up by a state-of-the-art formal framework, which is based on modal logic.

In the light of this work, it seems valid to reconsider the dichotomy between PS-based and dependency-based approaches to language. Very generally, it could be argued that PSG - besides the non-lexical categories - requires a notion of valency and additional machinery to cover order variation, whereas DG is already based on valency and only requires an ordering component such as the one sketched in this article. Perhaps one could eliminate $\os {the nonobservable linguistic construct that enjoys the widest acceptance}$ [Pollard & Sag1994, :9, referring to nonlexical categories] by investing more work in DG.

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