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<Paper uid="E95-1006">
  <Title>A Specification Language for Lexical Functional Grammars</Title>
  <Section position="3" start_page="39" end_page="39" type="metho">
    <SectionTitle>
2 Lexical Functional Grammar
</SectionTitle>
    <Paragraph position="0"> A lexical functional grammar consists of three main components: a set of context free rules annotated with schemata, a set of well formedness conditions on feature structures, and a lexicon. The role of these components is to assign two interrelated structures to any linguistic entity licensed by the grammar: a tree (the c-structure) and a feature structure (the f-structure). Briefly, the context free skeleton of the grammar rules describes the c-structure, the well-formedness conditions restrict f-structure admissibility, and the schemata synchronise the information contained in the cand f-structures.</Paragraph>
    <Paragraph position="1">  To see how this works, let's run through a simple example. Consider the grammar given in Figure 1. Briefly, the up- and down-arrows in the schemata can be read as follows: T Feature denotes the value of Feature in the f-structure associated with the tree node immediately dominating the current tree node, whereas ~ Feature denotes the value of Feature in the f-structure associated with the current tree node. For instance, in rule (1) the NP schema indicates that the f-structure associated with the NP node is the value of the SUBJ feature in the f-structure associated with the mother node. As for the VP schema, it requires that the f-structure associated with the mother node is identical with the f-structure associated with the VP node.</Paragraph>
    <Paragraph position="2"> Given the above lexical entries, it is possible to assign a correctly interrelated c-structure and f-structure to the sentence A girl walks. Moreover, the resulting f-structure respects the LFG well formedness conditions, namely the uniqueness, completeness and coherence principles discussed in section 5. Thus A girl walks is accepted by this grammar.</Paragraph>
  </Section>
  <Section position="4" start_page="39" end_page="40" type="metho">
    <SectionTitle>
3 Modeling the LFG ontology
</SectionTitle>
    <Paragraph position="0"> The ontology underlying LFG is a composite one, consisting of trees, feature structures and links between the two. Our first task is to mathematically model this ontology, and to do so as transparently as possible. That is, the mathematical entities we introduce should clearly reflect the intuitions important to LFG theorising -- &amp;quot;No coding!&amp;quot;, should be our slogan. In this section, we introduce such a representation of LFG ontology.</Paragraph>
    <Paragraph position="1"> In the following section, we shall present a formal language L: for talking about this representation; that is, a language for specifying LFG grammars.</Paragraph>
    <Paragraph position="2"> We work with the following objects. A model M is a tripartite structure (7.,zoomin,J:), where 7- is our mathematical picture of c- structure, 9 r our picture of f-structure, and zoomin our picture of the link between the two. We now define each of these components. Our definitions are given with respect to a signature of the form (Cat, Atom, Feat), where Cat, Atom and Feat are non-empty, finite or denumerably infinite sets. The intuition is that these sets denote the syntactic categories, the atomic values, and the features that the linguist has chosen for some language. For instance, Cat could be {S, NP, VP, V}, Atom might be {sg,pl, third, fem, masc} and Feat might be { subj, obj, pred, nb, case, gd}.</Paragraph>
    <Paragraph position="3"> Firstly we define 7&amp;quot;. As this is our mathematical embodiment of c-structure (that is, a category labeled tree) we take it to be a pair (T, Vt), where T is a finite ordered tree and Vt is a function from the set of tree nodes to Cat. We will freely use the usual tree terminology such as mother-of, daughter-of, dominates, and so on.</Paragraph>
    <Paragraph position="4"> Second, we take jr to be a tuple of the form (W, {fa}c, EFeat, initial, Final, VI) , where W is aftnite, non-empty set of nodes; f~ is a partial function from W to W, for all a E Feat; initial is a unique node in W such that any other node w' of W can be reached by applying a finite number  of fa to initial; Final is a non-empty set of nodes such that for all fa and all w E Final, f~(w) is undefined; and V! is a function from Final to Atom. This is a standard way of viewing feature structures, and is appropriate for LFG.</Paragraph>
    <Paragraph position="5"> Finally, we take zoomin, the link between c-structure and f-structure information, to be a partial function from T to W. This completes our mathematical picture of LFG ontology. It is certainly a precise picture (all three components, and how they are related are well defined), but, just as importantly, it is also a faithful picture; models capture the LFG ontology perspicuously.</Paragraph>
  </Section>
  <Section position="5" start_page="40" end_page="40" type="metho">
    <SectionTitle>
4 A Specification Language
</SectionTitle>
    <Paragraph position="0"> Although models pin down the essence of the LFG universe, our work has only just begun. For a start, not all models are created equal. Which of them correspond to grammatical utterances of English? Of Dutch? Moreover, there is a practical issue to be addressed: how should we go about saying which models we deem 'good'? To put in another way, in what medium should we specify grammars? Now, it is certainly possible to talk about models using natural language (as readers of this paper will already be aware) and for many purposes (such as discussion with other linguists) natural language is undoubtedly the best medium. However, if our goal is to specify large scale grammars in a clear, unambiguous manner, and to do so in such a way that our grammatical analyses are machine verifiable, then the use of formal specification languages has obvious advantages. But which formal specification language? There is no single best answer: it depends on one's goals. However there are some important rules of thumb: one should carefully consider the expressive capabilities required; and a judicious commitment to simplicity and elegance will probably pay off in the long run. Bearing this advice in mind, let us consider the nature of LFG grammars.</Paragraph>
    <Paragraph position="1"> Firstly, LFG grammars impose constraints on 7&amp;quot;. Context free rules are typically used for this purpose -- which means, in effect, that constraints are being imposed on the 'daughter of' and 'sister of' relations of the tree. Secondly, LFG grammars impose general constraints on various features in 2-. Such constraints (for example the completeness constraint) are usually expressed in English and make reference to specific features (notably pred). Thirdly, LFG grammars impose constraints on zoomin. As we have already seen, this is done by annotating the context free rules with equations. These constraints regulate the interaction of the 'mother of' relation on 7&amp;quot;, zoomin, and specific features in 2-.</Paragraph>
    <Paragraph position="2"> Thus a specification language adequate for LFG must be capable of talking about the usual tree relations, the various features, and zoomin; it must also be powerful enough to permit the statement of generalisations; and it must have some mechanism for regulating the interaction between 7&amp;quot; and 2-. These desiderata can be met by making use of a propositional language augmented with (1) modal operators for talking about trees (2) modal operators for talking about feature structures, and (3) a modal operator for talking about zoomin, together with a path equality construct for synchronising the information flow between the two domains. Let us build such a language.</Paragraph>
    <Paragraph position="3"> Our language is called Z: and its primitive symbols (with respect to a given signature (Cat, Atom, Feat)) consists of (1) all items in Cat and Atom (2) two constants, c-struct and f-struct,  (3) the Boolean connectives (true, false, -~, A, ~, etc.), (4) three tree modalities (up), (down) and ,,, (5) a modality (a), for each feature a E Feat, (6) a synchronisation modality (zoomin), (7) a  path equality constructor ~, together with (8) the brackets ) and (.</Paragraph>
    <Paragraph position="4"> The basic well formed formulas (basic wits) of PS are: {true, false, c-struct, f-struct}UCatUAtomU Patheq, where Patheq is defined as follows. Let t, t I be finite (possibly null) sequences of the modalities (up) and (down), and let f, f' be finite (possibly null) sequences of feature modalities. Then t(zoomin)f ~ t'(zoomin)f' is in Patheq, and nothing else is.</Paragraph>
    <Paragraph position="5"> Tim wffs of/:: are defined as follows: (1) all basic wffs are wffs, (2) all Boolean combinations of wffs are wffs, (3) if C/ is a wff then so is Me, where M E {(a) : a E Feat} U {(up}, (down), (zoomin)} and (4) if n &gt; 0, and C/1,...,C/n are wffs, then so is *(C/1,...,C/n)- Nothing else is a wff.</Paragraph>
    <Paragraph position="6"> Now for the satisfaction definition. We inductively define a three place relation ~ which holds between models M, nodes n and wffs C/. Intuitively, M, n ~ C/ means that the constraint C/ holds at (is true at, is satisfied at) the node n in model M. The required inductive definition is as follows:  n is a tree node with a mother node m and M,m~C/ M, n ~ (zoomin)C/ iff n is a tree node, zoomin(n) is defined, and</Paragraph>
    <Paragraph position="8"> n is a tree node with exactly k daughters nl ... nk and M, n I ~ ~1,... ,M, nk ~ C/k M, n ~ t(zoomin)f ~, t'(zoomin)f' iff n is a tree node, and there is a feature structure node w such that n(St;</Paragraph>
    <Paragraph position="10"> For the most part the import of these clauses should be clear. The constants true and false play their usual role, c-struct and f-struct give us 'labels' for our two domains, while the elements of Cat and Atom enable us to talk about syntactic categories and atomic f-structure information respectively. The clauses for --, and A are the usual definitions of classical logic, thus we have all propositional calculus at our disposal; as we shall see, this gives us the flexibility required to formulate non-trivial general constraints. More interesting are the clauses for the modalities. The unary modalities (a), (up), (down), and (zoomin) and the variable arity modality * give us access to the binary relations important in formulating LFG grammars. Incidentally, * is essentially a piece of syntactic sugar; it could be replaced by a collection of unary modalities (see Blackburn and Meyer-Viol (1994)). However, as the * operator is quite a convenient piece of syntax for capturing the effect of phrase structure rules, we have included it as a primitive in/3.</Paragraph>
    <Paragraph position="11"> In fact, the only clause in the satisfaction &amp;quot;definition which is at all complex is that for ~. It can be glossed as follows. Let St and St, be the path sequences through the tree corresponding to t and t ~ respectively, and let S I and Sf, he the path sequences through the feature structure corresponding to f and f' respectively. Then t(zoomin)f ~ t'(zoomin)f' is satisfied at a tree node tiff there is a feature structure node w that can be reached from t by making both the transition sequence St;zoornin; S! and the transition sequence S,,;zoomin; S!,. Clearly, this construct is closely related to the Kasper Rounds path equality (see gasper and Rounds (1990)); the principle difference is that whereas the Kasper Rounds enforces path equalities within the domain of feature structures, the LFG path equality enforces equalities between the tree domain and the feature structure domain.</Paragraph>
    <Paragraph position="12"> If M, n ~ C/ then we say that C/ is satisfied in M at n. If M, n ~ C/ for all nodes n in M then we say that C/ is valid in M and write M ~ C/. Intuitively, to say that C/ is valid in M is to say that the constraint C/ holds universally; it is a completely general fact about M. As we shall see in the next section, the notion of validity has an important role to play in grammar specification.</Paragraph>
  </Section>
  <Section position="6" start_page="40" end_page="42" type="metho">
    <SectionTitle>
5 Specifying Grammars
</SectionTitle>
    <Paragraph position="0"> We will now illustrate how/3 can be used to specify grammars. The basic idea is as follows. We write down a wff C/ a which expresses all our desired grammatical constraints. That is, we state in /3 which trees and feature structures are admissible, and how tree and feature based information is to be synchronised; examples will be given shortly.</Paragraph>
    <Paragraph position="1"> Now, a model is simply a mathematical embodiment of LFG sentence structure, thus those models M in which C/ a is valid are precisely the sentence structures which embody all our grammatical.</Paragraph>
    <Paragraph position="2"> principles.</Paragraph>
    <Paragraph position="3"> Now for some examples. Let's first consider how to write specifications which capture the effect of schemata annotated grammar rules. Suppose we want to capture the meaning of rule (1) of Figure 1, repeated here for convenience:</Paragraph>
    <Paragraph position="5"> Recall that this annotated rule licenses structures consisting of a binary tree whose mother node m is labeled S and whose daughter nodes nl and n2 are labeled NP and VP respectively; and where, furthermore, the S and VP nodes (that is, m and n2) are related to the same f-structure node w; while the NP node (that is, nl) is related to the node w ~ in the f-structure that is reached by making a SUBJ transition from w.</Paragraph>
    <Paragraph position="6"> This is precisely the kind of structural constraint that /3 is designed to specify. We do so as follows:</Paragraph>
    <Paragraph position="8"> This formula is satisfied in a model M at any node m iff m is labeled with the category S, has exactly two daughters nx and n2 labeled with category NP and VP respectively. Moreover, nl must be associated with an f-structure node w ~ which can also be reached by making a (sub j) transition from the f-structure node w associated with the mother node of m. (In other words, that part of the f-structure that is associated with the NP node is re-entrant with the value of the subj feature in  the f-structure associated with the S node.) And finally, n2 must be associated with that f-structure node w which m. (In other words, the part of the f-structure that is associated with the VP node is re-entrant with that part of the f-structure which is associated with the S node.) In short, we have captured the effect of an annotated rule purely declaratively. There is no appeal to any construction algorithm; we have simply stated how we want the different pieces to fit together. Note that . specifies local tree admissibility (thus obviating the need for rewrite rules), and (zoomin), (up) and ~ work together to capture the effect of ~ and T-In any realistic LFG grammar there will be several -- often many -- such annotated rules, and acceptable c-structures are those in which each non-terminal node is licensed by one of them. We specify this as follows. For each such rule we form the analogous PS wff Cr (just as we did in the previous example) and then we form the disjunction V Cr of all such wffs. Now, any non-terminal node in the c-structure should satisfy one of these disjunctions (that is, each sub-tree of c-struct must be licensed by one of these conditions); moreover the disjunction is irrelevant to the terminal nodes of c-struct and all the nodes in f-struct. Thus we demand that the following conditional statement be valid: (e-struct A (down)true) --~ V C/~&amp;quot; This says that if we are at a c-struct node which has at least one daughter (that is, a non-terminal node) then one of the subtree licensing disjuncts (or 'rules') must be satisfied there. This picks precisely those models in which all the tree nodes are appropriately licensed. Note that the statement is indeed valid in such models: it is true at all the non-terminal nodes, and is vacuously satisfied at terminal tree nodes and nodes of f-struct.</Paragraph>
    <Paragraph position="9"> We now turn to the second main component of LFG, the well formedness conditions on fstructures. null Consider first the uniqueness principle. In essence, this principle states that in a given fstructure, a particular attribute may have at most one value. In PS this restriction is 'built in': it follows from the choices made concerning the mathematical objects composing models. Essentially, the uniqueness principle is enforced by two choices. First, V! associates atoms only with final nodes of f-structures; and as V/ is a function, the atom so associated is unique. In effect, this hard-wires prohibitions against constantcompound and constant-constant clashes into the semantics of PS. Second, we have modeled features as partial functions on the f-structure nodes - this ensures that any complex valued attribute is either undefined, or is associated with a unique sub-part of the current f-structure. In short, as required, any attribute will have at most one value.</Paragraph>
    <Paragraph position="10"> We turn to the completeness principle. In LFG, this applies to a (small) finite number of attributes (that is, transitions in the feature structure). This collection includes the grammatical functions (e.g.</Paragraph>
    <Paragraph position="11"> subj, obj, iobj) together with some longer transitions such as obl; obj and to; obj. Let GF be a meta-variable over the modalities corresponding to the elements of this set, thus GF contains such items as (subj), (obj), (iobj), (obl)(obj) and (to)(obj).</Paragraph>
    <Paragraph position="12"> Now, the completeness principle requires that any of these features appearing as an attribute in the value of the PRED attribute must also appear as an attribute of the f-structure immediately containing this PRED attribute, and this recursively. The following wff is valid on precisely those models satisfying the completeness principle: (wed) GF true --* GF true.</Paragraph>
    <Paragraph position="13"> Finally, consider the counterpart of the completeness principle, the coherence principle. This applies to the same attributes as the completeness principle and requires that whenever they occur in an f-structure they must also occur in the f-structure associated with its PRED attribute.</Paragraph>
    <Paragraph position="14"> This is tantamount to demanding the validity of the following formula: ( GF true A (pred)true) ~ (pred) GF true</Paragraph>
  </Section>
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