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Turning now to the augmented part of the grammar, the augmented S > NP VP rule is represented in Figure 5, both under the form of a (declarative) rule with constraints and of the corresponding f-structure. The rule and the first three constraints say that the final analysis will give rise to a structure f0 having the category of S (sentence); the analysis will succeed only if the constituents of f0, f1, and f2, are characterized by the categories NP and VP, respectively. The fourth constraint states that the number feature, agreement, of the head of f1 (NP) must be the same as the number feature of the head of f2 (VP). This constraint will therefore rule out a sentence like "The boy eat." The fifth constraint says that the feature object of the head of f0 (S) must be equal to the head of f1 (NP); this means that, after application of the grammar, the head of f0 will be an f-structure with at least one feature, called subject and having as its value the head of f1. The sixth constraint states that the head of f0 (S) must contain all the features included in the head of f2 (VP), i.e., the global sentence will inherit all the features of the verbal constituent. Finally, a mood (modality) feature with value declarative must be inserted in the head of f0. In the f-structure part of Figure 5, the equalities imposed by the constraints are expressed by making use, as usual, of numerical coreference markers. If we now assign the symbolic labels f1 and f2 to, respectively, the f-structures representing the constituents NP and VP in Figure 4 above, we will obtain the global analysis of our example sentence by "unifying" these last f-structures with the f-structure of Figure 5. In the context of constraint-based grammars, "unification" must be understood mainly as "superposition." In a well-known paper on the subject, Martin Kay tried to visualize this particular type of unification by evoking the possibility of reconstructing some photographic transparencies that have been partially damaged using the technique of comparing homogeneous "...sets of pictures by superposing them and holding them up to the light." When unifying (superposing) two f-structures, we will check, first of all, that no contradictions exist (e.g., the presence of two identical attributes associated with contradictory values, as in the features "agreement: singular" and "agreement: plural"). If this check has been passed, the final f-structure is then built up by (1) introducing in it all the features (attribute + value) present in the original f-structures; (2) by assigning the most specific values to the common attributes; and (3) by taking into account the equalities imposed by the coreference markers.
In our case, the unification of the f-structures of the constituents with that of the rule f0 > f1 f2 will lead to the f-structure of Figure 6. The final result, Figure 7, i.e., the syntactical analysis of the example, will be obtained by retrieving from Figure 6 the specific f0 structure (the "sentence"). Within the general class of constraint-based grammars, Lexical Functional Grammars (LFG) are particularly popular. Even if they have been created by John Bresnan and Ron Kaplan, at the end of the 1970s, essentially for theoretical reasons (as a reaction against the supposed "lack of psychological reality" of Chomsky's Transformational Grammar), they are of a strong practical interest, essentially for two reasons:
According to LFG, the construction of the internal structure of a sentence (syntactic analysis) begins with the construction of a "classical" syntactic tree (constituent structure tree, or C-structure), which indicates the hierarchical structure of the phrasal constituents (NP, VP, etc.) of the original sentence. C-structures are produced making use of context-free grammars characterized by the presence of particular "annotations," or "functional schemata," on the rules; e.g., the standard S > NP VP rule will be written as:
When constructing the tree, the annotations are transferred onto the corresponding nodes of the tree, and the arrows take on the meaning of "pointers," where [up arrow] points toward the father-node, and [down arrow] toward the offspring-node. Taking also into account the fact that the notation (a p1 ... pn) represents something in the style of "p1 ... pn are the attributes (properties) of the element a," and remembering that the S node is the father-node for both NP and VP, the elementary tree corresponding to the above equation means, eventually, that the subject of the final S will be the first NP, and that the structure of this final S will be the same as the structure of the VP. In the case of the analysis of a concrete sentence, like "John loves Mary," the C-structure will be completed by introducing on the tree, information found in the lexical entries of the words of the sentence. As already stated, lexical entries in LFG are particularly structured, and they contain not only the usual information about the form of word and its grammatical category, but also a description, using the mechanism of functional schemata, of the syntactic functions of the word according to its meaning. For example, the lexical entry for "loves" will be as in Figure 8.
When the final, annotated C-structure has been built up, the next phase of the analysis is the "instantiation," which consists in (1) introducing a correspondence between each node of the C-structure tree and a variable fn; (2) using the structure of the tree to find the referents, under fn form, of all the [up arrow] and [down arrow] symbols found on the tree. After this, the set of functional schemata that appeared on the tree, where they had been introduced by the rules of the grammar, and the lexical entries, are now reduced to a set of constraints on the variables fn having the same form of the set reproduced above in Figure 5. This set of constraints is called a "functional description" (f-description). If we introduce now a correspondence between each fn variable and one of the f-structures proper to constraint-based grammars, we are reduced to the usual f-structure scenario, similar to that examined above for the "the boy eats noodles" example. The C-structure (the tree) is abandoned, and it will be no longer utilized in the final phases of the syntactic analysis.
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