<?xml version="1.0" standalone="yes"?>
<Paper uid="E93-1047">
  <Title>[ Type-Driven Semantic Interpretation of f-Structures \] ?.~,w),(njO)</Title>
  <Section position="3" start_page="405" end_page="408" type="metho">
    <SectionTitle>
2 A Simple Top-down Type-driven
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="405" end_page="406" type="sub_section">
      <SectionTitle>
Interpretation Algorithm
</SectionTitle>
      <Paragraph position="0"> In order to sketch how we can achieve a decomposition of an f-structure which is sufficient for its interpretation, we first introduce a simple top-down interpretation procedure which is restricted to certain special type systems. For the interpretation we generally assume from now on that types are assigned to all grammatical function values and semantically relevant atomic-valued features by a type assignment TY. Aside from the fact that grammatical functions and values and not c-structure constituents are typed, this assignment is similar to the one used in Montague grammar. The structure in figure 3 e.g.</Paragraph>
      <Paragraph position="1"> is an oversimplified typed f-structure 3 of the sentence  The typed f-structure of sentence (5).</Paragraph>
      <Paragraph position="2"> The typing of the f-structures can e.g. be established by additional type-assigning annotations within a grammar. Examples of such augmented rules and lexical entries are given in (6).</Paragraph>
      <Paragraph position="3">  (6) S ~ NP VP (T s~) =~ T=~ TY(I)=e TY(T)=t arrived: V, (1&amp;quot; PILED) ---~ 'arrive' &amp;quot;TY(T PREY) = (~, t&gt;  It is of course possible to think of more sophisticated type inheritance mechanisms for the specification of the f-structure typing. The investigation of such mechanisms, however, is beyond our present concerns.</Paragraph>
      <Paragraph position="4"> The restrictedness of the algorithm results from the fact that it operates under the assumption that we can recursively decompose each f-structure f into 3We drop the subcategorization frames in the following.</Paragraph>
      <Paragraph position="5"> a substructure or set element which corresponds to a one-place main functor and the rest structure of f which is interpreted as the argument of that functor. Although this restriction seems to be rather strong, this algorithm gives the right hierarchical semantic structures for the problematic flat f-structures containing sentence modifiers. And if we assume the usual type-raising for the subcategorized functions, it also describes all possible structural ambiguities for predicate modifiers, quantifiers, etc. 4 In detail, the algorithm works as follows.</Paragraph>
      <Paragraph position="6"> Proceeding from an initial mapping of a given f-structure f into an empty semantic structure the interpretation works top-down according to the following two principles: If trg is defined and h is a substructure (h = (g A)) or an element of a set-value of g (h E (g A)) and  (A1) TY(g) - r and TY(h) = {r', r) then (i) (g A) = h &amp;quot;-+ (fig FU) &amp;quot;- trh A (fig ARG) ---~ a(g\A) A TY(g\A) = r', (ii) h e (g A) ---* (#g FU) -&amp;quot; ah A</Paragraph>
      <Paragraph position="8"> The principle (A1) allows us to select a substructure or an element of a set value of type /v ~, r) from a structure g of type r, which is already mapped into the semantic representation, as a funetor and interpret the rest of g as its argument which becomes then of type r'. 5 If we apply principle (Alii) to the structure in figure 3 and choose b as the runefor we end up with the constellation in figure 4.</Paragraph>
      <Paragraph position="9"> For an interpreted structure g containing an immediate substructure or a set element h, principle (A2) drops the interpretation downwards if g and h are of the same type. This principle can then be applied e.g. to b of figure 4 and achieves the mapping in figure 5. Figure 6 gives a complete type-driven derivation of the functor-argument structure of (5) with wide scope of 'late'. One gets the other reading by first selecting b as described above.</Paragraph>
      <Paragraph position="10"> Note that the meanings are not constructed by our procedure. The complete semantic representation results then from the insertion of the meanings of the basic expressions which are assumed to be specified in the lexicon via equations like the following: late: ADV, (T BRED) = 'late' 4 For further illustration of the algorithms we give examples involving transitive verbs in the appendix.</Paragraph>
      <Paragraph position="11"> 5Note that a distinct re-interpretation of an already interpreted structure always fails, since predicates and predicate projections do not unify in LFG. Without this assumption, one would have to add to the principles the condition that g has no interpreted part.</Paragraph>
      <Paragraph position="13"> The result of applying principle (Alii) to b E (f ADJ) in figure 3.</Paragraph>
      <Paragraph position="15"> Figure 5 The result of applying principle (A2) to b in figure 4. The result is then the following structure L AP,-&lt;~ j l J and the meaning of the sentence can be calculated bottom-up by )t-conversion in the usual way.</Paragraph>
      <Paragraph position="17"> So, we end up with the expression L(T(A(j))) which corresponds to the wide scope reading of 'late'.</Paragraph>
    </Section>
    <Section position="2" start_page="406" end_page="408" type="sub_section">
      <SectionTitle>
Interpretation Algorithm
</SectionTitle>
      <Paragraph position="0"> In the following we sketch a more powerful mechanism which can also handle cases where the functor is not given by a substructure (f A) or a set element g E (f A) hut by a partial structure g subsumed by f (g C f) as e.g. in the typed structure for sentence (7) in figure 7. Here the part of the f-structure that comprises the modifiers and the predicate has to he interpreted as the main functor (either  The typed f-structure of sentence (7).</Paragraph>
      <Paragraph position="1"> Let &amp;quot;\[&amp;quot; be a new operator which is defined for a sub-structure (f A) off by f\[A = f\] {A} and for a set element g G (f A) by f\[(A g} = {(A, {g})}. The value is simply the f-structure subsumed by f which has only attribute A with its value in f or with the singleton set-value g. For every attribute or attribute-element pair x, f\z and fix are in fact complementary with respect to f, that is, f\x \[7 fix = 0.</Paragraph>
      <Paragraph position="2"> Proceeding from the interpretations of the basic expressions introduced by the lexical entries the algorithm works bottom-up according to the following principles:  (B1) If trh and irk are defined, h E g, k E_ g and</Paragraph>
      <Paragraph position="4"> terpreted structure h one level upwards to the partial structure of the given structure which contains only h as an attribute- or set-value and assigns to that partial structure the type of h. Note that principle (B2) can only be applied if glA resp. gl(A h) has no type assignment or is of the same type as h (otherwise the type assignment would not be a function).</Paragraph>
      <Paragraph position="5"> If a structure g contains two disjoint partial structures h and k, one of them being an appropriate argument for the other, then the structures are interpreted according to principle (B1) as the functor resp. argument of the interpretation of their unification. This is then assigned the value-type of the functor. Figure 8 shows how the semantic representation of one reading of sentence (7) is constructed. We represent here attribute-value paths in DAG form</Paragraph>
      <Paragraph position="7"> and depict the decomposition of the f-structure as a graph where each subtree dominated by a branching node represents the partial f-structure which comprises the attribute-value paths contained in that subtree. The construction starts with the mapping of the terminal nodes provided by the lexical entries of the basic expressions. Each mapping of a structure dominated by a non-branching non-terminal node results from an application of principle (B2). The interpretation of a partial substructure (a structure dominated by a branching node) is constructed by principle (B1).</Paragraph>
      <Paragraph position="8">  The restrictedness of the simple top-down algorithm results from the fact that the main functor was always assumed to take exactly one argument which is represented by the semantics of the rest of the f-structure. The algorithm fails in cases where the type of the substructure representing the main functor indicates that more than one argument is needed by the main functor in order to yield a meaning of the type of the entire f-structure. If we choose e.g. the '3times' modifier in the structure of figure 7 as the main functor (having widest scope), then we need a first argument of type (e,t) and a second argument of type e to get a meaning of type t. So, the rest of the structure corresponds in the general case to a list or set of arguments.</Paragraph>
      <Paragraph position="9"> In order to overcome this difficulty, we assign to  the rest structure now a separate semantic structure. This structure is a set that contains typed variables for all those arguments which are still needed to saturate previously processed (complex) functors. If we start with the '3times' modifier this set contains the typed variables ae and a(e,t). In detail the algorithm works as follows* If TY(f) = r the algorithm starts from the initial assignment ~rf = fr and proceeds top-down according to the following principles: If ag is defined and h is a substructure (h = (9 A)) or an element of a set-value of g (h E (g A)),</Paragraph>
      <Paragraph position="11"> n &gt; O, then o'h -&amp;quot; (kr FU n) = h(r,,(r,_,...(rt,r)..)), (kr FU i-1 ARG) = h i (for each i : 1, n)  and (i) if (g A) ---- h then \[{h~,..,h'~} if kr =~g o'(g \A) = t(og -- {k,}) LI {hr,,.. ,1 hr,~ }n else,6 (ii) ifh G (g A) then o'(g\(A h)) is determined as in case (i), (C2) ag = k~ and n = O, then ~g = ah.  In contrast to the simple top-down algorithm, each application of (C1) creates a new semantic structure which includes typed variables for all missing arguments. The new structure is linked to structures previously constructed either by explicit reentrancies or because they share common substructures. (The latter is enforced, since all those arguments (typed variables) which remain to be found after selecting kr are passed on to the semantic representation of the next restriction by (Cli,ii).) Reentrancies are used to link the (new) arguments to their right positions which are encoded in a functor-argument matrix in ~rg by applying (C1). Figure 9 gives three steps of a derivation of one reading of (7). (We omit in the example the upper indices of the typed variables provided by (C1), since no funetor needs more than one argument of the same type.)</Paragraph>
    </Section>
  </Section>
  <Section position="4" start_page="408" end_page="409" type="metho">
    <SectionTitle>
5 Completeness, Conservativity,
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="408" end_page="409" type="sub_section">
      <SectionTitle>
Constraints and Compositionality
</SectionTitle>
      <Paragraph position="0"> Since the meaning of a sentence with an f-structure f is given by the formula described by the semantic representation or f, the bottom-up construction is successful if we have constructed a value for ~rf.</Paragraph>
      <Paragraph position="1"> Within the top-down approaches the meaning of each basic expression represented in the f-structure has to be connected with the root or f, otherwise the semantic representation would not be a description of a well-formed formula.</Paragraph>
      <Paragraph position="2"> eI.e, if k~ E ag.</Paragraph>
      <Paragraph position="3">  Some steps of the interpretation of (7).</Paragraph>
      <Paragraph position="4"> } In LFG, we can ensure that all syntactically well-formed utterances are also semantically well-formed by mechanisms which are already part of the theory. By the completeness and coherence conditions it can first be guaranteed that the different kinds of nuclei (consisting of a predicate and the functions it subcategorizes for) will get an interpretation of the right type. Since all free grammatical functions (ADJ) are homogeneous functors (argument and value are of the same type) and it is clear from the c-structure rules which type of argument they modify (a modifier on S-level is either a sentence or predicate modifier,  etc.), f-structures with free functions can also be ensured to be interpreted.</Paragraph>
      <Paragraph position="5"> On the other hand particular readings can be excluded by global binding and/or scoping principles, similar to the ones formulated in \[Dalrymple et al., 1990\]. These principles constrain the interpretation of the f-structures and their parts if special conditions are satisfied. By combining outside-in and inside-out functional uncertainty we can express by the following constraint e.g. that under some conditions E the substructure (T A) of an f-structure has wide scope over (T B): Z -&amp;quot;* ((FU 0&amp;quot;(T A)) ARG + FU) &amp;quot;-C/ o-(T B). Due to the interpretation function (~r) between a typed f-structure and its semantic representation it is also possible to formulate a compositionality principle very similar to the classical one. The classical compositionality principle (Frege principle) says roughly that the meaning of an expression is a function of the meaning of its components and their mode of combination and is normally specified in terms of c-structure categories and their immediate constituents. As is well-known, the attractiveness of this principle gets lost to some degree if we have to handle phenomena which can only be described by assuming transformations on the constituent structures.</Paragraph>
      <Paragraph position="6"> In LFG, the f-structures describe explicitly the underlying predicate-argument structures of well-formed expressions, and the components of an expression are taken to be the sets of string elements that are mapped via C/ (the structural correspondence between c- and f-structure) to the units of its type-driven decomposed f-structure. On this view, the meaning of an expression remains a function of the meaning of its components. Thus, the reading of sentence (5) given in figure 6, e.g., is composed of the meanings of the components {(1, John), (2, arrived), (4, today)} and {(3,1ate)} associated with f \ (ADJ a) and a by C/, respectively. Their mode of combination (determined by the type assignment) is encoded in the functor-argument matrix as function-application of aa to a(f \ (ADJ a)) (i.e. the meaning is o'a(a(f \ (ADJ a)))). Ambiguities result then from the indeterminism of the type-driven decomposition of the f-structure of a sentence. Thus, we can state for LFG a compositionality principle without assuming any kind of transformations, since all information relevant for the interpretation is locally available (cf. e.g. \[Bresnan et al., 1982\]).</Paragraph>
    </Section>
  </Section>
class="xml-element"></Paper>