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<Paper uid="C90-3052">
  <Title>Typed Unification Grammars</Title>
  <Section position="3" start_page="0" end_page="293" type="intro">
    <SectionTitle>
2 Typed t~ature structures and unification
</SectionTitle>
    <Paragraph position="0"> Tlle basic data structure of tile language is a typed featm'e structure: a feature structure (FS in the following) with which a type can be associated. Cornpared to untyped FSs (as presented in \[16\] for example), the TFS system offers the possibility to name complex FSs, and to associate constraints with these names, thus defining a type.</Paragraph>
    <Paragraph position="1"> We write feature names in small caps letters (v, ~, u), type symbols in upper case letters (A, B), and we use symbols inside a box \[~, called tags, for denoting shared values. For cxarnple, the typed FS, written in a linear form A\[F: \[\[\]B\[H: C\], a: ~\]\], is an FS of type A with two features f.&amp;quot; and e, v having as a value tile typed FS B\[H: A\] and G having the same shared value aS F.</Paragraph>
    <Paragraph position="2"> In the system, one can specify type definitions which can, as a first approximation, be seen as a kind of template definition like in e.g. PATR-II. There is, however, a major difference. The system uses a type inference mechanism to derive new types dynamically during computation whereas templates in PATR-II are expanded statically at compile time.</Paragraph>
    <Paragraph position="3"> A type that encodes agreement features can be written: AGR = \[num: NUM,gender: GEN\] and types NtJM and GEN being themselves defined as NUM = SING V PLUR (where the symbol &amp;quot;y&amp;quot; denotes the logical OR) and GEN : MASC V FEM V NEU. The types NUM, SG,... do not have definitions: they are called atomic types. AGR, NUM and GEN are called complex types.</Paragraph>
    <Paragraph position="4"> From a set of type definitions, one can extract, a partial order on type symbols. For example, from the *Researdl reported in this paper is partly supported by the German Ministry of Research and Technology (BMFT, Bundesminister ffir Forschung und Technologie), under grant No. 08 B3116 3. The views and conclusions contained herein are those of the authors and should not be interpreted as representing official policies.  set of definitions above, we can derive the following partial order on type symbols (Fig. l) where T represents the greatest element (no information) and 3_ the smallest element (inconsistent information, leading to failure in unification). This partial order is in turn used to derive a lattice of type syml)ols, which is then exl.ended to typed FSs or(lere(1 by (typed) substunption, forming a lattice on which the interpreter works (see a formal account in \[2\]).</Paragraph>
    <Paragraph position="6"> SIN(; PLUR MASC FEM NEU For exainple, the FS fl AGR\[nnm: NUM\] subsumes the FS f9 AGR\[num: PLUR, gen(ler: FEM\] because f2 has more specifc information than fl : no gender is specified in fl, and the number value of f2 PLUR is more specific than the number value offl, NUM.</Paragraph>
    <Paragraph position="7"> Typed unification proceeds as ordinary unification for FSs, recnrsively corot)thing substructures at the same paths. When two (typed) FSs are unified, frst the type symbols are unified, and if this unification succeeds, the FSs are unified. Unification of two types X and Y is defined . .(. . . . as the set of) most general type(s) which is smaller than both x and Y: ~t is the greatest lower bound (GLB) of these two symbols in the lattice of type syml)ols. If taso types are directly comparable, the smallest iv the result of the unification: HUM M PLUR = PLUR. This extension is consistent with the definition of the unifier of two FSs ~s the GLB of these structures (see, for exampie, \[16\]).</Paragraph>
    <Paragraph position="8">  A list of words will be defined in a LISP-like fashion as either the END of a list or a CONS with two attributes first and rest:</Paragraph>
    <Paragraph position="10"> WORD denotes the set of word forms, and the list. of words &amp;quot;John likes Mary&amp;quot; will be encoded as</Paragraph>
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