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<Paper uid="P87-1033">
  <Title>A Unification Method for Disjunctive Feature Descriptions</Title>
  <Section position="3" start_page="0" end_page="236" type="intro">
    <SectionTitle>
2 Data Structures
</SectionTitle>
    <Paragraph position="0"> The most common unification methods for non-disjunctive feature structures use a directed graph (DG) representation, in which arcs are labeled by names of features, and nodes correspond to values of features. For an introduction to these methods, the reader is referred to Shieber's survey \[11 I. In the remainder of this section we will define a data structure for disjunctive descriptions, using DG structures as a basic component.</Paragraph>
    <Paragraph position="1"> In the following exposition, we will carefully observe the distinction between feature structures and their descriptions, as explained in \[4\]. Feature structures will be represented by DGs, and descriptions of feature structures will be represented by logical formulas of the type described in \[4 I. The  denoting no information; denoting inco~istent information; where a E A, to describe atomic values; where l E L and ~ E FDL, to describe structures in which the feature labeled by I has a value described by ~; where each pC E L deg, to describe an equivalence class of paths sharing a common value in a feature structure; where @, C/ E FDL; where @, C/ E FDL.</Paragraph>
    <Paragraph position="2"> Figure I: Syntax of FDL Formulas.</Paragraph>
    <Paragraph position="3"> syntax for formulas of this feature description logic (hereafter called FDL) is given in Figure 1. I Note, in particular, that disjunction is used in descriptions of feature structures, but not in the structures themselves. As we have shown (see \[9\]) that there is a unique minimal satisfying DG structure for any nondisjunctive FDL formula, we can represent the parts of a formula which do not contain any disjunction by DGs. DGs are a more compact way of representing the same information that is contained in a FDL formula, provided the formula contains no disjunction.</Paragraph>
    <Paragraph position="4"> Let us define an unconditional conjunct to be a conjunct of a formula which contains no occurrences of disjunction. After path expansion any formula can be put into the form: uco~j ^ disj~ A... A disy,,, where uconj contains no occurrences of disjunction, and each disjC/, for 1 ~ i ~ m, is a disjunction of two or more alternatives. The ,~conj part of the formula is formed by using the commutative law to bring all unconditional conjuncts of the formula together at the front. Of course, there may be no unconditional conjuncts in a formula, in which case ucoaj would be the formula NIL.</Paragraph>
    <Paragraph position="5"> Each disjunct may be any type of formula, so disjuncts can also be put into a similar form, with aLl unconditional conjuncts grouped together before all disjunctive components. Thus the disjunctions of a formula can be put into the form (~conj~ ^disA ~ ^...^disA,) v... v (uconj,, ^disj,, ~ ^...^ dlsj,, ). The embedding of conjuncts within disjuncts is preserved, but the order of conjuncts may be changed.</Paragraph>
    <Paragraph position="6"> The unconditional conjuncts of a formula contain information that is more definite than the information contained in disjunctions. Thus a formula can be regarded as having a definite part, containing only unconditional conjuncts, and an indefinite part, containing a set of disjunctions. The definite part contains no disjunction , and therefore it may be represented by a DG structure. To encode these parts of a formula, let us define a feature-description as a type of data structure, having two components: ILet A and L be sets of symbols which are used to denote atomic values and feature labels, respectively.</Paragraph>
    <Paragraph position="7">  indefinite: a SET of disjunctions, where each disjunction is a SET of feature.descriptlon.s.</Paragraph>
    <Paragraph position="8"> It is poesibh to convert any FDL formula into a feature-description structure by a simple automatic procedure, a.s described in \[5\]. This conversion does not add or subtract any information from a formula, nor increase its size in any significant way. It simply identifies components of the formula which may be converted into a more el~cient representation as DG structures.</Paragraph>
    <Paragraph position="9"> A feature-descriptlon is conceptually equivalent to a special kind of AND/OR graph, in which the terminal nodes are represented by DG structures. For example, an AND/OR graph equivalent to the formula, 4,0 ^ (C/1 v ,/,2) ^ (~,3 v C/4 v (Ca ^ (C/o v C/7))) is shown in Figure 2. In the AND/OR graph representatlon, each AND-node represents a feature-description. The first outgoing arc from an AND-node represents the definite component of a feature-description, and the remaining outgo-Lug arcs represent the indefinite component, Each OR-node represents a disjunction.</Paragraph>
    <Paragraph position="10">  Function I/N\]FY-DESC (f, g) Returns feature.description: where f and g are feature-descriptions.</Paragraph>
    <Paragraph position="11"> I. Unify definite components.</Paragraph>
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