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<Paper uid="P88-1029">
  <Title>Conditional Descriptions in Functional Unification Grammar</Title>
  <Section position="3" start_page="235" end_page="236" type="intro">
    <SectionTitle>
2 Definitions
</SectionTitle>
    <Paragraph position="0"> The feature description logic (FDL) of Kasper and Rounds \[Kas86\] provides a coherent framework to give a precise interpretation for conditional descriptions. As in previous work, we carefully observe the distinction between feature structures and their descriptions. Feature structures are represented by directed graphs (DGs), and descriptions of feature structures are represented by logical formulas. The syntax for formulas of FDL is given in Figure 6. We define several new types of formulas for conditional descriptions and negations, but the domain of feature structures remains DGs, as before.</Paragraph>
    <Section position="1" start_page="235" end_page="236" type="sub_section">
      <SectionTitle>
2.1 Satisfaction and Compatibility
</SectionTitle>
      <Paragraph position="0"> In order to understand how conditional descriptions are used, it is important to recognize two relations that may hold between a particular feature structure and a description: satisfaction and compatibility. Satisfaction implies compatibility, so there are three possible states that a particular structure may have with respect to a description: the structure may fully 8ati~/X/the description, the structure may be i.eompat.</Paragraph>
      <Paragraph position="1"> isle with the description, or the structure may be C/ompatiMe with (but not satisfy) the description. To define these terms more precisely, consider the state of an arbitra~ 7 structure, /~, with respect to an atomic feature description, f : e: satisfies f : e if f occurs in A with value e; is incompatible with f : e if j' occurs in g with value z, for some z ~ ~; /~ is (merely) compatible with f : e if f does not occur inA.</Paragraph>
      <Paragraph position="2"> Because feature structures are used to represent partial information, it is possible for a structure that is merely compatible wlth a description to be extended (i.e., by adding a value for some previously nonexistent feature) so that it either satisfies or becomes incompatible with the description. Consider, for example, the structure (~z) shown in Figure 7, and the three descriptions: aubj : (perao. : 3 A .umber : ai.g) (I) subj : (perao. : 1 A .umber : .i.g) (2)  i,eompatible with Az, because Az has a different value for the feature aubj : person. Description (3) is merely compatible with Az (but not satisfied by Az), because Az has no value for the feature aubj : e~se.</Paragraph>
      <Paragraph position="3"> In the following definitions, the notation A ~ ~5 means that the structure A satisfies the description ~, and the notation A ~ ~ means that the structure A is compatible toith the description ~.</Paragraph>
      <Paragraph position="4"> Logical combinations of feature descriptions are evaluated with their usual semantics to determine whether they are satisfied by a structure. Thus, a conjunction is satisfied only  when every conjunct is satisfied, and a disjunction is satlsfied if any disjunct is satisfied. The formal semantics of the satisfaction relation has been specified in our previous work describing FDL \[Kas86\]. The semantics of the compatibility relation is given by the following conditions: I. ~ -- NIL always; 2. A .~ * C/=C/. /~ is the atomic structure ~; 3. A ~ \[&lt; Pz &gt;,-.-,&lt; P. &gt;\] ~=~ all DGs in the set {~q/ &lt; Pz &gt; ..... 4/ &lt; p. &gt;} can be unified (any member of this set may be undefined; such members are equivalent to null DGs); 4. /~ ~ I : ~ C/=~ /~/! is undefined or ~/1 ~ ~; 5. A~~V~ C/=~ ~~~or~~~0; 6. ~ N ~bA~, C/ffiffi~ .~, canonical form of~bA~.</Paragraph>
      <Paragraph position="5">  Unlike satisfaction, the semantics of compatibility cannot be defined by simple induction over conjunctive formulas, because of a subtle interaction between path equivalences and  nonexistent features. For example, consider whether A,, shown LU Figure 7, is compatible with the description: nurnber: pl A |&lt; ~*~mber &gt;, &lt; aubj number &gt;\].</Paragraph>
      <Paragraph position="6"> A, is compatible with r~urnber : pl, and d, k also compatible with ~&lt; nurnber &gt;,&lt; subj n~mber &gt;l, but #~, is not compatible with the conjunction of these two descriptions, because it requires aub\] : r~mber : pl and ,~, has si~,g as the value of that feature.</Paragraph>
      <Paragraph position="7"> Thus, in order to determine whether a structure is compatible wlth a conjunctive description, it is generally necessary to unify all conjuncts, putting the description into the canonical form described in \[Kas87c\]. This canonical form (i.e. the feature.description data structure) contains definite and indefinite components. The definite component contains no disjunction, and is represented by a DG structure that satisfies all non-disjunctive parts of a description. The indefinite component is a list of disjunctions. A structure is compatible with a description in canonical form if and only if it is unifiable with the definite component and it is compa!;ible wlth each disjunction of the indefinite component.</Paragraph>
    </Section>
    <Section position="2" start_page="236" end_page="236" type="sub_section">
      <SectionTitle>
2.2 Conditional Description
</SectionTitle>
      <Paragraph position="0"> We augment FDL with a new type of formula to represent conditional descriptions, using the notation, n -. ~, and the standard interpretation given for material implication:</Paragraph>
      <Paragraph position="2"> This Luterpretatlon of conditionals presupposes an interpretation of negation over feature descriptions, which is given below. To simpLify the interpretation of negations, we exclude formulas contaiuing path equivalences and path values from the antecedents of conditlonak.</Paragraph>
    </Section>
    <Section position="3" start_page="236" end_page="236" type="sub_section">
      <SectionTitle>
2.3 Negation
</SectionTitle>
      <Paragraph position="0"> We use the classical interpretation of negation, where /~ ~ -~b C/=~ /~ ~: #. Negated descriptions are defined for the following types of formulas:  1. A~-~ C/=~ A is not the atom ~; 2. A ~ -~(l : ~) ~ Jl ~= l : &amp;quot;-~ or .~/! is not defined; 3. ,~ ~ -~(~ v ,/,) ~:~ A ~ -,~ ^ -,,p; 4. ,~ M -,(~ ^ ,p) ~ ,~ M -,~ v -,,p.</Paragraph>
      <Paragraph position="1">  Note that we have not defined negation for formulas containing path equivalences or path values. Thls restriction makes it possible to reduce all occurrences of negation to a boolean combLuatlon of a fiuite number of negative constraints on atomic values. While the classical interpretation of negation is not strictly monotonic with respect to the normal subsumptlon ordering on feature structures, the restricted type of negation proposed here does not suffer from the inefficiencies and order-dependent uuificaticn properties of general negation or intuiticnistic negation \[Mosh87,Per87\]. The reason for this is that we have restricted negation so that all negative information can be specified as local constraLuts on single atomic feature values. Thus, these constraints only come into play when specific atomic values are proposed for a feature, and they can be checked as efficiently as positive atomic value constraints.</Paragraph>
    </Section>
    <Section position="4" start_page="236" end_page="236" type="sub_section">
      <SectionTitle>
2.4 Feature Existence Conditions
</SectionTitle>
      <Paragraph position="0"> A special type of conditional description k needed when the antecedent of a conditional is an existence predicate for a particular feature, and not a regular feature description. We call this type of conditional a \[eature ezistence condition, and use the notation: B/ -+ ~, where A ~ 3\[ 4==~ A/\[ is defined.</Paragraph>
      <Paragraph position="1"> Thk use of B/is essentially equivalent to the use of f = ANY in Kay's FUG, where ANY lsa place-holder for any substantive (i.e., non-NIL) value.</Paragraph>
      <Paragraph position="2"> The primary effect of a feature existence condition, such as 3f --, ~, is that the consequent is asserted whenever a substantive value is introduced for a feature labeled by f. The treatment of feature existence conditions differs slightly from other conditional descriptions in the way that an uusatisfiable consequent is handled. In order to negate the antecedent of 3f --~ #, we need to state that f may never have any substantive value. This is accomplished by unifying a special atomic value, such as NONE, with the value of f. This special atomic value is incompatible with any other real value that might be proposed as a value for f.</Paragraph>
      <Paragraph position="3"> Feature existence conditions are needed to model the second type of implication expressed by systemic input conditions - namely, when a constituent has one of the feature alternatives described by a system, it must also have the feature(s) specified by that system's input condition. Generally, a system named f with input condition a and alternatives described by/~, can be represented by two conditional descriptlons: null  1. a .--. p; 2. Bf -* a.</Paragraph>
      <Paragraph position="4">  For example, recall the BenfactiveVoice system, which is represented by the two conditionals shown in Figure 5.</Paragraph>
      <Paragraph position="5"> It is important to note that feature existence conditions are used for systems with simple input conditions as well as for those with complex input conditions. The use of feature existence conditions is essential in both cases to encode the bidirectional dependency between systems that is implicit in a systemic network.</Paragraph>
    </Section>
  </Section>
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