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<Paper uid="J88-1001">
  <Title>CATEGORY STRUCTURES</Title>
  <Section position="16" start_page="11" end_page="11" type="concl">
    <SectionTitle>
APPENDIX
</SectionTitle>
    <Paragraph position="0"> In this appendix we restate the semantic rules for L c more precisely. All well-formed expressions of L c have Computational Linguistics, Volume 14, Number 1, Winter 1988 15 Gerald Gazdar et aL Category Structures the same kind of denotation~they denote truth values (i.e., members of 2) relative to the category structure and a category a determined by E. If 05 is a well-formed expression of L o then we use f105flx,~ to stand for the denotation of 05 with respect to the category structure and category a. If 005(\]z~,, ~ = 1 then we shall say that t~ SATISFIES 05. The formal statement of our semantic rules is the following, where a, f, 05, and q~ are as above.</Paragraph>
    <Paragraph position="1"> (AI) a. 0fl~.~ = 1 iff s0') is defined.</Paragraph>
    <Paragraph position="2">  b. Of:aD~,,~ = 1 iff a(J) = a.</Paragraph>
    <Paragraph position="3"> c. Uf:05fl~.~ = 1 iff 0050~.,~0~ = I.</Paragraph>
    <Paragraph position="4"> d. D~050~.~ = 1 iff 0050~.~ = 0.</Paragraph>
    <Paragraph position="5"> e. 005 V q~.~ = 1 iff D05D~,~ = 1 or 0q~:,,~ = 1.</Paragraph>
    <Paragraph position="6"> f. 005 A ~,,~ = 1 iff 005~,,~ = 1 and 0q~,,~ = 1.</Paragraph>
    <Paragraph position="7"> g. 005 ---* q~,,~ = 1 iff 005~,,~ = 0 or 0q~x,,~ = 1. h. 105 &lt;-&gt; ~1:~,~ = 1 iff 1050:~,~ = lqA~,~.</Paragraph>
    <Paragraph position="8"> i. 1\[\]051~,~ = 1 iff 1051:~,~ = 1 and for all fin F ~ n</Paragraph>
    <Paragraph position="10"> j. I 0 051:, ~ = 1 iff 1051:, ~ = 1 or for some fin F ~ n ~(a),l ~ 4&gt;0:~,=00 = 1.</Paragraph>
    <Paragraph position="11"> Note that if a ~_ fl and a satisfies 05, it does ~0T follow in L c that 3 satisfies 05 (compare Rounds and Kasper (1986), Theorem 6). For example, we have ~ ~_ {(F, a)} and ~ satisfies --1 F, but {(F, a)} does not. Likewise, the fact that both a and/3 satisfy some constraint 05 does not entail that a U/3 will satisfy 05, even if a IA/3 is defined. The desire to incorporate negation whilst maintaining an upward closure property lead Moshier and Rounds (1987) to set aside a classical semantics for their feature description language and postulate an intuitionistic se mantics that, in effect, quantifies over possible extensions. null We will write ~ 05 to mean that for every category structure \]i and category a in 11, a satisfies 05. Given this, we can list some valid formula: and valid formula schemata of the logic of category constraints.</Paragraph>
    <Paragraph position="12"> (A2) a. ~a) --&gt; f (for all a E p(\]), f E F deg) This simply says that if a feature has an atomic value, then it has a value. We also have all the valid formula: of the standard propositional calculus, which we will not list here. Furthermore, we have the following familiar valid modal formula:.</Paragraph>
    <Paragraph position="13">  Here, (A2h) shows us that our logic at least contains $4 (we follow the nomenclature of Hughes and Cresswell (1968) throughout). But we do not have ~ &lt;&gt; 05--~ \[\] 0 05, and so our logic does not contain $5. To see this, consider the following category, assuming F is a category-valued feature: {(F, O)}. This category satisfies 0 F but not \[\] O F.</Paragraph>
    <Paragraph position="14"> The category {(F, {(G, a)}), (H, {(G, b)})} (graphically represented in (50), below) provides us with an analogous falsifying instance for ~ O 1-105 ~ \[\] O 05 when we set 05 = (~:a).</Paragraph>
    <Paragraph position="15"> This shows that our logic does not contain $4.2. Interestingly, the converse of this constraint zs valid, hence: (A2) i. ~F-1005---~ OD05 This is easy to demonstrate: if o~ satisfies \[\] O 05 then 0 05 must hold in all the categories that terminate a, and if O 05 holds in those categories, then 4, and I-\]05 hold in them as well. So r-\]05 holds in at least one category in o~, and thus a must satisfy O D05. This shows that our logic at least contains K1 and, as a consequence, is not contained by SS.</Paragraph>
    <Paragraph position="16"> However, our logic cannot contain K2, since the latter contains S4.2. Nor does it contain K1.2 since the latter's characteristic axiom, namely ~ 05 ~ 1-1( O 05 ~ 05) is shown to be invalid by the category {(G, a), (F, {(G, b), (F, {(G, a)})})} (shown in (51), below) when set set 05 = (c: a).</Paragraph>
    <Paragraph position="17"> I In fact, our logic does not merely contain K1, it also contains KI.1, whose characteristic axiom is: (A2) j. ~Fq(D(05 --&gt; D05) ---&gt; 05) -o 05) Hughes and Cresswell note that KI.1 'is characterized by the class of all finite partial orderings, i.e., finite frames in which R \[the accessibility relation\] is reflexive, transitive, and antisymmetrical' Hughes and Cresswell ((1984), p. 162). So it should be no surprise, given the basis for our semantics, that our logic turns out to include KI.1. This logic, also known as S4Grz (after Grzegorczyk (1967)), 'is decidable, for every nontheorem of S4Grz is invalid in some finite weak partial ordering' (Boolos (1979, p. 167).</Paragraph>
    <Paragraph position="18"> Two further valid formula schemata of Lc have some interest, before we conclude the list of valid formula: in (A2): (A2) k. ~0 -Tf (for allfE F I) 1. ~(f.'05)--&gt; 005 (forallfEF 1) The first of these follows from the fact that categories are finite in size and thus ultimately grounded in categories that contain no category-valued features: f must be false of these terminating embedded categories, and hence O --1 f must be true of the category as a whole. 16 Computational Linguistics, Volume 14, Number 1, Winter 1988 Gerald Gazdar et al. Category Structures The second states that if a category is defined for a category-valued feature whose value satisfies 4,, then the category as a whole satisfies O 4'.</Paragraph>
    <Paragraph position="19"> (A2) m. ~(f:th) ---~f (for allfE F I)</Paragraph>
    <Paragraph position="21"> It is worth considering the valid formulae one would get in certain restricted classes of category structures.</Paragraph>
    <Paragraph position="22"> Suppose we consider category structures which contain only atom-valued features (i.e., F = Fdeg). In this case, as one would expect, the modal logic collapses into the propositional calculus and the relevant notion of validity (call it Po) gives us the following: (A5) moth ,o ruth The converse case, where we only permit category-valued features (i.e. F = F1), is uninteresting, since it is not distinct from the general case. We can always encode atom-valued features as (sets of) category-valued features and subject the latter to appropriate constraints, as follows. For every feature specification (f, a) such thatfE F deg and a E p(f), we introduce a new type 1 feature fa and use the presence of 0Ca, 0) to encode the presence of (f, a) and likewise absence to encode absence. Then, for each pair of atoms a and b in p(f), we require the new features to satisfy \[\] -7 (fa A fb). And to constrain each new feature fa to have the empty set as its value, we stipulate \[\] -7 (fa:g) for every feature g.</Paragraph>
    <Paragraph position="23"> However, consider validity in category structures containing at most one category-valued feature (call this kind of validity ~ 1)- With this restriction, the $4.2 axiom considered earlier becomes valid: (A6) ~10\[N~b--~ \[\]O4, In addition, we get (A7).</Paragraph>
    <Paragraph position="24"> (A7) ~ ~\[\](\[\]t h ~ \[-\]~) V \[-l(f--\]q~--~ \[~th) This means that this restricted logic at least contains K3, but it cannot contain K4, since ~1~)~ (0\[~(~ &amp;quot;--&gt; D~b) is falsified by the category {(G, a), (~&amp;quot; {(G, b), (~', {(G, a)})})} when we set ~b = (G: a).</Paragraph>
    <Paragraph position="25"> In fact it must also contain K3.1, in view of the validity of (A2j) above, and this logic, also known as S4.3Grz, is characterized by finite linear orderings Hughes and Cresswell (1984). This is the characterization we would expect given the character of the ~1 restriction on the form of permissible categories, since with only one category-valued feature, there is at most one path through the structure of a category and so the partial order becomes a linear order. These observations concerning the logic induced by category structures where IFll = 1 are of some potential relevance to the study of indexed grammars whose categories can be- construed as being restricted in just this way (see section 4.9, above).</Paragraph>
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