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<Paper uid="P86-1012">
  <Title>CATEGORIAL AND NON-CATEGORIAL LANGUAGES</Title>
  <Section position="2" start_page="0" end_page="75" type="intro">
    <SectionTitle>
INTRODUCTION
</SectionTitle>
    <Paragraph position="0"> Categorial grammars have recently been the topic of renewed interest, stemming in part from their use as the underlying formalism in Montague grammar. While the original categorial grammars were early shown to be equivalent to context-free grammars, 1, 2, 3 modifications to the formalism have led to systems both more and less powerful than context-free grammars.</Paragraph>
    <Paragraph position="1"> Motivated by linguistic considerations, Ades and Steedman 4 introduced categorial grammars with some additional cancellation rules. Full cancellation rules correspond to application of functions to arguments.</Paragraph>
    <Paragraph position="2"> Their partial cancellation rules correspond to functional composition. The new backward combination rule is motivated by the need to treat preposed elements. They also modified the formalism by making category symbols parenthesis-free, treating them in general as governed by a convention of association to the left, but violating this convention in certain of the rules.</Paragraph>
    <Paragraph position="3"> This treatment of categorial grammar suggests a family of eategorial systems, differing in the set of cancellation rules that are allowed. Earlier, we began a study of the mathematical properties of that family of systems, s showing that some members are fully equivalent to context-free grammars, while others yield only a subset of the context-free languages, or a superset of them.</Paragraph>
    <Paragraph position="4"> In this paper we continue with these investigations.</Paragraph>
    <Paragraph position="5"> We characterize the rule systems that can obtain context-sensitive languages, and compare the sets of categorial \]ar~guages with the context-free languages.</Paragraph>
    <Paragraph position="6"> Finally, we discuss the linguistic relevance of these results, and compare categorial grammars with TAG systems i, this regard.</Paragraph>
    <Paragraph position="7"> A categorial grammar under a set R of reduction rules is a quadruple CGR (VT, VA, S, F), whose elements are defined as follows: VT is a finite set of morphemes. VA is a finite set of atomic category symbols. S EVA is a distinguished element of VA. To define F, we must first define CA, the set of category symbols.</Paragraph>
    <Paragraph position="8"> CA is given by:i) ifAEVA,thenA ECA;ii) ifX EUA and A EVA, then X/A ECA; andiii) nothing elselsin CA . F is the lexicon, a function from VT to 2 ea such that for every aEVT, F(a) is finite. We often write CGR to denote a categorial grammar with rule set R, when the elements of the quadruple are known.</Paragraph>
    <Paragraph position="9"> Notation: Morphemes are denoted by a, b; morpheme strings by u,v,w. The symbols S,A,B,C denote atomic category symbols, and U. V, X, Y denote arbitrary (complex) category symbols. Complex category symbols whose left-most symbol is S (symbols &amp;quot;headed&amp;quot; by S) are denoted by Xs, Ys. Strings of category symbols are denoted by z, y.</Paragraph>
    <Paragraph position="10"> The language of a categorial grammar is determined in part by the set R of reduction rules. This set can include any subset of the following five rules. In each  statement, A EVA, and U/A,A/U,A/V, VIA E CA.</Paragraph>
    <Paragraph position="11"> (1) (F Rule) The string of category symbols U/A A can be replaced by U. We write: U/A A---*U; (2) (FP Rule) The string U/A A/V can be replaced by U /V. Wewrite: U /A A/V-*U/V; (3) (B Rule) The string A V/A can be replaced by U. We write:A U/A~U; (4) (Bs Rule) Same as B rule, except that U is headed by S.</Paragraph>
    <Paragraph position="12"> (5) (BP Rule) The string A/U V/A can be replaced by V/U. We write: A/U V/A--*V/U.</Paragraph>
    <Paragraph position="13">  If XY ---,Z by the F-rule , XY is called an F-redex. Similarly, for the other four rules. Any one of them may simply be called a redex.</Paragraph>
    <Paragraph position="14"> The reduction relation determined by a subset of these rules is denoted by =&gt; and defined by: if X Y --* Z by one of the rules of R, then for any a, /~ in CA* , aXY/3 &gt;aZ/3. The reflexive and transitive closure of the relation -&gt; is =&gt;*. A morpheme string w=wlu,~&amp;quot; &amp;quot;'w, is accepted by CGR(VT, VA,S,F) if there is a category string z = X1X2 &amp;quot;&amp;quot; * X, such that XiEF(w,) for each i=l,2,'--n, and x =&gt;* S. The language L(CGR) accepted by CGR(VT, VA,S,F) is the set of all morpheme strings that are accepted.</Paragraph>
  </Section>
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