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<Paper uid="P03-1045">
  <Title>k-valued Non-Associative Lambek Categorial Grammars are not Learnable from Strings</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
2 Background
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
2.1 Categorial Grammars
</SectionTitle>
      <Paragraph position="0"> The reader not familiar with Lambek Calculus and its non-associative version will find nice presentation in the first ones written by Lambek (Lambek, 1958; Lambek, 1961) or more recently in (Kandulski, 1988; Aarts and Trautwein, 1995; Buszkowski, 1997; Moortgat, 1997; de Groote, 1999; de Groote and Lamarche, 2002).</Paragraph>
      <Paragraph position="1"> The types Tp, or formulas, are generated from a set of primitive types Pr, or atomic formulas by three binary connectives &amp;quot; = &amp;quot; (over), &amp;quot; n &amp;quot; (under) and &amp;quot; &amp;quot; (product): Tp ::= Pr j Tp n Tp j Tp = Tp j Tp Tp. As a logical system, we use a Gentzen-style sequent presentation. A sequent ' A is composed of a sequence of formulas which is the antecedent configuration and a succedent formula A.</Paragraph>
      <Paragraph position="2"> Let be a fixed alphabet. A categorial grammar over is a finite relation G between and Tp. If &lt; c;A &gt;2 G, we say that G assigns A to c, and we write G : c 7! A.</Paragraph>
      <Paragraph position="3">  The relation 'L is the smallest relation ' between Tp+ and Tp, such that for all ; 0 2 Tp+; ; 0 2 Tp and for all A;B;C 2 Tp :</Paragraph>
      <Paragraph position="5"> We write L; for the Lambek calculus with empty antecedents (left part of the sequent).</Paragraph>
      <Paragraph position="6">  In the Gentzen presentation, the derivability relation of NL holds between a term in S and a formula in Tp, where the term language is S ::= Tpj(S;S). Terms in S are also called G-terms. A sequent is a pair ( ;A) 2 S Tp. The notation [ ] represents a G-term with a distinguished occurrence of (with the same position in premise and conclusion of a rule). The relation 'NL is the smallest relation ' between S and Tp, such that for all ; 2 S and for all A;B;C 2 Tp :</Paragraph>
      <Paragraph position="8"> We write NL; for the non-associative Lambek calculus with empty antecedents (left part of the sequent). null  Cut elimination. We recall that cut rule can be eliminated in 'L and 'NL: every derivable sequent has a cut-free derivation.</Paragraph>
      <Paragraph position="9"> Type order. The order ord(A) of a type A of L or</Paragraph>
      <Paragraph position="11"> Let G be a categorial grammar over . G generates a string c1 ::: cn 2 + iff there are types A1;::: ;An 2 Tp such that: G : ci 7! Ai (1 i n) and A1;::: ;An 'L S: The language of G, written LL(G) is the set of strings generated by G.</Paragraph>
      <Paragraph position="12"> We define similarly LL;(G), LNL(G) and LNL;(G) replacing 'L by 'L;, 'NL and 'NL; in the sequent where the types are parenthesized in some way.</Paragraph>
      <Paragraph position="13">  In some sections, we may write simply ' instead of 'L, 'L;, 'NL or 'NL; . We may simply write L(G) accordingly.</Paragraph>
      <Paragraph position="14">  Categorial grammars that assign at most k types to each symbol in the alphabet are called k-valued grammars; 1-valued grammars are also called rigid grammars.</Paragraph>
      <Paragraph position="15"> Example 1 Let 1 = fJohn;Mary;likesg and let Pr = fS;Ng for sentences and nouns respectively. Let G1 = fJohn 7! N; Mary 7! N; likes 7! N n (S = N)g. We get (John likes Mary) 2 LNL(G1) since ((N; N n (S = N)); N) 'NL S.</Paragraph>
      <Paragraph position="16"> G1 is a rigid (or 1-valued) grammar.</Paragraph>
    </Section>
    <Section position="2" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
2.2 Learning and Limit Points
</SectionTitle>
      <Paragraph position="0"> We now recall some useful definitions and known properties on learning.</Paragraph>
      <Paragraph position="1">  A class CL of languages has a limit point iff there exists an infinite sequence &lt; Ln &gt;n2N of languages in CL and a language L 2 CL such that: L0 ( L1 ::: ( ::: ( Ln ( ::: and L = Sn2N Ln (L is a limit point of CL).</Paragraph>
      <Paragraph position="2">  The following property is important for our purpose. If the languages of the grammars in a class G have a limit point then the class G is unlearnable. 1</Paragraph>
    </Section>
    <Section position="3" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
2.3 Some Useful Models
</SectionTitle>
      <Paragraph position="0"> For ease of proof, in next section we use two kinds of models that we now recall: free groups and pregroups introduced recently by (Lambek, 1999) as an alternative of existing type grammars.</Paragraph>
      <Paragraph position="1">  Let FG denote the free group with generators Pr, operation and with neutral element 1. We associate with each formula C of L or NL, an element in FG written [[C]] as follows:</Paragraph>
      <Paragraph position="3"> We extend the notation to sequents by: [[C1;C2;::: ;Cn]] = [[C1]] [[C2]] [[Cn]] The following property states that FG is a model for L (hence for NL): if 'L C then [[ ]] =FG [[C]]  Pregroup. A pregroup is a structure (P; ; ;l;r;1) such that (P; ; ;1) is a partially ordered monoid2 and l;r are two unary operations on P that satisfy for all a 2 P ala 1 aal and aar 1 ara.</Paragraph>
      <Paragraph position="4"> Free pregroup. Let (P; ) be an ordered set of primitive types, P (a0 ) = fp(i) j p 2 P;i 2 Zg is the set of atomic types and T(P; ) = P(a0 ) = fp(i1)1 p(in)n j 0 k n;pk 2 P and ik 2 Zg is the set of types. For X and Y 2 T(P; ), X Y iif this relation is deductible in the following system where p;q 2 P, n;k 2 Z and X;Y;Z 2 T(P; ):  CL of languages has infinite elasticity iff 9 &lt; ei &gt;i2N sentences 9 &lt; Li &gt;i2N languages in CL 8i 2 N : ei 62 Li and fe1; ::: ;eng Ln+1 .</Paragraph>
      <Paragraph position="5"> 2We briefly recall that a monoid is a structure &lt; M; ; 1 &gt;, such that is associative and has a neutral element 1 (8x 2</Paragraph>
      <Paragraph position="7"> q p if k is even, and p q if k is odd This construction, proposed by Buskowski, defines a pregroup that extends on primitive types P to T(P; )3.</Paragraph>
      <Paragraph position="8"> Cut elimination. As for L and NL, cut rule can be eliminated: every derivable inequality has a cut-free derivation.</Paragraph>
      <Paragraph position="9"> Simple free pregroup. A simple free pregroup is a free pregroup where the order on primitive type is equality.</Paragraph>
      <Paragraph position="10"> Free pregroup interpretation. Let FP denotes the simple free pregroup with Pr as primitive types. We associate with each formula C of L or NL, an</Paragraph>
      <Paragraph position="12"> We extend the notation to sequents by: [A1;::: ;An] = [A1] [An] The following property states that FP is a model for L (hence for NL): if 'L C then [ ] FP [C].</Paragraph>
    </Section>
  </Section>
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