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<Paper uid="J98-2003">
  <Title>nian Academy</Title>
  <Section position="4" start_page="248" end_page="248" type="intro">
    <SectionTitle>
2. Definitions
</SectionTitle>
    <Paragraph position="0"> In this section, we introduce the classes of grammars we shall investigate in this paper.</Paragraph>
    <Paragraph position="1"> As usual, given an alphabet V (which we also call vocabulary), we denote by V* the set of all words (equivalently: strings) over V, including the empty one, which is denoted by A. The set of all nonempty words over V, hence V* - {A}, is denoted by V +. The length of x c V* is denoted by Ix\[ and its mirror image (also called the reversal) by mi(x). The families of finite, regular, linear, context-free, context-sensitive, and recursively enumerable languages are denoted by FIN, REG, LIN, CF, CS, RE, respectively. For the elements of formal language theory we use, we refer to Harrison (1978), Rozenberg and Salomaa (1997), and Salomaa (1973). 3 A contextual grammar (with choice) is a construct:</Paragraph>
    <Paragraph position="3"> where V is an alphabet, A is a finite language over V, $1 ..... Sn are languages over V, and C1 ..... Cn are finite subsets of V* x V*.</Paragraph>
    <Paragraph position="4"> The elements of A are called axioms (starting words), the sets Si are called selectors, and the elements of sets Ci, written in the form (u, v), are called contexts. The pairs (Si, Ci) are also called productions. The intuition behind this construction is that the contexts in Ci may be adjoined to words-in the associated set Si. Formally, we define the direct derivation relation on V* as follows: X ::=:=-kin y iff x = XlX2X3, y =Ill, lX2VX3, where x2 E Si,(u,v) E Ci, for some i, 1 &lt; i &lt; n. Denoting by ~Tn the reflexive and transitive closure of the relation ==-&gt;'in, the language generated by G is: Lin(G) -~ {z E V* I w ===&gt;'i~ z, for some w E A}.</Paragraph>
  </Section>
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