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<Paper uid="C92-1021">
  <Title>Hopfield Models as Nondeterministic Finite-State Machines</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
2 A Bounded Push-Down Au-
</SectionTitle>
    <Paragraph position="0"> tomaton Although it is not an t.~tablished fact, it is asstoned here that natural languages are context-free, and consequently that sentences in a natural language can be recognized, by a push-down atttomaton (PDA). ilowever, we are not interested in modeling the competence of natural language users, but in modeling their performance. The human performance in natural language use is also characterized by a very limited degree of center-embedding. In terms of PDAs this means that there is n bound on the,number of items on the stack of a PI)A for a natural language. A bounded push-down automaton M = (Q,Y~,I',6, qs, Zo, F) is a PDA that has an upper limit k E ~ on the number of items on its stack, i.e. H ~&lt; k for every instantaneous description (ID) (q, w, a) of M. The set of stack states of this PDA is delined to be: QST =- {a I (qo,w, Zs) P*~t (q,e,C/~)}.</Paragraph>
    <Paragraph position="1"> QsT is finite: IQsT\[ &lt;_ (IFI) ~, therefore we may define a nondeterministic finite~state aeceptor (NDA) M ~ that has QST ms its set of states.</Paragraph>
    <Paragraph position="2"> The class of PDAs of which we would like to map bounded versions onto NDAs is constrained, among others to the class of v-free PDAs. By this constraint we anticipate the situation that grammars are stored in a neural network by self-organization. In that sitnation a neural network will store e-productions only if examples of applications of e-productions are repeatedly presented to it. This requires e to have a representation in the machine, in which case it fails to accommodate its definition.</Paragraph>
    <Paragraph position="3"> Another restriction we would like to introduce is to grammars in 2-standard form with a minimal number of quadratic productions: productions of the form ACrEs DE COL~G-92, NANTES, 23-28 AOOT 1992 I I 3 PROC. OF COLING-92, NA~rEs, AIJG. 23-28, 1992 A ~ bCD where b is a terminal and C and D are variables. Such a grammar can be seen as a minimal extension of a right-linear grammar. Within such grammars, quadratic productions provide for the center-embedding. Since such grammars have a minimal number of quadratic productions, acceptance by a PDA defined for such grammars requires a minimal use of (stack) memory, and titus generates a minireal Qs'r. To maintain this minimal use of memory a restriction to one-state PDAs that accept by empty stack is also required: when a PDA is mapped onto an NDA, the information concerning its states is h)st, unless it was stored on the stack.</Paragraph>
    <Paragraph position="4"> An e-free, one-state PDA that simulates a context-free grammar in 2-standard form with a minimal number of quadratic productions (and that accepts by empty stack) satisfies all our criteria. For every such PDA we can define an NDA, for which we can prove \[4\] that it accepts the same language ms the</Paragraph>
    <Paragraph position="6"> Theorem 2.2 (correctness of the NDA) Let M be an e.free one-state PDA with bound k, if M' is an NDA a.~ defined in definition 2.1, then M accepts a string by empty stack tf and only if M' accepts it by accepting state.</Paragraph>
    <Paragraph position="7"> In as far as a natural language is context-free, we claim that there is an instance of our aeeeptor that recognizes it.</Paragraph>
  </Section>
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