Mind, Body, World- Foundations of Cognitive Science, 2013a

Whether a learner is undergoing informant learning or text learning, Gold
(1967) assumed that learning would proceed as a succession of presentations of
expressions. After each expression was presented, the language learner would
generate a hypothesized grammar. Gold proposed that each hypothesis could be
described as being a Turing machine that would either accept the (hypothesized)
grammar or generate it. In this formalization, the notion of “learning a language”
has become “selecting a Turing machine that represents a grammar” (Osherson,
Stob, & Weinstein, 1986).
According to Gold’s (1967) algorithm, a language learner would have a current
hypothesized grammar. When a new expression was presented to the learner, a test
would be conducted to see if the current grammar could deal with the new expression.
If current grammar succeeded, then it remained. If the current grammar
failed, then a new grammar—a new Turing machine—would have to be selected.
Under this formalism, when can we say that a grammar has been learned? Gold
defined language learning as the identification of the grammar in the limit. When
a language is identified in the limit, this means that the current grammar being
hypothesized by the learner does not change even as new expressions are encountered.
Furthermore, it is expected that this state will occur after a finite number of
expressions have been encountered during learning.
In the previous section, we considered a computational analysis in which different
kinds of computing devices were presented with the same grammar. Gold (1967)
adopted an alternative approach: he kept the information processing constant—
that is, he always studied the algorithm sketched above—but he varied the complexity
of the grammar that was being learned, and he varied the conditions under
which the grammar was presented, i.e., informant learning versus text learning.
In computer science, a formal description of any class of languages (human or
otherwise) relates its complexity to the complexity of a computing device that could
generate or accept it (Hopcroft & Ullman, 1979; Révész, 1983). This has resulted in
a classification of grammars known as the Chomsky hierarchy (Chomsky, 1959a). In
the Chomsky hierarchy, the simplest grammars are regular, and they can be accommodated
by finite state automata. The next most complicated are context-free grammars,
which can be processed by pushdown automata (a device that is a finite state
automaton with a finite internal memory). Next are the context-sensitive grammars,
which are the domain of linear bounded automata (i.e., a device like a Turing
machine, but with a ticker tape of bounded length). The most complex grammars are
the generative grammars, which can only be dealt with by Turing machines.
Gold (1967) used formal methods to determine the conditions under which
each class of grammars could be identified in the limit. He was able to show that
text learning could only be used to acquire the simplest grammar. In contrast, Gold
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