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

What is meant by the claim that an information processing device can “accommodate”
a grammar? In the formal analysis of the capabilities of information processors
(Gold, 1967), there are two answers to this question. Assume that knowledge
of some grammar has been built into a device’s machine head. One could then ask
whether the device is capable of accepting a grammar. In this case, the “question” on
the tape would be an expression, and the task of the information processor would
be to accept the string, if it is grammatical according to the device’s grammar, or to
reject the expression, if it does not belong to the grammar. Another question to ask
would be whether the information processor is capable of generating the grammar.
That is, given a grammatical expression, can the device use its existing grammar to
replicate the expression (Wexler & Culicover, 1980)?
In Chapter 2, it was argued that one level of investigation to be conducted by
cognitive science was computational. At the computational level of analysis, one
uses formal methods to investigate the kinds of information processing problems a
device is solving. When one uses formal methods to determine whether some device
is capable of accepting or generating some grammar of interest, one is conducting
an investigation at the computational level.
One famous example of such a computational analysis was provided by Bever,
Fodor, and Garrett (1968). They asked whether a finite state automaton was capable
of accepting expressions that were constructed from a particular artificial grammar.
Expressions constructed from this grammar were built from only two symbols, a
and b. Grammatical strings in the sentence were “mirror images,” because the pattern
used to generate expressions was b N ab N where N is the number of bs in the
string. Valid expressions generated from this grammar include a, bbbbabbbb, and
bbabb. Expressions that cannot be generated from the grammar include ab, babb,
bb, and bbbabb.
While this artificial grammar is very simple, it has one important property: it
is recursive. That is, a simple context-free grammar can be defined to generate its
potential expressions. This context-free grammar consists of two rules, where Rule
1 is S a, and Rule 2 is a bab. A string is begun by using Rule 1 to generate an a.
Rule 2 can then be applied to generate the string bab. If Rule 2 is applied recursively
to the central bab then longer expressions will be produced that will always be consistent
with the pattern b N ab N .
Bever, Fodor, and Garrett (1968) proved that a finite state automaton was not
capable of accepting strings generated from this recursive grammar. This is because
a finite state machine can only move in one direction along the tape, and cannot
write to the tape. If it starts at the first symbol of a string, then it is not capable
of keeping track of the number of bs read before the a, and comparing this to the
number of bs read after the a. Because it can’t go backwards along the tape, it can’t
deal with recursive languages that have embedded clausal structure.
Elements of Classical Cognitive Science 71