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Mathematical models and methods 561
which would be obtained by applying one or other group of restrictions to the
rules of grammar. To take now a more precise example of these formal investigations,
let us imagine that the rules of grammar are restricted by the following: 'if
x -+ y is a rule of grammar, then x is a single member of the auxiliary vocabulary,
and y is a nonnul segment'.
Let us suppose that the items of auxiliary vocabulary are termed A, B, C, . and the items of the terminal vocabulary a, b, c, ...; then the above condition
means that in the grammar under consideration such rules as :
AB+a
AB + ac, etc.
are prohibited. For in fact, when the above condition is applied, two items of the
auxiliary vocabulary cannot stand to the left of +. Similarly, if # stands for
boundary, a rule such as A -+ # is not permitted.
The following formal problem can then be tackled : given the restrictive
requirements set out above, is it possible, from a grammar observing this
requirement, to generate 'sentences' with this structure:
or, more generally, with the structure:
aaaabbbaaaabbb,
anbmanbm? (1)
Another instance of what might be asked: is it possible from this type of
grammar (which Chomsky calls conte xt-free grammar) to generate sentences
with the structure:
aaaadbbdacadb bdaaaa, (2)
or, more generallly, with the structure:
a "zk- Id.. .db"~da"~~a"idb"zd. ..bnzk-zdanzk- i,
where k, n,, nz,. . .nZk- are some arbitrary positive integers?
Evidently, these are purely formal questions. One theorem of the formal
theory of grammars states that sentences of type (2) can be generated by the
grammar in question.
Let us postulate the following rules :
rule I S + adAda
rule z S +aSa
rule 3 S +aca
rule 4 A + bAb
rule 5 A + bdSdb
Rule z permits the formation of segment aSa. Reapplications of the same
rules successively form segments aaSaa and aaaSaaa. By next applying rule I,
the following segment is formed:
aaaadArlaaa or ddAda4.
Going on now to apply rules 4, 5 and 3 the desired segment is obtained:
a4dbzdacadbzda4