Course Notes on Formal Languages and Compilers

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2) P is a finite set of rules (replacement rules, productions) each of which
has the form x→y where x, y∈(N ∪ T)*.
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3) S is the starting set (initial set) of the grammar and is a finite subset of
(N ∪ T)* that is S ⊆ (N ∪ T)*.
Derivations and language of a Grammar
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Let G=(N,T,P,S) be any phrase structure grammar and let u,v∈(N∪T)*. We
write u⇒v and say v is derived in one step from u by the rule x→y, providing
that u = pxq and v = pyq. (Here the rule x→y is used to replace x by y in u
to produce v. Note that p,q∈(N∪T)*.)
If u 1 ⇒u 2 ⇒u 3 ......⇒ u n we say u n is derived from u 1 in G and write u 1 ⇒ + u n .
Also if u 1 =u n or u 1 ⇒ + u n we write u 1 ⇒ * u n
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L(G) the language of G is defined by:
L(G) = {t∈T*: Z⇒ * t for some Z∈S}
= {t∈T*:t∈S or Z⇒ + t for some Z∈S}.
So the elements of L(G) are those elements of T* which are elements of S
or which are derivable from elements of S.
Classification of Phrase Structure Grammars
The following table describes the various types of grammars, and their
relative advantages and disadvantages. Historically this classification of
grammars arose because of the need to find efficient recognition algorithms
for the languages of grammars.