Chapter 3 Context-Free Grammars, Context-Free Languages, Parse ...

3.4. REGULAR LANGUAGES ARE CONTEXT-FREE 47
by the symbol |. Thus, a group of productions
may be abbreviated as
A → α1,
A → α2,
···→···,
A → αk,
A → α1 | α2 | ··· | αk.
An interesting corollary of the CNF is the following decidability result. There is an
algorithm which, given a context-free grammar G, given any string w ∈ Σ ∗ , decides whether
w ∈ L(G). Indeed, we first convert G to a grammar G ′ in Chomsky Normal Form. If w = ɛ,
we can test whether ɛ ∈ L(G), since this is the case iff S ′ → ɛ ∈ P ′ .Ifw = ɛ, letting n = |w|,
note that since the rules are of the form A → BC or A → a, wherea ∈ Σ, any derivation
for w has n − 1+n =2n − 1 steps. Thus, we enumerate all (leftmost) derivations of length
2n − 1.
There are much better parsing algorithms than this naive algorithm. We now show that
every regular language is context-free.
3.4 Regular Languages are Context-Free
The regular languages can be characterized in terms of very special kinds of context-free
grammars, right-linear (and left-linear) context-free grammars.
Definition 3.4.1 A context-free grammar G =(V,Σ,P,S)isleft-linear iff its productions
are of the form
A → Ba,
A → a,
A → ɛ.
where A, B ∈ N, anda ∈ Σ. A context-free grammar G =(V,Σ,P,S)isright-linear iff its
productions are of the form
where A, B ∈ N, anda ∈ Σ.
A → aB,
A → a,
A → ɛ.
The following lemma shows the equivalence between NFA’s and right-linear grammars.