1.54)
Consider the language
F
= {a
i
b
j
c
k

i
,
j
,
k
≥ 0 and if
i
= 1 then
j
=
k
}
(A)
Show that
F
is not regular.
Assume that
F
is regular and conforms appropriately.
If
F
is regular, then the reverse of
F
is regular; Use the pumping lemma, and let
p
be its
hypothetical pumping number.
Consider
w
= c
p
b
p
a ,
w
є
F
R
.
Attempting to pump
w
will yield some c
p
+
i
b
p
a, which is not in the
reverse of
F
, as
p
+
i
>
p
.
Thus
F
R
is not regular as it violates the pumping lemma, so
F
is not regular.
(B)
Show that
F
acts like a regular language in the pumping lemma
For any string
w
in
F
, there are four possible forms: aaa*b*c* (two or more as, followed by any
number of as, followed by any number of bs, followed by any number of cs), ab
i
c
i
(exactly one a,
followed by some number of bs, possibly zero, followed by an equal number of cs), bb*c* (no as,
followed by at least one b, followed by any number of cs, possibly zero), or c*(no as or bs, followed by
any number of cs, possibly zero).
We must now show that for each of these cases,
w
is suitable for
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 Spring '08
 STOUT
 Formal languages, Regular expression, Regular language, Lemma, Pumping lemma for regular languages

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