CMSC 330 Homework #2– Automata Fall 2005
1. For each of the following problems construct a deterministic finite automaton which describes or recog-
nizes the language given. Write all DFAs in the form of a transition diagram. The underlying alphabet
is Σ = {a, b}.
Be sure to give a DFA for each part, and not an NFA unless the problem asks for one. Do not use any
notational conveniences or shortcuts given in lecture.
The notation #a(w) is used below to refer to the number of a’s occurring in the string w. For example,
#a(bbaba) = 2.
Note: the first seven parts are the same languages which were questions on Homework #1. It would
be very instructive to compare the DFA and the regular expression for these languages– it is suggested
that you do so.
a. {w|wbegins with abab }
b. {w|wends with abab }
c. {w|wbegins with ab and ends with ba }
Note: The string aba is in this language.
d. {w|#a(w)≡2 (mod 5) }
Recall that i≡j(mod k) if and only if i−jis divisible by k.
e. {w|#a(w) is even or |w|is even }
f. {w|aaa is a substring of w}
g. {w|aaa is not a substring of w}
h. {w|wcontains exactly one occurrence of the substring aaa }
Note: the string aaaa has two occurrences of aaa.
i. {w|either #a(w) is divisible by 3 or wbegins with bbb }
Also write a regular expression which describes or recognizes this language.
j. {w|#a(w)≡1 (mod 3) and #b(w) is odd }
k. {w|neither aa nor bb is a substring of w}
l. Write an NFA for the language in part #1i.
2. Consider the following language:
{w|w∈ { 0,1}∗and wcontains an even number of 0s, and wdoes not contain three consecutive 1s }
Determine whether each of the following DFAs correctly describes or recognizes this language or not.
Identify why each incorrect DFA is wrong– give a string which the DFA doesn’t give the right results
for, and identify what result the DFA should give for that string, and what result it actually gives.
a. dead
state
0
1
0
1
0
1
0
1
1
