CS 172: Computability and Complexity, Spring 2008 S. A. Seshia & A. Kulkarni
HW 2: Equivalence of DFAs & NFAs; Regular Expressions
Assigned: January 31, 2008 Due in 283 Soda box, 5pm: February 7, 2006
Note: Take time to write clear and concise solutions. Confused and long-winded answers
may be penalized. Consult the course webpage for course policies on collaboration.
1. (8 points)
(a) Give an NFA recognizing all binary strings with a 1 in the third position from
the end and a 0 in the fourth position from the end.
(b) Following the equivalence proof, convert this to a DFA.
(c) Give a regular expression for this language. You are not required to use the
equivalence proof.
(d) Simplify the DFA given in part (b) so that every state is reachable by some
possible computation of the machine.
2. (6 points) If w=w1w2. . . wnis a string, define w1/2to be the first half of w. I.e.
w1/2=w1w2w3. . . wbn
2c. For any language L, define L1/2=w1/2:w∈L.
Prove that if Lis regular, then L1/2is regular.
Hint: If a DFA recognizing Lhas state space Q, construct an NFA with state space
Q×Qwhere the first coordinate goes forward and the second coordinate goes back-
ward.
3. (4 points)
(a) Prove that every NFA can be converted to an equivalent NFA that has a single
accept state.
(b) Give an example to support the falsity of the following statement: every DFA
can be converted into an equivalent DFA with a single accept state.
(c) (no points, optional) Give an NFA with a single accept state that recognizes
L=∅. Also, prove formally that the statement in part (b) is false.
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