Theory of Computation
Sample Final.
This exam is closed book. Answer all questions.
1. (10 points.) Give a regular expression for the following language A.
A={uv |u, v ∈ {a, b, c}∗, u, v 6=, and uand vhave no character in common}.
e.g. abc ∈A,cacbb ∈A,abcabc /∈A,abab /∈A.
2. (10 points.) Give a DFA or NFA to accept the following language B.
B={w|w∈ {0,1}∗and wincludes a 1 but not as its third character}.
3. (10 points.) a. State the Pumping Lemma for regular languages.
b. Show that the following language Cis not regular.
C={wcuwRv|w, u, v ∈ {a, b}∗}.
4. (10 points.) Give a context free grammar to generate language Cfrom Problem
3b.
5. (10 points.) a. State the Pumping Lemma for context free languages.
b. Show that the following language Dis not context free.
D={aibajbakbal|i=kand j=l}.
6. (10 points.) Give a PDA to accept the following language E.
E={x1#x2#· · · #xk|k≥2, xh∈ {a, b}∗,1≤h≤k, and xi=xR
jfor some i < j}.
7. (10 points.) Let L={G|Gis a CNF grammar with no useless variables},
where a useless variable is one that is not used in any derivation of a string of variables.
Give an algorithm to show that Lis decidable. That is, give an algorithm that takes
as input a CNF grammar Gand determines whether for each variable Vin Gthere
is some derivation of a string of terminals or the empty string that uses the variable
V.
Hint: A useful first step is to determine those variables that can derive a string of
terminals or the empty string.
8. (10 points.) Suppose that you are given an algorithm AFto decide the following
language F:
F={hPi | Phalts on at least one input}.
Using AFas a subroutine, give an algorithm AAPr og to decide AP rog . Recall that
AP rog ={hP, w i | Phalts on input w}.
