The document is a postgraduate final exam for a computer science course at Cairo University, focusing on theories of computation. It consists of various questions related to languages, determinism, automata, grammar, and complexity classes, with specific tasks to perform such as constructing regular expressions and proving languages are not regular. Students are required to solve five questions, including two mandatory ones, within a two-hour time limit.

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Cairo University
Faculty of Computers and Information
Postgraduate Final Exam (Pre-masters)
Department : Computer science
Lecturer : Dr. Hussien Sharaf
Course : Theory of computation
Course Code : CS611 Marks : 60 marks
Date : 1-June-2013 Time : 2 hours
Solve only five questions, given that questions one and two are
mandatory:
Question 1[mandatory] [15 marks]
For each of the following languages do the following:
i. Understand the languages without any assistance;
ii. Write sample words for the language where you show the smallest possible
word.
iii. Construct a regular expression that represents the language.
Note: Draw and fill the following table in your answer sheet:
Sample Regular Expression
a. Σ ={a, b} Only words with exactly two consecutive b's [1 marks]
Solution:
Sample = {bb, bba, abb, abba, aabbaaa …..}
a*bba*
b. Σ ={a, b} Empty words or words that do not end with ab. [2 marks]
Solution:
Sample= { Λ, a,b, abb, aa, ba, bbb,….. }
Λ + b+ ((a + b)*(a + bb))
c. Σ ={a, b} Only words such that { an
bm
| n is even and m is odd} where n and m indicate number
of “a”s and “b”s respectively. [2 marks]
Solution:
Sample = {b,aab,aabbb,..}
(aa)* b(bb)*
(aa+bb)* b
d. Σ ={a, b} Only words such that { an
bm
| (n+m) is even }; where n and m indicate number of “a”s
and “b”s respectively. Note that an even number could be sum of two even numbers or of two
odd numbers. [3 marks]
Solution:
Sample = {Λ, ab, aabb, aaabbb...}
(aa)* (bb)* + (aa)* a (bb)* b
e. Σ ={a, b} Words where “a” appears ONLY in even positions or doesn’t appear at all.
[3 marks]
Solution:
Sample = { Λ b ba bbb bbba baba …..}
- b+(bb+ba)*b*
- (ba+b)*
- (b(bb)* a)* + ( b(bb)*)*
f. Σ ={a, b} Construct a regular expression for all strings that have exactly one double letter in
them. “One double letter” implies two equal touching letters; triples or more are excluded.
[4 marks]
Solution:
Sample = { aa, baa, …..}
(b + Λ)(ab)*aa(ba)*(b + Λ) + (a + Λ)(ba)*bb(ab)*(a + Λ)

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Question 2[mandatory] [10 marks]
For each of the following languages do the following:
i. Understand the languages without any assistance;
ii. Write sample words for the language where you show the smallest possible
word.
iv. Draw a deterministic finite automaton that represents the language and
handles incorrect inputs.
a. Σ = {a, b} words where “a” appears ONLY in the second position or doesn’t appear at all.
[2 marks]
Solution:
Sample = { Λ b ba bbb bab babbb …..}
b. Σ = {a, b} Words that begin and ending with the same letter. [4 marks]
Solution:
Sample = { Λ b ba bbb bab babbb …..}
a
b
b
-+1 2
a,b
+3
a
a
-+1
b
4
a
b
3
a
+2
b
b
a
a
5
b
+4

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c. Prove that the language an
bn
is not regular using pumping lemma. [4 marks]
Solution:
If L is regular, then by Pumping Lemma (P.L.) ∃ n such that . . .
Now let x = 0n
1n
s ∈ L and |s| ≥ n, so by P.L. ∃x, y , z
then x = 0n
1n
= xyz with |xy| ≤ n and |y| ≥ 1.
But then (4) fails for i = 0:
So, x = 0s
, y = 0t
, z = 0p
1n
with
s + t ≤ n, t ≥ 1, p≥ 0, s+ t + p = n.
fails for i = 0: mean that t = 0 , then s+p not = n and number of 0 will be greater than
number of 1 and language will be not regular
Solve three of the next four questions
Question 3 [12 marks]
a. Σ ={a, b}. Describe “anything aa anything” “(a+b)*aa(a+b)*” using CFG. [6 marks]
Solution:-
(a+b)*aa(a+b)*
1 S à XaaX
2) X à aX
3) X à bX
4) X à Λ
Textbook Reference: Page 234
b. Derive “baabaab” from the above grammar. [6 marks]
Note: Draw and fill the following table in your answer sheet:
Derivation Rule Number
Solution:-
S à XaaX [by 1]
X à bXaaX [by 3]
X à bᴧaaX [by 4]
X à baabX [by 3]
X à baabaX [by 2]
X à baabaaX [by 2]
X à baabaabX [by 3]
X à baabaabΛ [by 4]
Textbook Reference: Page 234
Question 4 [12 marks]
PDA superceed FA by having a memory in form of stack hence it can handle
recursive structures.
a. Create a diagram for a PDA that can parse the CFG: [6 marks]
S→(S)|a
b. Trace the PDA on input “(a)” [6 marks]

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State Stack Tape
Start Δ….. (a)Δ…..
READ1 Δ…… (a)Δ…..
PUSH ( (Δ……. (a)Δ…..
READ1 (Δ……. (a)Δ…..
POP ) (Δ……. (a)Δ…..
Question 5 [12 marks]
a. Construct a TM that accepts all words with ‘b’ as a second letter. [6 marks]
Solution:
b. Write down the tracing table of aba for the above TM. [6 marks]
Note: Draw and fill the following table in your answer sheet:
Input string
(underline the letter that the head is reading)
State Number
2Start 1
Reject 5
Accept 4
(a, a, R)
(b, b, R)
(b, b, R)
(a, a, R)
PUSH ( READ2READ1
POP3
POP2
START
ACCEPT
( a )
(
(
(a, a, R)
(b, b, R)
)

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Solution:
Input string
(underline the letter the the head is reading)
State Number
aba 1
aba 2
aba 4 Accept
Question 6 [12 marks]
a) Given the following table for some problems, write down the class of the problem [Class
P, Class NP]: [6 marks]
Problem complexity Class
heap sort n log n
linear sort n3
Linear search n2
quick sort n log n
Creating a schedule for m football matches for n teams. 2m*n
Traveling sales man visiting n cities n! (n factorial)
3 Cars need to visit n customers only once with the least cost. 3n
check if a number is prime for a number n n/4
Composite Sum for a number n n-1
Does m clauses with n boolean variables has an assignment that makes
the whole expression TRUE
2m*n
Solution:
Problem complexity Class
heap sort n log n N
linear sort n3
N
Linear search n2
N
quick sort n log n N
Creating a schedule for m football matches for n teams. 2m*n
NP
Traveling sales man visiting n cities n! (n factorial) NP
3 Cars need to visit n customers only once with the least cost. 3n
NP
check if a number is prime for a number n n/4 N
Composite Sum for a number n n-1 N
Does m clauses with n boolean variables has an assignment that makes
the whole expression TRUE
2m*n
NP
b) Given the following instance of SAT: [6 marks]
SAT= (a ν
b)
Λ
(¬b ν
c)
Λ
(¬a ν¬b)
i) Construct a table that solves the problem.
ii) Does this problem have a valid solution?
Solution:
a b c a∨b ¬b∨c ¬a∨¬b SAT
T T T T T F F
T T F T F F F
T F T T T T T
T F F T T T T
F T T T T T T
F T F T F T F
F F T F T T F
F F F F T T F
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