View
214
Download
1
Category
Preview:
DESCRIPTION
theory of computation
Citation preview
Jaypee Institute of Informtion Technology University Noida
Computer Science Deprtment Problem Set 1
PART A
PROBLEM 1 (2+2+2+2 points)
Suppose A,B ⊆ Σ∗ are languages. In each of the following cases, briefly prove or give a counterex-ample to (i) C ⊆ D and (ii) D ⊆ C.
(A) C = (A ∩B)∗ and D = A∗ ∩B∗
(B) C = A∗
and D = A∗
(C) C = (A ∪B)∗ and D = A∗ ∪B∗
(D) C = (AB)∗ and D = A∗B∗
Solution.
PROBLEM 2 (4 points, suggested length of 1/4 of a page)
Show that if A,B ⊆ Σ∗ are regular languages, then the language A tB defined by
A tB = {u1w1u2w2 · · ·unwn : u1, w1, . . . , un, wn ∈ Σ∗, u1 · · ·un ∈ A, and w1 · · ·wn ∈ B}
is also regular by providing the 5-tuple representation of an NFA recognizing A t B in terms ofDFAs for A and B. (Hint: look at the cross-product construction.)
Solution.
PROBLEM 3 (Challenge! 3 points)
An equivalence relation ∼ on Σ∗ is called a congruence if for any two strings u,w ∈ Σ∗ and anyletter σ ∈ Σ, if u ∼ w then uσ ∼ wσ. The relation ∼ is said to be closed with respect to thelanguage A ⊆ Σ∗ if for any strings u ∈ A and w ∈ Σ∗, if u ∼ w then w ∈ A.
Show that A is regular if and only if there is a congruence with finitely many equivalence classesthat is closed with respect to A.
Solution.
PART B
PROBLEM 4 (3+3+3 points, suggested length of 1/3 of a page)
Are the following statements true or false? Justify your answers with a proof or counterexample.
(A) If L ⊆ {a, b}∗ is a regular language, then the language that consists of all the strings in L whichdo not contain ba is regular.
(B) Let D be a DFA. Then, D contains a cycle if and only if L(D) is infinite.
(C) Complementing all states in a DFA M (making the final states non-final and vice-versa) willresult in a new DFA M ′ such that L(M ′) = Σ∗ − L(M).
Solution.
PROBLEM 5 (6 points, suggested length of 1/3 of a page)
Let Sn ⊆ {a, b, c}∗ be the language of strings of length n where b appears an even number of times.If sn = |Sn|, show that
sn = 3n−1 + sn−1
Use this fact to find the value of sn in closed form (i.e. in terms of only of n).
Solution.
PART C
PROBLEM 6 (3+3+3 points)
Describe informally the language represented by the following regular expressions and DFA.
(A) ((a ∪ b ∪ c)(a ∪ b ∪ c)b)∗(a ∪ b ∪ c)(a ∪ b ∪ c)
(B) b∅∗a ∪ ab ∪ ba∅ba
(C) ll l
lh
l..........................
���
�������
����@@R
@@R@@I
@@I
a
a
b
b
b
baa
Solution.
PROBLEM 7 (6 points)
Convert the following NFA into a DFA using subset construction. Provide a formal description(5-tuple) and digram for full credit.
0 1 3
2
a
a
a
b
b
b
Solution.
Recommended