Tasks of theoretical computer science/ Main source Sipser’s book

All tasks are in files. Introduction to the Theory of ComputationIs the following statement true or false? Then, provide a justification for your answer,either by providing a valid reduction, or a proof showing that it is not possible. Here the languageDIV-3 is defined as { n | n is an integer that is evenly divisible by 3}.
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CS Track Core Theory
Fall 2017
Part 1 (20 pts. total)
1. (5 pts.) For the following function, express the T complexity in relation to n in the most concise
way. Then, provide a complete proof for your answer – you are encouraged to use the limit
lemma from the in-class homework.
7?2 v2?5 + 17
10?4 – 12? + 6
2. (5 pts.) Create a Deterministic Finite Automaton (DFA) which recognizes the language with
alphabet ? = {a, b, c} that includes exactly those strings that contain exactly two b’s, and also
contains the substring abc. You may first want to construct an NFA for the language, and then
convert it into a DFA, although this approach is not required.
3. (5 pts.) Determine whether the following language is regular or non-regular. If regular, provide a
DFA, NFA, or a regular expression for the language. If non-regular, prove it using the pumping
lemma or closure properties for regular languages.
L = { ai bj ck | where i, j, and k are non-negative integers such that i mod j = k }
4. (5 pts.) Is the following statement true or false? Then, provide a justification for your answer,
either by providing a valid reduction, or a proof showing that it is not possible. Here the language
DIV-3 is defined as { n | n is an integer that is evenly divisible by 3}.
DIV-3 =m ¯¯¯¯¯¯
ATM
Part 2
Deadline: Monday, 04.12.2017
The quiz must be made individually. Please hand in your solutions to HdN
in room 7.222, on Monday 04.12.2017, not later than 17.00. Total number of
points that can be obtained is 40.
1. (10 pts.) In the following questions, you must provide proofs for your answers. If you believe that some statement is false, give a counter example.
You may make use of commonly believed assumptions, like P 6= N P, or
NP =
6 co-N P.
Let L1 and L2 be arbitrary languages in N P. Which of the following
statements are true?
• L1 ? L2 is in N P.
• L1 n L2 is in N P.
• L1 L2 is in N P.
2. (10 pts.) Now assume that L1 and L2 are N P-complete languages.
Which of the following statements are true?
• L1 ? L2 is N P-complete.
• L1 n L2 is N P-complete.
• L1 L2 is N P-complete.
3. (5 pts.) Let Si be the language of Boolean SAT, restricted to i-variables.
For each i ? N ,
Si is a different language.
What do you think of the complexity of the Si ?
• Are the Si in N P?
• Are Si in P?
• Are the Si N P-complete?
4. (15 pts.) Consider the following decision problem.
Given a Boolean formula F (not necessarily in CNF), and a natural number k, represented in binary, are there at least k interpretations (only using
variables in F ) that make formula F true?
The restriction ‘only using variables in F ’ is necessary, because without
it, the problem becomes trivial.
1
(a) Explain why the problem becomes trivial without the domain restriction.
(b) Prove N P-completeness of the decision problem.
2

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