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 bs, 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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