Exam Computability and Complexity

Transcription

1 Total number of points:... Number of extra sheets of paper:... Exam Computability and Complexity by Jiri Srba, January 2009 Student s full name CPR number Study number Before you start, fill in the three identification fields in the table above. Do not fill in the heading at the top (Total number of points and Number of extra sheets of paper). You are allowed to bring and use only your pen, nothing else. Questions 1 to 11 should be answered directly into the framed boxes. Any calculations/drafts can be done on the extra sheets of paper (do not hand them in). You may request additional sheets from the present guards. Questions 12 and 13 should be answered and handed in on the extra sheets of paper. Remember to write your name on every extra sheet you will use. Note that there are several variants of the exam. Even though the exam questions of your neighbours might look identical, they in fact differ in small details. This has a significant influence on what are the correct answers. Question 1 (4 points, each correct string +1, wrong answer 0, no answer 0) Consider the TM M drawn below (all missing transitions go implicitly to the state q reject and move the head to the right). The input alphabet is Σ = {0} and the start state of M is q 0. Give three different strings (over Σ ) that do not belong to L(M) and one string that belongs to L(M). 0,R M: 0 R q 0 q 1 0 L R 0 R q accept q 2 0,R L(M) 1

2 Question 2 (4 points, each correct answer +2, wrong answer 0, no answer 0) Finish the definitions below and be formally precise and complete. Write only into the framed boxes. Definition: A language L Σ is recognized by a Turing machine M iff... Definition: A language L Σ is co-recognizable iff... Question 3 (8 points, each correct answer +2, wrong answer 0, no answer 0) Consider the following languages: L 1 = { M M is a TM which loops on the string aabbab } L 2 = { M, w M is a TM that halts on w } L 3 = { M M is a TM which accepts the string aabbab in less then 572 steps } L 4 = { M M is a TM and L(M) is regular} Assign the languages into the categories below (some of the fields might be empty). Decidable: Undecidable and recognizable: Undecidable and co-recognizable: Neither recognizable nor co-recognizable: Question 4 (4 points, correct answer +4, wrong answer and no answer 0) What is the statement of Church-Turing Thesis? Write the answer into the framed box. 2

3 Question 5 (8 points, point distribution +2,+1,+1,+2,+1,+1, wrong answer 0, no answer 0) Fill in the details into the empty boxes in the proof below. Theorem: The emptiness problem for nondeterministic finite automata (NFA), i.e. the question whether the language of a given NFA is empty, is decidable. Proof: We shall first formulate the problem as a language E NFA. E NFA ={ } We want to construct a decider M for E NFA. Such a Turing machine M, when run on an input string w Σ, should behave (accept/reject/loop) as follows: M w if w E NFA, and M w if w E NFA. Here is the machine M: M = On input w, check if w is of the form A for some NFA A, if not M rejects. 2. Mark all accept states of A. 3. Repeat until no new states are marked: Mark any state s which satisfies that 4. If the start state is marked, then M, else M. Question 6 (4 points, each correct answer +1, wrong answer 0, no answer 0) Finish the following sentences so that the statements about mapping reducibility are true. If A m B and B is recognizable then A is. If A m B and A is undecidable then B is. If A m A and A is recognizable then A is. If A m B and A is not recognizable then B is. 3

4 Question 7 (4 points, each correct definition +2, wrong answer 0, no answer 0) Finish the definitions below and be formally precise and complete. Write only into the framed boxes. Definition: A language L Σ belongs to the complexity class NP iff... Definition: A language L Σ belongs to the complexity class PSPACE iff... Question 8 (4 points, each correct answer +2, each wrong answer -1, no answer 0) Mark true/false for each of the claims by putting a cross in the corresponding field. If L belongs to the complexity class co-np then it belongs also to the class PSPACE. True False If L belongs to the complexity class P then L belongs to the complexity class NP. True False Question 9 (4 points, each correct answer +2, wrong answer 0, no answer 0) In this question give the tightest upper-bounds as proved during the course. Assume a nondeterministic TM M running in time O(n 3 ). What is the running time of the corresponding deterministic TM simulating M and deciding the same language? Assume a nondeterministic TM M running in space O(n 3 ). What is the space complexity of the corresponding deterministic TM simulating M and deciding the same language? 4

5 Question 10 (4 points, each correct part +2, wrong answer 0, no answer 0) Complete the definition below. Definition: A language L is PSPACE-complete if 1. and 2.. Question 11 (8 points, satisfiability +2, correct construction +6, wrong answer 0, no answer 0) Consider the following Boolean formula φ in cnf. φ = (x 1 x 3 ) (x 1 x 2 x 3 x 4 ) Find one satisfying truth assignment of the formula φ. x 1 x 2 x 3 x 4 Using the reduction described in the proof of CNF-SAT P 3SAT construct a formula φ in 3-cnf such that φ is satisfiable if and only if φ is satisfiable. Question 12 (22 points, formulation +4, proof strategy +6, correct proof +12, wrong answer 0) This question should be answered on a separate sheet of paper. Consider the T OT AL TM language which contains the encodings of Turing machines that accept all strings from Σ. Prove that T OT AL TM is undecidable. First, (i) give a precise definition of the language T OT AL TM, then (ii) describe the proof strategy you plan to use in your proof, and finally (iii) give the full proof. Question 13 (22 points, formulation +4, construction +12, complexity analysis +6, wrong answer 0) This question should be answered on a separate sheet of paper. Prove that the complexity class P is closed under intersection. First, (i) formulate what you assume and what you want to show, then (ii) write down the proof, and finally (iii) provide the complexity analysis of your construction. 5