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TDDC95 Introduction to the Theory of Computation 2013-08-31, kl. 14-18, Room TER1 Materials allowed: A dictionary from English to any other language is allowed. No other books, notes etc. are allowed and no electronic equipment (calculators, computer, mobile phones etc.) are allowed. Questions: Christer Bäckström will show up after approx one hour and is otherwise available on phone 0705-840889 Grading: The maximum number of points is 30 and 15 points are required to pass the examination. At least 15 p is required for grade 3, at least 20 p is required for grade 4 and at least 25 p is required for grade 5. Results: When the exams are graded there will be an opportunity to see the exams and discuss the result with the examiner (this is called a tentavisning in swedish). When and where this will happen will be announced on the course homepage as soon as the grading is finished. Please observe the following: • Use only one side of each paper. • Each problem must be solved on a separate paper (or several papers, if necessary. Subproblems of a problem (a, b, c etc.) may be solved on the same page. • Properly justify all your answers. If you give only an answer without justification, you may get zero points even if the answer is correct. • Make sure your answers are readable. • Try to leave space for comments on every page. Good luck! Problems 1. Define the alphabet Σ = {0, 1, 2, 3} and the language L that consists (4 p) of all strings over Σ such that the sum of the symbols in the string is even. For example, the strings 022 and 12313 are both in L but 012 and 333 are not in L. (a) Give a regular expression defining L. (2) (b) Draw the transition graph for a DFA that accepts L. (2) 1 2. Consider the language L = {ak b∗ ck | k > 0} over the alphabet Σ = (6 p) {a, b, c}. (a) Prove that L is not regular. (4) (b) Prove that L is a CFL by providing a CFG for L. (2) 3. For any string s define the reverse string R(s) of s as the same string as (6 p) s but with the symbols in opposite order. For example R(abab) = baba and R(01231) = 13210. For any language L define the corresponding reverse language R(L) as R(L) = {R(s) | s ∈ L}, i.e. R(L) consists of the reverse strings for all strings in L. (a) Prove that if L is a regular language, then also R(L) is a regular (3) language, by explaining how to modify a regular grammar for L to a regular grammar for R(L). (b) Prove that if L is a CFL, then also R(L) is a CFL, by explaining (3) how to modify a CFG for L to a CFG for R(L). 4. (4 p) (a) Is it meaningful to consider algorithms that run in polynomial (2) time but exponential space? (b) NP-complete is an important and useful concept. Is it meaningful (2) to consider a corresponding concept P-complete for the class P? 5. Adding a stack to a DFA gives a pushdown automaton, which is a (4 p) strictly more powerful concept. What if we add a stack to a deterministic Turing machine, would that give a more powerful concept? 6. For each of the following 3 problems, prove that the problem is in NP (6 p) or explain why this cannot be proven (with our current knowledge). (a) AndSat Input: Two 3CNF formulae ϕ and ψ. Question: Are both ϕ and ψ satisfiable? (2) (b) OrSat Input: Two 3CNF formulae ϕ and ψ. Question: Is at least one of ϕ and ψ satisfiable? (2) (c) XorSat Input: Two 3CNF formulae ϕ and ψ. Question: Is exactly one of ϕ and ψ satisfiable? (2) 2