Download ECS 120: Midterm Section 2 - Theory of Computation - Prof. Phillip W. Rogaway and more Exams Computer Science in PDF only on Docsity! ECS 120: Theory of Computation Handout M2 UC Davis — Phillip Rogaway February 21, 2002 Midterm — Section 2 Instructions: Please answer the questions succinctly and thoughtfully. Good luck. — Phil Rogaway Name: Signature: On problem you got out of 1 45 2 30 3 25 Σ 100 ECS 120 Handout M2: Midterm — Section 2 1 1 Short Answer [45 points] Let M1 be an n1-state NFA and let M2 be an n2-state NFA. Using the procedures given in class and in your text, how many states will be in the DFA M the language of which is L(M1) ∪ L(M2)? Explain your reasoning. (2) An n-operation regular expression is a regular expression that uses a total of n operations—union, concatenation, or star. For example, 001∗ ∪ 1 is 4-operation regular expression. Let f(n) be the maximum number of states that you get in your NFA when, using the procedure given in class, you convert an n-operation regular expression into an NFA. Give a formula for f(n) (3) Complete the following, mathematically precise, definition, according to the conventions of our text a CFG is a 4-tuple G = ( ) where: ECS 120 Handout M2: Midterm — Section 2 4 2 Justified True or False [30 points] Put an X through the correct box. Where it says “Explain” provide a brief (but convinc- ing) justification. No credit will be given to correct answers that lack a proper justification. Where appropriate, make your justification a counter-example. Throughout, we use L to denote a language (maybe regular, maybe not). 1. If L is regular then L is context free. True False Explain: 2. If M = (Q,Σ, δ, q0, F ) is an NFA and F = Q then L(M) = Σ ∗. Explain: True False 3. If M = (Q,Σ, δ, q0, F ) is an NFA and M ′ = (Q,Σ, δ, q0, F ′), where F ′ = Q−F , then L(M ′) = L(M). True False Explain: 4. If L is context free then L is context free. True False Explain: ECS 120 Handout M2: Midterm — Section 2 5 5. There is an algorithm to decide if a CFG G generates the string abbaa. Explain: True False 6. If L is accepted by an NPDA then L is accepted by a 3-state NPDA. (Assume the conventions on PDAs adopted in lecture.) True False Explain: ECS 120 Handout M2: Midterm — Section 2 6 3 A Little Algorithm [25 points] A black box for a language L is a device (a subroutine) B that, when called on a string x, answers Yes if x ∈ L and No if x 6∈ L. Suppose I give you a black box B for some language L ⊆ {0, 1}∗. I don’t tell you what is L, but I do tell you that L = L(M) for some DFA M having 50 or fewer states. Describe an algorithm A to determine if L(M) is finite or infinite. Explain why your algorithm works. You will be calling B as a subroutine. You may make any number of calls.
