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Home12UniversalTMsandClosurePCPandReductionEasy

# Home12UniversalTMsandClosurePCPandReductionEasy - CS 341...

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Unformatted text preview: CS 341 Automata Theory Elaine Rich Homework 12 Due Wednesday, November 29 at noon (Slide it under my office door.) 1) Construct a standard one-tape Turing machine M to enumerate the language: { w : w is the binary encoding of a positive integer that is divisible by 3}. Assume that M starts with its tape equal to ❑ . Also assume the existence of the printing subroutine P , defined in Section 20.5.1. As an example of how to use P , consider the following machine, which enumerates L ′ , where L ′ = { w : w is the unary encoding of an even number}: > P R 1 R 1 You may find it useful to define other subroutines as well. 2) Encode the following Turing Machine as an input to the Universal Turing Machine. M = ( K , Σ , Γ , δ , q , { h }), where K = { q , q 1 , h }, Σ = { a , b }, Γ = { a , b , c , ❑ }, and δ is given by the following table: q σ δ (q, σ ) q a ( q 1 , b , → ) q b ( q 1 , a , → ) q ❑ ( h , ❑ , → ) q c ( q , c , → ) q 1 a ( q , c , → ) q 1 b ( q , b , ← ) q 1 ❑ ( q , c , → ) q 1 c ( q 1 , c , → ) 3) Consider the language L = {< M > : M accepts at least two strings}. a) Describe in clear English a Turing machine M that semidecides L . b) Suppose we changed the definition of L just a bit. We now consider: L ′ = {< M > : M accepts exactly 2 strings>. Can you tweak the Turing machine you described in part a to semidecide L ′ ? 4) Prove that D is closed under: a) union b) concatenation c) Kleene star d) Intersection 5) Prove that SD is closed under: a) union b) concatenation c) Kleene star d) Intersection 6) Show that H ALL is not in D by reduction from H . 7) Let L = {< M > : M is a Turing machine and a ∈ L ( M )}. Show that L is not in D . 8) Consider the Post Correspondence Problem. Construct a nontrivial instance of PCP , different from the ones in the book, for which a solution exists and show the solution. By nontrivial we mean one in which the X and Y lists are not identical. 9) Prove that A ANY is not in D. 10) Use Rice’s Theorem to prove that L = {< M > : M accepts at least two odd length strings} is not in D. CS 341 Automata Theory Elaine Rich Homework 12 Answers 1) Construct a standard one-tape Turing machine M to enumerate the language: { w : w is the binary encoding of a positive integer that is divisible by 3}. is the binary encoding of a positive integer that is divisible by 3}....
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Home12UniversalTMsandClosurePCPandReductionEasy - CS 341...

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