cs373 fa10 hw8sol - Solutions for Problem Set 8 CS 373...

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Solutions for Problem Set 8 CS 373: Theory of Computation Assigned: October 26, 2010 Due on: November 2, 2010 at 10am Homework Problems Problem 1 . [Category: Proof] Read about Post’s Correspondence Problem from section 5.2. Solve problem 5.18. Solution: Given an instance of PCP over alphabet Σ = { a 1 ,...a n } , we will reduce it to an instance of PCP over the binary alphabet. The idea will be to apply the homomorphism h that maps the symbol a i to string 10 i . The crucial property about this mapping the following: for u,v Σ * , u = v iff h ( u ) = h ( v ). Let P be an instance of PCP over Σ. Take h ( P ) to be the PCP instance over the binary alphabet, where we replace any string w appearing in any tile of P by h ( w ). Given that u = v iff h ( u ) = h ( v ), we can conclude that P is in PCP iff h ( P ) is in PCP. Since we have reduced PCP over Σ to PCP over { 0 , 1 } and PCP over Σ is undecidable, we conclude PCP over { 0 , 1 } is undecidable. Problem 2
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