PPT-Reducibility

Sipser 51 pages 187198 Reducibility Driving directions Western Mass Cambridge Boston If you cant drive to London If somethings impossible Theorem 411 A TM lt. Today •We’ve seen several undecidability proofs. •Today we’ll extract some of the key ideas of those •Two main ideas: –A formal definition of –Rice’s Theorem 1. 2. Text book Pages 187– 199. Reducibility. A . reduction. . is a way of converting one problem to another problem in such . a way . that a solution to the second problem can be used to solve the first problem. Sipser. 5.3 (pages 206-210). Computable functions. Definition 5.17: A function . f:Σ. *→. Σ. *. is a . computable function. if some Turing machine . M. , on every input . w. , halts with just . K (i) halts } { i | i



(x) halts for all x } x[x ] } Note that by showing (i) halts if and only if Ms(a, i)x) halts for all x Based on . M. . Sipser. , “Introduction to the Theory of Computation,” Second Edition, Thomson/Course Technology, 2006, Chapter 5.. Review. Recall the . halting problem. :. . HALT. TM. = { . . The class P is the class that contains all the problems that are solved in polynomial time on the size of the input by a deterministic Turing Machine.. . For n being the size of the input, the running time is O(.