PPT-Reducibility

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. 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 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 Sipser. 5.1 (pages 187-198). Reducibility. Driving directions. Western Mass. Cambridge. Boston. If you can’t drive to London…. If something’s impossible…. Theorem 4.11: . A. TM. = {. <. 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(.