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Unformatted text preview: Jeff Edmonds Jeff Edmonds York University York University COSC 4111 Lecture Lecture 2 2 Computable Computable Halting Halting Acceptable, Witness, Enumerable Acceptable, Witness, Enumerable Computable Computable Ackermann's Time Ackermann's Time Double Exponential Time Double Exponential Time E: Exponential Time E: Exponential Time Exp Time Exp Time PSpace PSpace PH: PolyTime Hierarchy PH: PolyTime Hierarchy NP & CoNP NP & CoNP Complexity Classes Complexity Classes PolyTime PolyTime NC: Polylog depth Circuits NC: Polylog depth Circuits NC2 NC2 NL: NonDet Log Space NL: NonDet Log Space L: LogSpace L: LogSpace AC AC 0, Thres , Thres 0, Arith , Arith 0: Constant Depth : Constant Depth NC NC 0: Constant Fanout : Constant Fanout Computable Exp Poly A complexity class is a set of computational problems that have a similar difficulty in computing them. Design a new class: 1. Choose some model Java or Circuits 1. Deterministic or Nondeterministic 2. Limit some resource Time or Space to Log, Poly, Exp, . Complexity Classes Complexity Classes NP CoNP Computable Exp Poly A complexity class is a set of computational problems that have a similar difficulty in computing them. We will start with a small weak class and work our way up. Each will be a super set of the previous Proving C 1 C 2 is not too hard. Prove C 2 can simulate C 1 . Proving C 1 C 2 is hard. Prove P C 2 and P C 1 . Complexity Classes Complexity Classes NP CoNP (unless pointed out) Computable Exp Poly A complexity class is a set of computational problems that have a similar difficulty in computing them. A problem is complete for a class if being able to solve this problem fast means that you can solve every problem in the class fast. Complexity Classes Complexity Classes complete NP CoNP NC = Set of computational problems computed by circuits of and/or/not gates each with two inputs of constant depth. Each output bit can only depend on a constant number of the input bits. But can compute any function of these bits. NC NC x 31 x 25 x 43 OR OR AND AND OR NOT constant constant NC is short for Nicks Class after Nick Pippenger NC = Set of computational problems computed by circuits of and/or/not gates each with two inputs of constant depth. Eg: It can compute the next configuration of a TM. NC NC AC = Set of computational problems computed by circuits of and/or/not gates each arbitrary fanin of constant depth and polysize. = circuits of and/or/not gates each with two inputs any depth but constant # of alternations between and & or . Arbitrary fanin means can depend on all the input bits....
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This note was uploaded on 02/13/2012 for the course CSE 4111 taught by Professor Edmonds during the Winter '12 term at York University.
 Winter '12
 Edmonds

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