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Class Note for CMPSCI 601 at UMass(36)

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Date Created: 02/06/15
CMPSCI 601 Recall From Last Time Lecture 7 Th 62 The busy beaver function 0n is eventually larger than any total recursive function Th 63 There is a Universal Turing Machine U such that UPnam Mum Thm 64 Unsolvability 0f Halting Problem Let HALT Pn m 1 TM eventually halts Then HALT is re but not recursive Listing of all re sets W0 W1 W2 W in 1 Mil 1 Cor 66 Let K in Mltngt1 in mam 1 n I n E Wn Then K E re Recursive Notation Mna means that TM Mn converges on in put 33 ie Mnai gt MnaEN gt Fundamental Th of re Sets Let S g N TFAE 1 S is the domain of a partial recursive function ie 3W5 d0mMn39 93 EN 1 Mn93i 2 S Z or S is the range of a total recursive function ie S Z or S rangeMn MAN for some total recursive function 3 S is the range of a partial recursive function ie S MAN for somen E N 4 S is re ie S W for some n E N Proof Please learn this proof 1 gt 2 Assume 1 S 2 I case 1 S Z Thus 5 satis es 2 case 2 5 Z let a0 6 S From Mn compute MT which on input z does the follow ing 1 a Lz 31 Rz ie z Pay 2 run for y steps 3 if it halts then returna 4 else returna0 Claim 5 MAN I a E N MTltNgt g s MAN 2 5 Suppose a E 5 Thus converges in some number y of steps Therefore MTP1 13 Noncomputable step in construction no way to tell if we are in case 1 or case 2 2 gt 3 Assume 2 HS 2 thhen S M0N where M0 is a Turing machine that halts on no inputs Otherwise S MAN ie S is the range of the partial recursive function Note Even though is total it is still considered a partial recursive function However of course is not strictly partial 3 gt 4 Assume 3 S From Mn construct Md which on input a does the fol lowing l fori ltooo 2 run MAO Mn1 for 239 steps each 3 if any of these output 13 then returnl above construction called dovetailing Claim Md p5 Ifa E S then a E rangeMn MHU 2 computation takes k steps for some 339 k Thus at round 239 maxj k Mda will halt and output 177 If a e S then Mda will never halt ThusS Wd 2 13 I Mda l 4 gt 1 Assume 4 and thus 5 W S 2 i Z 1 From Mn construct Md which on input a does the fol lowing 1 run 2 if 1 then return1 3 else run forever 5 93 I Md93i ThusS domMd 2 I Mda Q CMPSCI 601 Reductions Lecture 7 De nition 71 We say that S is reducible to T S g T iff 3 total recursive f N gt N VwEN 1065 ltgt gamer A Note Later we require f E FDSPACElog 14017 in Mn017 Claim K g A047 Proof De ne f as follows erase input if 1 then write 17 Mfm write n Mn else 100p Tl E K gt 2 gt Z gt E Aggy Q Fundamental Th of Reductions If S g T then 1 lfT is re then S is re 2 lfT is core then S is core 3 If T is Recursive then S is Recursive Moral Suppose S g T Then lfT is easy then so is S HE is hard then so is T Proof Let f S g T ie E S gt fa E T 1 Suppose T W 2 I 1 From M compute the TM Mir which on input a does the following a compute f b run MAM Miquot f M w E 5 ltgt x 6 T ltgt Mzf93 1 ltgt Maw 1 Therefore 5 Wiz and S is re as desired 8 f S lt T 16 E S gt fa E T 2N0teS T gt T T E cone T E re g E re S E core 3 T E Recursive gt T E re T E cone gt S E re S E core gt S E Recursive De nition 72 Let C lt N C is recomplete iff 1 C E ne and 2 VA 6 re A g C Intuition C is a hardest re set In the g ordering it is above all other re sets Q 10 Theorem 73 K is 716 complete Proof Let A E re ie A W6 for some 239 Wanted Vnn E A gt f E K De ne the recursive function f which on input n com putes the following TM M Erase input Write n Mi n E A gt 1 gt V23Mfna 1 ltgtMfnltfltngtgt1 ltgt MM 11 Proposition 74 Suppose that C is icecomplete and the following hold I S E re and 2 C 3 S then S is 1 16 complete Proof Show VA 6 reA g 5 Know VA 6 reA g C Follows by transitivity of g A g C 3 S Q Corollary 75 A047 is icecomplete Every icecomplete set is 1 16 and hot recursive 12 HALT Pn m 1 TM eventually halts Proposition 76 HALT is Icecomplete Proof We have already seen that HALT is re It thus suf ces to show that K g HALT We want to build a total recursive f such that for all 10 E N w E K gt fw E HALT 1 gt halts That is we want 1 gt MAT halts where fw P r Given 10 let M aw Erase input Write 111 M w igl e igellgg Letting f PE39w 0 we have that 1 gt Mgw0 halts gt fw E HALTQ l3 mm Arlthmetlc H1erarchy W c0re Recursive 139 e re complete Primitive Recursive EXPTIME PSPACE c0NP PolynomialTime Hierarchy complete c0NP NP NP 1 c0NP NP complete quottruly feasiblequot NC NC2 logCFL SAC NSPACE log n 3 DSPACE log n Regular LogarithmicTime Hierarchy 14

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