Moed_A_Spring_2010_Solution

Moed_A_Spring_2010_Solution - (236343 zeiaeyigd zxez...

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Unformatted text preview: (236343) zeiaeyigd zxez r"yz aia` '` cren oexzt '` wlg 1 dl`y 1 sirq . L 1 ∈ RE \ R . | L ( M ) | ≥ |h M i|-k zehyta gqpl ozip L 1 ly i`pzd z`y al miyp ziy`x w ∈ Σ * milind lk lr zxwean dvxda M z` dvixn `id h M i hlw lr :oldlck zlret L z` zlawnd M 1 dpekn f` h M i ∈ L 1 m` xexiaa .zlawn M 1 ,milin |h M i| zegtl elawzd m` . M ici lr elawzd milin dnk zxteqe dvxda aly meya f` h M i / ∈ L 1 m`e ;milin |h M i| zegtl elawzi ok` dvxda edylk alya ik dze` lawz M 1 . L ∈ RE okle L ( M 1 ) = L 1 okl .lawz M 1-y ick milin witqn elawzi `l hlw lk lry dpekn M x-y jk f (( h M i ,x )) = h M x i : L 1 / ∈ R-y ziciin raep dpnny HP ≤ L 1 divwecx d`xp ik gipdl ozip .(zxver dpi` dnvr `id mb zxver dpi` M m` m`e) dxvr M m` zlawne x lr M z` dvixn .(0-n lecb jxe`n `ed ziweg dpekn ly ceciw ik) cinz |h M x i| > m` :dzetwz z` d`xp .dheyt divlitnew zlert zrvan `id oky aeyigl zpzipe ,dzxcbdn d`ln divwecxd h M x i ∈ L 1 okle | L ( M x ) | = ∞ > |h M x i| xnelk ,hlw lk zlawn M x okle x lr zxver M f` ( h M i ,x ) ∈ HP | L ( M x ) | = okle hlw s` lr zxver dpi` M x mb okle x lr zxver dppi` M f` ( h M i ,x ) / ∈ HP m` ipy cvn .yxcpk .yxcpk h M x i / ∈ L 1 ok lre < |h M x i| 2 sirq . L 2 ∈ R w z` zazekd M w dpeknd w ∈ Σ * lkl lynl jk) L ( M ) = ∅ zeniiwnd M zepekn seqpi` zeniiw ik al miyp M zniiw irah n lkly o`kn .(dl`ky zepekn seqpi` yie L ( M w ) = ∅ zniiwnd dpekn `id dgec f`e hxqd lr zepekn ly iteq xtqn wx yiy jkn raep did efk zniiw dziid `l m` ixdy) |h M i| ≥ n sqepae L ( M ) = ∅-y jk df z`e ,iaeig irah xtqn bviin w m`d wecal wx yi w ∈ L 2 m` rixkdl zpn lr ok lr .( L ( M ) = ∅ zeniiwnd .rval ozipy oaenk 3 sirq . L 3 / ∈ RE dztyy dpekn ly ilnipind lcebd `ed k xy`k f (( h M i ,x )) = ( h M x i ,k ) :z`f dgikeny HP ≤ L 3 divwecx d`xp ,dxvr m`e x lr M z` dvixn :`ad ote`a zlretd dpekn M x-e ( k = min {|h M i|| L ( M ) = ∅} :zilnxet) dwix .oldl x`ezzy dpekn `id M L xy`k , M x ly w hlwd lr M L z` dvixn 1 zety ly iteq xtqn mb okle , k xzeid lkl lcebn ceciw zelra zepekn ly iteq xtqn wx yi ,irah k-y oeeikn zlra dxear dpekn s` oi`y L dty zniiwy o`kn . k-l deey e` ohw lcebn ceciw zelra zepekn odl yiy RE-a .ef L xear dpekn M L `dz ; k lcebn ceciw M x-n wlgk M ceciw :dheyt divlitnew zlert wx zrvan `id oky aeyigl zpzip `id .d`ln xexiaa divwecxd . ( h M i ,x ) divwecxd hlwa ielz epi` it lr) recie L ( M x ) = ∅ okle hlw s` lr zxver dpi` M x f` ( h M i ,x ) ∈ HP m` :divwecxd zetwz z` gikep M x f` ( h M i ,x ) / ∈ HP z`f znerl m` . ( h M x i ,k ) ∈ L 3 okle k lcebn M x-l dlewy M zniiw ok`y ( k zxcbd zniiw `l ok lre , L ( M x ) = L xnelk , M L dpeknd enk weica zbdpzn okle x lr M ziivleniq z` zniiqn .yxcpk , ( h M x i ,k ) / ∈ L 3-y jk , k lcebn dlewy dpekn M x-l 2 dl`y 1 sirq . L S / ∈ RE ik gikep jkae HP ≤ L S divwecx d`xp .idylk ziteq L ∈ S dty dlikn `id zil`ieeixh `l dpekz S-e xg`n ozip) w ∈ L m`d zwcea ziy`x w hlw lr xy` dpekn `id M x-y jk f (( h M i ,x )) = h M x i :jk xcbez divwecxd .dxvr M m` zlawne x lr M z` dvixn w / ∈ L m`e (drixk hxtae ziteq...
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Moed_A_Spring_2010_Solution - (236343 zeiaeyigd zxez...

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