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CDHYJML OQT{<B{\[{\v]B^<_{` a bcd`$$$$"$"?OB]&"$H@*ߝ"$~lMPhE4{ę`"$˰ HI7=#H}! 0AA`33333___@ s ? ʚ;ʚ;g4cdcd8 0ppp@  <4dddd l 0T<4BdBd l 0T4___PPT10D[SOaldx}0<@?6TTx 0xDArialdx}0<@?6TTx 0x T___PPT96.D22&?  %O  =37VOn the Relationship Between Concurrent Separation Logic and Assume-Guarantee ReasoningWW W Xinyu Feng Yale University   Motivation  (Concurrency verification is challenging!)) ) Two kinds of modularity  Control modularity a.k.a. thread-modularity each thread is verified independently of others Data modularity a.k.a. information hiding each thread only cares about data it usesLIDID   Existing work  Assume-Guarantee (A-G) reasoning [Misra&Chandy 81, Jones 83] thread modularity general: no restriction over synchronization pattern spec of A&G requires global data invariants Concurrent Separation Logic (CSL) [O Hearn, Brookes 2004] thread modularity data modularity: local reasoning restrictive synchronization pattern shared resources can be accessed only inside critical regions${ZIJZ-LZ;Z5JZ%LZ>LZ ?33I-33"35%  3>3&   Our Contributions  SAGL: combining CSL and A-G reasoning extend A-G logic with local reasoning better data modularity (than A-G reasoning) thread modularity no restriction over synchronization pattern A formal study of the relationship between CSL and A-G reasoningX&&mJB&&mB   Outline of this talk  Background Concurrent Separation Logic Assume-Guarantee reasoning SAGL: combination of CSL & A-G reasoning Interpretation of CSL in SAGL6 8H 8H  &Concurrent Separation Logic  Assertions capture ownerships of resources Cannot access resource without ownership Shared resources are protected by critical regions (CRs) Transfer of ownership at boundary of CRZ&  + CSL assertions   (Locks and Critical Regions  !Lock-based critical regions (CR):"" " rGConcurrent Separation Logic  ?. Example: List   ,Concurrent Separation Logic  Thread modularity Data modularity by local reasoning do not need knowledge of other threads data Ownership transfer is bound with CRs requires built-in critical regions hard to support ad-hoc synchronizationsZZ#Z.Z&L3ZKZ#.&3K  3&  L Outline of this talk  Background Concurrent Separation Logic Assume-Guarantee reasoning SAGL: combination of CSL & A-G reasoning Interpretation of CSL in SAGL6 8H 8H  Assume-Guarantee Reasoning  lThread T and its environment Environment: the collection of all other threads except T A: assumption about environment s transition G: guarantee to the environment a: precondition;_;33,33 33  !Assume-Guarantee Reasoning  To certify each thread:  $ A-G reasoning   ' A-G reasoning  Thread modular separate verification of threads Not require CRs but need to know smallest granularity of transitions A, G: global invariants about all resources hard to define lack data modularity Need to check A and G at every stepZ#ZZ6Z,LZ$ZZ$LZ#6,3$3%3&>   T>Example: data modularity broken (    Outline of this talk  Background Concurrent Separation Logic Assume-Guarantee reasoning SAGL: combination of CSL & A-G reasoning Interpretation of CSL in SAGL6 8H 8H  -SAGL: Separated A-G Logic  hPartition of each thread s resource shared and private shared can be accessed at any time governed by A and G exclusive access to private resources follows local reasoning in CSL not specified in A and G better memory modularity Dynamic change of partition may occur at any point, not tied with CR boundaries$6&Q4$6&Q  4 5 :)SAGL: Specification of Threads  A, G: assumption and guarantee a: precondition about shared resources p: precondition about private resources Spec for Ti: ((ai, pi), Ai, Gi)h333333 3333E3333M33E3333M33E3333M33E3333M33E33@      {J<SAGL  Access Private Resource(  V?!Example: regained data modularity""( "  |I:SAGL  Access Shared Resource(   }HSAGL - Redistribution   9(SAGL  Thread modular separate verification of threads Not require CRs but need to know smallest granularity of transitions A, G: only specifies shared resources better modularity Need to check A and G at every step but the check is trivial if only private resource is used Z#ZZ6Z&ZZ%Z;Z#6&%  ;&>    Outline of this talk  Background Concurrent Separation Logic Assume-Guarantee reasoning SAGL: combination of CSL & A-G reasoning Interpretation of CSL in SAGL6 8H 8H  KConcurrent Separation Logic  3"Implicit Invariants in CSL  >Shared resources are well-formed outside of critical regions. "?=33 ? =,Specialization of SAGL  At each program point in SAGL: (p1 * & * pn) * (a1 & an)& 3333333333333333333333&1   >-Interpretation of CSL in SAGL  zIf G%CSL {p} C {q}, then (AG,GG)%SAGL {(aG,p)} C {(aG,q)}(G33E33M33 E33A3333333333E33M33E3333 E3333E33@,     M: Example: List     Conclusion  SAGL combination of local reasoning with A-G reasoning improved modularity than A-G reasoning more flexible than CSL Embedding of CSL in SAGL formalization of invariants: shared resources are well-formed outside of CRs a new way to prove the soundness of CSLhqvqu33&  ) O< Thank you!   L Example: GCD   M Example: GCD   PProblem with CSL  bNeeds to rewrite code using CRs then needs to prove semantic preservation CSL is a program logic what if the language does not support CRs? only use  cas to do synchronizationx *+% *+  A  Z  i      NExample: malloc&     OExample: malloc&     XASoundness of SAGL  {Safety: progress & preservation Partial correctness: assertions assigned to program points hold when we reach these points&6F6F | /IP@:   0` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> ldp(    6l˜ " `}  X Click to edit Master title style!! (  0XŘ " `  RClick to edit Master text styles Second level Third level Fourth level Fifth level!    S   0$̘ "^ `  V*b    0Xј "^   X*b    0՘ "^ `  X*b  H  0޽h ? 33g80___PPT10.N Default Design 0 0.(    0}d P   d X*b    0Ld    d Z*b  d  c $ ?  d  0d  0 d RClick to edit Master text styles Second level Third level Fourth level Fifth level!    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B"` )   ~v0___PPT106___PPT9  G(l2)Zemd  T < /# x R r t2 0 6"`t2 1 6"`<l 2B <G0*HI7)1 3 B"`_ 0p4___PPT10 B___PPT9$ G(l1) * p2 eld`h  >T  0 4# 8 BH t2 5 6"`  t2 6 6"`@0rb 7 BG}HuIv %UrR 8 BG>HiI>h  Ht2 9 6"` prR : BZGHIpHx ; c $t  `}   F P  < P  = <A H??X  HZ p 0  >3 '  t2 ? 6"`p P0 t2 @ 6"`p 0 l A <GHIP  NZ  0 B3  't2 C 6"` 0t2 D 6"`0rb E BGH4ILMrb F BZGcH IW Z   p  G3  tt2 H 6"`@ t2 I 6"`  p t2 J 6"`   l K <G0*HI7 \ l LB <G0*HI7 \ xZ   M3 @Kgt2 N 6"`@` t2 O 6"`0` t2 P 6"`@  t2 Q 6"`p0pZ R s *@ 0 Z S s *D L Z T s *TLh U B/"`LCM*~v0___PPT106___PPT9 Zp1(`h   V B6"`~v0___PPT106___PPT9  G(l2)Zemd  y W Bd"`{~v0___PPT106___PPT9 kp2 8`h   X B="`<P74___PPT10 B___PPT9$  G(l1)held  R Y@ BZGeH9Iem|  ,$@ 0R Z <G H1I mx  ,$@ 0R [ <G;H I;md ,$@  0 \ B@E"`I 10 $ 0v0___PPT106___PPT9 l1.acq()X eld   ] BC"` $ 0v0___PPT106___PPT9 l1.rel()X eld   ^ BQ"` P T $ 0v0___PPT106___PPT9 f& 4e  z ; 0 _  % ,$D 0 `2  `A I? ?; c0 I Z p 0  a3 K pt2 b 6"`p P0 t2 c 6"`p 0 l d <GHIP  NZ  0 e3  s t2 f 6"` 0t2 g 6"`0rb h BGH4ILMrb i BZGcH IW Z   p  j3  \ t2 k 6"`@ t2 l 6"`  p t2 m 6"`   l n <G0*HI7 \ l oB <G0*HI7 \ xZ   p3  { t2 q 6"`@` t2 r 6"`0` t2 s 6"`@  t2 t 6"`p0pZ u s *@ 0 Z v s *D L Z w s *TLh x B_"` ~v0___PPT106___PPT9 Zp1(`h   y Bd"`x ~v0___PPT106___PPT9  G(l2)Zemd   z Bf"`  4___PPT10 B___PPT9$ G(l1) * p2  eld`h  H  0޽h ??`?@ACDEDCFHJKHILNORPOSQOT Y Z [ bcd fghgfikmnkloqrusrvtrw"#$&'*('+)',56795801269:    !"# 33___PPT10.m+-D' d= @B D' = @BA?%,( < +O%,( < +D' =%(DJ' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-6B'blinds(horizontal)*<3<*\DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*Y%(D' =-6B'blinds(horizontal)*<3<*YDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*_%(D' =-6B'blinds(horizontal)*<3<*_D' =%(DJ' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*^%(D' =-6B'blinds(horizontal)*<3<*^DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*Z%(D' =-6B'blinds(horizontal)*<3<*ZDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DJ' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*]%(D' =-6B'blinds(horizontal)*<3<*]DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*[%(D' =-6B'blinds(horizontal)*<3<*[DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*++0+\0 ++0+]0 ++0+^0 +?1   0ӊL0 )>|(  |r | S  `}   X  | 0; X  | 0 X  | 0;`X | 0`X | 0; 0 X | 0 0 L | c $0 RB | s *D>LRB | s *D>`^b | 6G0*H!I@[RB | s *Dk` k | 0  Ix 2`  RB | s *D | 0 ; @" N&  2`  RB | s *Dp`` | 0  &G = { l List(x) } 233333 f3&     | 0 p |getNode(): l.acq(); & l.rel(); ? 2bf33c bf33 bf33c bf33N        * #| 0@1: ,$ 0  -{emp} B 2f33cf33&   K $| 0  1z ,$ 0 -{emp * List(x)};B 2f33f3f33@     N &| 0X 6@ ,$ 0 -{Node(y) * List(x)}B 2 f33f3f33@      '| 0(6,$ 0 r -{Node(y)} 2 f33&   .l 0   =| 0  ,$D 0f *| 6  f +| 6 z f ,| 6 p f -| 6 pz f .| 6 0 f /| 6 0 p `B 1| 0D>L  `B 2| 0D>  lb 3| <G0*H!I@pB  `B 4| 0D G G  5| 6  P  Ix 2`  `B 6| 0DC C  7| 6$   N&  2`  `B 8| 0DA A `B 9| 0D 9 9  :| 6(0  Iy 2`   <| 6ZAoX ,$D  0H | 0޽h ??`| || ||,|3| 33gJ TIMING.|27.9|11.3|7.9|8.1|21.3___PPT10}.Pf+>>Dq' = @B D,' = @BA?%,( < +O%,( < +D3 ' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<|%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*<|D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*<|D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*<|D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*<|D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*=|%(D' =-o6Bdissolve*<3<*=|D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*#|%(D' =-6B'blinds(horizontal)*<3<*#|D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*$|%(D' =-6B'blinds(horizontal)*<3<*$|D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*&|%(D' =-6B'blinds(horizontal)*<3<*&|D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*'|%(D' =-6B'blinds(horizontal)*<3<*'|++0+#|0 ++0+$|0 ++0+&|0 ++0+'|0 +   0HL0 x@((  (r ( S lS `}   r ( S 0T `   ( 0hU@`  G % {p} C {q}: 233 g33  H ( 0޽h ? 33g___PPT10i.JP0 +D=' d= @B +   0L0 (  X  0x  c $@u `}   x  c $v    H  0޽h ? 33___PPT10i.N+D=' d= @B +   0&L0 $(  r  S @} `}   r  S $    H  0޽h ? 33___PPT10i.N]+D=' d= @B +H    0)L0 G ?   (  r  S  `}   r  S p0 0   V  0` @  8" S, S'. a S A S S' a S' 233 f3333f333333    0@  p `Non-Interference of threads:  f  0Դ  DG1 & Gi-1 Gi+1 & Gn Aix# 2f33n3333f33n3333n33f33n3333f33n33n33&   T   0ñ  DGi A1 & Ai-1 Ai+1 & Anr# 2f3n3ff33n3333f33n3333n33f33n3333f33n33 !    0б  `  cpreservation of precondition:      0Ա G S Nextc(S)h 2 33333333&      0۱   h"transitions satisfy the guarantee:## # H  0޽h ? 33___PPT10i.O@Y+D=' d= @B +Xq   0UL0 ^>V>_`:(  x  c $ `}    F PD   d z2  <Ԕ"`PD Z p 0  3 0` t2  6"`p P0 t2  6"`p 0 l  <GHIP  NZ  0  3 00t2   6"` 0t2   6"`0rb   BGH4ILMrb   BZGcH IW Z   p  3 xdt2  6"`@ t2  6"`  p t2  6"`   l  <G0*HI7 \ l B <G0*HI7 \ xZ   3 \`0|t2  6"`@` t2  6"`0` t2  6"`@  t2  6"`p0pZ  s *@ 0 Z  s *D L Z  s *TL  B"`0P7~v0___PPT106___PPT9  a1a2J`h   z P`   ` P,$@ 0z2  <Ԕ"`P` Z p 0  3  @pt2   6"`p P0 t2 ! 6"`p 0 l " <GHIP  NZ  0 #3  d@ t2 $ 6"` 0t2 % 6"`0rb & BGH4ILMrb ' BZGcH IW Z   p  (3 p t t2 ) 6"`@ t2 * 6"`  p t2 + 6"`   l , <G0*HI7 \ l -B <G0*HI7 \  N p   . p  t2 / 6"`$ L t2 0 6"`$  t2 1 6"`L  t2 2 6"`pp  Z 3 s *  Z 4 s * 9  5 B"``` YG ~v0___PPT106___PPT9  a1a2J`h   z  $ 6  $,$@  0z2 7 <Ԕ"` $Z p 0  83 t2 9 6"`p P0 t2 : 6"`p 0 l ; <GHIP  xZ   <3 <  t2 = 6"`@` t2 > 6"`0` t2 ? 6"`@  t2 @ 6"`p0pZ A s *@ 0 Z B s *D L Z C s *TL D B| "` w ~v0___PPT106___PPT9  a1a2Z`h  T  X  E#  X t2 F 6"` p t2 G 6"`T p rb H BGH4I i rb I BZGcH IW [ i\ t2 J 6"` X t2 K 6"`  t2 L 6"` ` l M <G0*HI7kK  l NB <G0*HI7k ) T O c $[   P B"`$  0v0___PPT106___PPT9 xG2Fd   Q B\"`  $ 0v0___PPT106___PPT9 vA1Ddl   R B"`0 1 $ 0v0___PPT106___PPT9 zA2H   S B""`p$ 0v0___PPT106___PPT9 vG1Ddl  R T HGݮHIݮԔp  ,$@  0R U NGFH IFԔ]` ,$@ 0 V H5G5H"`p-0 ,$@ 0 W HG5H"`p`0 ,$@ 0 X HOSG5H"`Y a ,$D 0 Y B*"`70 $ 0v0___PPT106___PPT9 za2H   Z HOSG5H"` P ,$D  0 [ B/"` `Y $  0v0___PPT106___PPT9 za2H   \ HS6G5H"`pQ ,$D 0 ] HS6G5H"` [b ,$@ 0 ^ B7"` $ 0v0___PPT106___PPT9 za1H   _ BD<"` `Y $ 0v0___PPT106___PPT9 za1H  H  0޽h ?@       !" $%& %$' )+, )*-1032049:;FGHGFIJLMJKN=>A?>B@>CLGODT5U gg32J TIMING.|17.7|34.5|6.7|16.4|2.502___PPT102.7pE'+N1D$0' = @B D/' = @BA?%,( < +O%,( < +D' =%(DJ' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*U%(D' =-6B'blinds(horizontal)*<3<*UDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D' =-6B'blinds(horizontal)*<3<*SDS' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*V%(D' =-6B'blinds(horizontal)*<3<*VDT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*R%(D' =-6B'blinds(horizontal)*<3<*RD ' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*Y%(D' =-6B'blinds(horizontal)*<3<*YD' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-6B'blinds(horizontal)*<3<*XD' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*Z%(D' =-6B'blinds(horizontal)*<3<*ZD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*[%(D' =-6B'blinds(horizontal)*<3<*[DM ' =%(D ' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =-6B'blinds(horizontal)*<3<*TDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-6B'blinds(horizontal)*<3<*6DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-6B'blinds(horizontal)*<3<*PDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*W%(D' =-6B'blinds(horizontal)*<3<*WDT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*Q%(D' =-6B'blinds(horizontal)*<3<*QD ' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*^%(D' =-6B'blinds(horizontal)*<3<*^D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-6B'blinds(horizontal)*<3<*\D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*]%(D' =-6B'blinds(horizontal)*<3<*]DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*_%(D' =-6B'blinds(horizontal)*<3<*_++0+P0 ++0+Q0 ++0+R0 ++0+S0 ++0+Y0 ++0+[0 ++0+^0 ++0+_0 +   0L0 (  r  S  _ `}   r  S {4 `    0_"`0  "(A,G)% {a} C {a }2 2 c  H  0޽h ? 33___PPT10i.O7z+D=' = @B +    0'WL0 nf  (  x  c $p `}     0hf  j & [100] := m; &  2b  8 @ pp   @ p p TB  c $D@ p@ TB  c $Dp pp   0p   h& [101] := n; &  2b  X   00 LB   c $Da a    0p   O100 2b    0  O101 2b  3  6   'G1: [101] = [101]' A1: [100] = [100]' v( 2d3l3d3l3d3` ( 7  6@  'G2: [100] = [100]' A2: [101] = [101]' z( 2d3l3d3l3d3d3 ( H  0޽h ? 33g& TIMING |23.6___PPT10i.P +D=' = @B +   0 L0 0(  X  0:  x  c $3 `}  3 x  c $3   3 H  0޽h ? 33___PPT10i.N+D=' = @B +   0L0 P,$(  ,r , S @ 3 `}  3 r , S  3 ` 3 H , 0޽h ? 33g___PPT10i.PP'+D=' = @B +   0BfL0 @h0(  hx h c $d3 `}  3 x h c $<3 ` 3 H h 0޽h ? 33g___PPT10i.QP=G6+D=' = @B +*   0JL0 U$M$;;!(  Cz ,mT   g  ,$D 0 2  `A O? ?,mT  O 3Z p 0  3  z t2  6"`p P0 t2  6"`p 0 l  <GHIP  NZ  0 3  B t2   6"` 0t2   6"`0rb   BGH4ILMrb   BZGcH IW xZ    3 6o R t2  6"`@` t2  6"`0` t2  6"`@  t2  6"`p0pZ  s *@ 0 Z  s *D L Z  s *TLp  B$73"`. ~v0___PPT106___PPT9 bp10dl    B8:3"`*  ~v0___PPT106___PPT9  a1a2^h    BH;3"`  ~v0___PPT106___PPT9 up2'Bdld  T DpD  #  "g t2  6"`$  t2  6"`D D t2  6"`Dp l B <G0*HI7 1 T  c $  F ,kT   $N  2  `A P? ?,mT  P 3Z p 0   3  w t2 ! 6"`p P0 t2 " 6"`p 0 l # <GHIP  NZ  0 $3  B t2 % 6"` 0t2 & 6"`0rb ' BGH4ILMrb ( BZGcH IW Z   p  )3  "d t2 * 6"`@ t2 + 6"`  p t2 , 6"`   l - <G0*HI7 \ l .B <G0*HI7 \ xZ   /3 6l R t2 0 6"`@` t2 1 6"`0` t2 2 6"`@  t2 3 6"`p0pZ 4 s *@ 0 Z 5 s *D L Z 6 s *TLp 7 B|O3"`. ~v0___PPT106___PPT9 bp10dl   8 BT3"`'  ~v0___PPT106___PPT9  a1a2^h  p 9 BU3"` k R~v0___PPT106___PPT9 bp20dl  x : c $`3 `}  3 b ; <G0*HI΁Ԕ@t ,$@ 0H  0޽h ? ;!"#%&'&%(*,-*+.014215 316         33___PPT10.m ?3+D' 3= @B DL' = @BA?%,( < +O%,( < +D' =%(D+' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*;%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*+!   0$]L0    (  x  c $d3 `}  3   0r3f  j & [100] := m; &  2b  F @ pp    @ p p TB  c $D@ p@ TB  c $Dp pp   0w3p   h& [101] := n; &  2b  X  00 LB   c $Da a    0s3   O100 2b     03 P  O101 2b    03P  &-{(emp , 100 _) }J 2`d3`&     0Ј3P  $-{(emp , 101 _)}J 2`d3`&    88 Q Qt  6"`[t  6"`Qtz  @   P ,$D 0r  03  G1: emp A1: emp  2`h`d3`h`d3`@     p  03 @ G2: emp A2: emp  2`h`d3`h`d3`@     H  0޽h ? 33g. TIMING|17.8|5.3w___PPT10W.P +LnhD+' 3= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D1' =4@BB BB%(D' =-o6Bdissolve*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+K   0FL0 ..AA>,(  z ,jT   P /{ ,$D 0 2  `A M? ?,mT  M 3t2  6"` Q t2  6"` q NZ  0 3  J t2  6"` 0t2  6"`0rb   BGH4ILMrb   BZGcH IW Z   p   3  * t2   6"`@ t2   6"`  p t2  6"`   l  <G0*HI7 \ l B <G0*HI7 \ xZ   3 >e Z t2  6"`@` t2  6"`0` t2  6"`@  t2  6"`p0pZ  s *@ 0 Z  s *D L Z  s *TLh  Bt3"`Z A~v0___PPT106___PPT9 Zp1(`h    Bp3"`!  ~v0___PPT106___PPT9  a1a2'd  h  B 3"` j Q~v0___PPT106___PPT9 Zp2(`h  F ,kT     2  `A N? ?,mT  N 3Z p 0  3  | t2  6"`p P0 t2   6"`p 0 l ! <GHIP  NZ  0 "3  B t2 # 6"` 0t2 $ 6"`0rb % BGH4ILMrb & BZGcH IW Z   p  '3  "i t2 ( 6"`@ t2 ) 6"`  p t2 * 6"`   l + <G0*HI7 \ l ,B <G0*HI7 \ xZ   -3 6q R t2 . 6"`@` t2 / 6"`0` t2 0 6"`@  t2 1 6"`p0pZ 2 s *@ 0 Z 3 s *D L Z 4 s *TLh 5 B3"`. ~v0___PPT106___PPT9 Zp1(`h   6 B3"`,  ~v0___PPT106___PPT9  a1a2rl  h 7 B3"` k R~v0___PPT106___PPT9 Zp2(`h  x 8 c $3 `}  3  9 B3"`I ! 0 $ 0v0___PPT106___PPT9 zA1H   : B`3"` $ 0v0___PPT106___PPT9 vG2Ddl   ; H3G5H"` F w,$@ 0b < <G0*HIԔ h ,$@ 0 = BG5H"` 1 ,$D 0 > B43"`@ ' $ 0v0___PPT106___PPT9 za1H   ? B3"`I 0 $  0v0___PPT106___PPT9 za1H   @ BG5H"` ,$@  0B A 0L@"` `,$D  0 XA-G reasoning: a special case where p1 and p2 are emp.Y 2#b#f33f3n3bbbf3n3bbbf3b&T   H  0޽h ?< !#$%$#&(*+(),./20/3 1/4        33c[___PPT10;.mQGJ+D' 3= @B D' = @BA?%,( < +O%,( < +D' =%(D+' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =-6B'blinds(horizontal)*<3<*:DS' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =-6B'blinds(horizontal)*<3<*;DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*9%(D' =-6B'blinds(horizontal)*<3<*9D ' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*>%(D' =-6B'blinds(horizontal)*<3<*>D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*=%(D' =-6B'blinds(horizontal)*<3<*=D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*@%(D' =-6B'blinds(horizontal)*<3<*@DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D' =-6B'blinds(horizontal)*<3<*?DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*A%(++0+90 ++0+:0 ++0+>0 ++0+?0 ++0+A0 +Z   0L0 d=\=[u 9(   Hz ,mT    } ,$D  0  2  `A J? ?,mT  J @NZ  0  3 (  t2   6"` 0t2   6"`0rb   BGH4ILMrb   BZGcH IW T dp  #  6 zt2   6"`<$t2   6"`dpl   <G0*HI7Q9xZ    3 >d Z t2   6"`@` t2   6"`0` t2   6"`@  t2   6"`p0pZ   s *@ 0 Z   s *D L Z   s *TLh   B/@"`f M~v0___PPT106___PPT9 Zp1(`h     B6@"` 1j ~v0___PPT106___PPT9 a1a2r  ~   B\1@"`[  ~v0___PPT106___PPT9 pp28`h  T   # $ Z t2   6"`t2   6"`l   <GHIXXt2   6"``T   c $`XXz ,mT     ,$@ 0  2  `A K? ?,mT  K @Z p 0   3 I b t2 !  6"`p P0 t2 "  6"`p 0 l #  <GHIP  NZ  0 $ 3 E r   t2 %  6"` 0t2 &  6"`0rb '  BGH4ILMrb (  BZGcH IW t2 )  6"`Y  b T  l  * #   6t2 +  6"` d l t2 ,  6"` @ l -  <G0*HI7 + y xZ   . 3 fi t2 /  6"`@` t2 0  6"`0` t2 1  6"`@  t2 2  6"`p0pZ 3  s *@ 0 Z 4  s *D L Z 5  s *TLh 6  BPK@"`^ E~v0___PPT106___PPT9 Zp1(`h   7  BO@"` " ~v0___PPT106___PPT9  a1a2r  | 8  BQ@"`y  i~v0___PPT106___PPT9 np28`h  TF ,jT  9    : 2  `A L? ?,mT  L @t2 ;  6"` L t2 <  6"` l NZ  0 = 3  J t2 >  6"` 0t2 ?  6"`0rb @  BGH4ILMrb A  BZGcH IW Z   p  B 3  *| t2 C  6"`@ t2 D  6"`  p t2 E  6"`   l F  <G0*HI7 \ l G B <G0*HI7 \ xZ   H 3 >` Z t2 I  6"`@` t2 J  6"`0` t2 K  6"`@  t2 L  6"`p0pZ M  s *@ 0 Z N  s *D L Z O  s *TLh P  Ba@"`Z A~v0___PPT106___PPT9 Zp1(`h   Q  B`f@"`  ~v0___PPT106___PPT9  a1a2'd  h R  Bxi@"` j Q~v0___PPT106___PPT9 Zp2(`h  x S  c $Hm@ `}  @  T  BDt@"`  $ 0v0___PPT106___PPT9 zA1H   U  Bz@"`  $ 0v0___PPT106___PPT9 vG2Ddl  R V @ NZGx;H*Ix;Ԕ  ,$@ 0 W  H3G5H"` "] ,$@ 0 X  B@"`  $  0v0___PPT106___PPT9 zA1H   Y  BЂ@"` Tm $ 0v0___PPT106___PPT9 vG2Ddl  R Z  HGH3IԔ  ,$@ 0 [  HiG5H"` T% ,$@  0H   0޽h ?V Z > ? @ ? > A C E F C D G I J M K J N  L J O  ! " #  % & '  & % (  + , - / 0 3 1 0 4 2 0 5                          33D<___PPT10.m@{+P 3D' @= @B D' = @BA?%,( < +O%,( < +D' =%(DJ' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*V %(D' =-6B'blinds(horizontal)*<3<*V DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*U %(D' =-6B'blinds(horizontal)*<3<*U DS' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*W %(D' =-6B'blinds(horizontal)*<3<*W DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*T %(D' =-6B'blinds(horizontal)*<3<*T D' =%(DJ' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*Z %(D' =-6B'blinds(horizontal)*<3<*Z DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*Y %(D' =-6B'blinds(horizontal)*<3<*Y DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* DS' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*[ %(D' =-6B'blinds(horizontal)*<3<*[ DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*X %(D' =-6B'blinds(horizontal)*<3<*X ++0+T 0 ++0+U 0 ++0+X 0 ++0+Y 0 +   0L0  `p(  `x ` c $ĭ@ `}  @ x ` c $@ ` @ 8 ` 0@"`p  4(A,G)% {(a,p)} C {(a ,p )}2 2c@      H ` 0޽h ? 33___PPT10i.O7z+D=' @= @B +   0 L0 @(  X  0 n x  c $@ `}  @ x  c $H@   @ H  0޽h ? 33___PPT10i.N+D=' @= @B +ok   0UL0 QQzz'M(  z ,QT     ,$D  0 2  `A R? ?,mT  R @Z p 0  3 B = t2  6"`p P0 t2  6"`p 0 l  <GHIP  T < #  - M t2   6"`t2   6"`<l  B <G0*HI7)1xZ    3 a } t2   6"`@` t2  6"`0` t2  6"`@  t2  6"`p0pZ  s *@ 0 Z  s *D L Z  s *TLh  B@"`e L~v0___PPT106___PPT9 Zp1(`h    B@"` ? & ~v0___PPT106___PPT9  G(l2)Zemd  |  B@"`  ~v0___PPT106___PPT9 np28`h  >T  0 # N Y^ t2  6"`  t2  6"`@0rb  BG}HuIv %UrR  BG>HiI>h  Ht2  6"` prR  BZGHIpH  BI"` Q1 8~v0___PPT106___PPT9  G(l1)Zemd  z ,m0   P x ,$D 0  2  `A S? ?,mT  S IZ p 0  !3 P p t2 " 6"`p P0 t2 # 6"`p 0 l $ <GHIP  xZ   %3 h t2 & 6"`@` t2 ' 6"`0` t2 ( 6"`@  t2 ) 6"`p0pZ * s *@ 0 Z + s *D L Z , s *TLh - B I"`{ b~v0___PPT106___PPT9 Zp1(`h   . BTI"` )   ~v0___PPT106___PPT9  G(l2)Zemd  T < /# x R r t2 0 6"`t2 1 6"`<l 2B <G0*HI7)1 3 BI"`_ 0p4___PPT10 B___PPT9$ G(l1) * p2 eld`h  >T  0 4# 8 BH t2 5 6"`  t2 6 6"`@0rb 7 BG}HuIv %UrR 8 BG>HiI>h  Ht2 9 6"` prR : BZGHIpHx ; c $dI `}  I F P  < P  = <A T??X  TZ p 0  >3 '  t2 ? 6"`p P0 t2 @ 6"`p 0 l A <GHIP  NZ  0 B3  't2 C 6"` 0t2 D 6"`0rb E BGH4ILMrb F BZGcH IW Z   p  G3  tt2 H 6"`@ t2 I 6"`  p t2 J 6"`   l K <G0*HI7 \ l LB <G0*HI7 \ xZ   M3 @Kgt2 N 6"`@` t2 O 6"`0` t2 P 6"`@  t2 Q 6"`p0pZ R s *@ 0 Z S s *D L Z T s *TLh U B`@"`LCM*~v0___PPT106___PPT9 Zp1(`h   V B/I"`~v0___PPT106___PPT9  G(l2)Zemd  y W B0I"`{~v0___PPT106___PPT9 kp2 8`h   X B:I"`<P74___PPT10 B___PPT9$  G(l1)held  R Y@ BZGeH9Iem|  ,$@ 0R Z <G H1I mx  ,$@ 0R [ <G;H I;md ,$@  0 \ B\BI"`I 10 $ 0v0___PPT106___PPT9 l1.acq()X eld   ] BLI"` $ 0v0___PPT106___PPT9 l1.rel()X eld   ^ B$PI"` P T $ 0v0___PPT106___PPT9 f& 4e  z ; 0 _  % ,$D 0 `2  `A U? ?; c0 U IZ p 0  a3 K pt2 b 6"`p P0 t2 c 6"`p 0 l d <GHIP  NZ  0 e3  s t2 f 6"` 0t2 g 6"`0rb h BGH4ILMrb i BZGcH IW Z   p  j3  \ t2 k 6"`@ t2 l 6"`  p t2 m 6"`   l n <G0*HI7 \ l oB <G0*HI7 \ xZ   p3  { t2 q 6"`@` t2 r 6"`0` t2 s 6"`@  t2 t 6"`p0pZ u s *@ 0 Z v s *D L Z w s *TLh x BHJI"` ~v0___PPT106___PPT9 Zp1(`h   y BxbI"`x ~v0___PPT106___PPT9  G(l2)Zemd   z BdI"`  4___PPT10 B___PPT9$ G(l1) * p2  eld`h  H  0޽h ??`?@ACDEDCFHJKHILNORPOSQOT Y Z [ bcd fghgfikmnkloqrusrvtrw"#$&'*('+)',56795801269:    !"# 33___PPT10.m+-D' I= @B D' = @BA?%,( < +O%,( < +D' =%(DJ' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-6B'blinds(horizontal)*<3<*\DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*Y%(D' =-6B'blinds(horizontal)*<3<*YDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*_%(D' =-6B'blinds(horizontal)*<3<*_D' =%(DJ' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*^%(D' =-6B'blinds(horizontal)*<3<*^DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*Z%(D' =-6B'blinds(horizontal)*<3<*ZDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DJ' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*]%(D' =-6B'blinds(horizontal)*<3<*]DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*[%(D' =-6B'blinds(horizontal)*<3<*[DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*++0+\0 ++0+]0 ++0+^0 +c    0 1L0 b Z H (  Hr H S I `}  I r H S HI  I  H 0I)  >Invariants of shared resources: aG Inv(l1) * & * Inv(ln)fH 2 bf3333f33333333g33o33f33g333333333333&@   L H 04I p0 tAt each program point: (p1 * & * pn) * aG S.; 2bf3333f33n33f33333333f3333f33f3333&/     H 0I 0  lfInv(li) (free(li) G(li)) ( free(li) emp)`4 233g33o33f33g3333333333g33o33333333g33o333333333333g33o33333333          H H 0޽h ? 33g___PPT10i.uP0_+D=' I= @B +(%   0??L0 p"Ztr(  tF ,kT  ?t  (J  @t BA Q? ?,mT  Q IZ p 0  At3  | t2 Bt 6"`p P0 t2 Ct 6"`p 0 l Dt <GHIP  NZ  0 Et3  B t2 Ft 6"` 0t2 Gt 6"`0rb Ht BGH4ILMrb It BZGcH IW Z   p  Jt3  "i t2 Kt 6"`@ t2 Lt 6"`  p t2 Mt 6"`   l Nt <G0*HI7 \ l OtB <G0*HI7 \ xZ   Pt3 6q R t2 Qt 6"`@` t2 Rt 6"`0` t2 St 6"`@  t2 Tt 6"`p0pZ Ut s *@ 0 Z Vt s *D L Z Wt s *TLh Xt BI"`. ~v0___PPT106___PPT9 Zp1(`h   Yt BI"`,  ~v0___PPT106___PPT9  a1a2rl  h Zt B`I"` k R~v0___PPT106___PPT9 Zp2(`h  r t S hI `}  I r t S I ` I  "t 0M  ` ,$ 0 L|Specialize ai into aG (p1 * & * pn) * (aG & aG) l' 33333333 3333333333333333333333@      V #t 0 M  ``,$ 0 Specialize A and G to enforce aG at every step AG S S' aG S aG S' GG S S' aG S aG S' /D 3333 3333 3333 333333333333 3333 333333333333 s  $t 0!M"` 0 ,$D 0 V* (p1 * & * pn) * aG(2bbjbb&    H t 0޽h ?BtCtDtFtGtHtGtFtItKtMtNtKtLtOtQtRtUtStRtVtTtRtWt 33g : TIMING|29.8|15.2|11.2 ___PPT10d .P0+y< ?D' M= @B DK' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"t%(D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$t%(D' =-o6Bdissolve*<3<*$tDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#t%(++0+"t0 ++0+#t0 ++0+$t0 +   0L0 |tx (  xr x S 0mM `}  M r x S nM `  M  x 0=M"` P0,$D 0 >Also a new way to prove the soundness of CSL! The soundness of SAGL is proved following the syntactic approach (by proving progress & preservation). Proofs have been formalized in Coq.H 2-bbqb$b  H x 0޽h ? 33g___PPT10.P+ksDO' M= @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x%(+8+0+x0 +A   0}L0 ""2X!(    0LZB < s *D>`fb = 6G0*H!I@[ZB > s *D` k k ? 0\M  Ix 2`  ZB @ s *D A 0M ;@" N&  2`  ZB B s *Dp`` C 0  ` D 0 z ` E 0 p ` F 0 pz ` G 0 0 ` H 0 0 p ZB I s *D>L  ZB J s *D>  fb K 6G0*H!I@pB  ZB L s *D G G  M 0M P  Ix 2`  ZB N s *DC C  O 0M   N&  2`  ZB P s *DA A ZB Q s *D 9 9  R 0M0  Iy 2`  f S 6ZoAX |l 0 U0,$@  0z  <___"`0 D z  <___"`8&z  <___"`0 p z   <___"`0  8 V 6`M P ,$ 0  aG d 23333  8 W 6M P ,$ 0  aG d 23333  8 X 6XM6`0,$ 0  aG d 23333  H  0޽h ??`84:6=EK 33gd TIMINGH|32.3|12.1|8.3|11.1|6.5|7.8|12.1|4.7Q___PPT101.Pf+iD' M= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*V%(D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*W%(D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =%(Dd' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*U%(D' =-6B'blinds(horizontal)*<3<*UD' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*V%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*W%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*X%(+0+0+0 ++0+0 ++0+0 ++0+0 ++0+V0 ++0+V0 ++0+W0 ++0+W0 ++0+X0 ++0+X0 +   0L0 `$(  r  S R `}  R r  S R ` R H  0޽h ? 33___PPT10i.NИ+D=' M= @B +   0 L0 (  r  S R `  R H  0޽h ? 33g___PPT10i.{QP+D=' M= @B +   0gL0 z r   (  x  c $\R `}  R :  0R  dwhile(x <> y){ if (x > y) x = x  y; }83 2bbb , J  0\ R   dwhile(x <> y){ if (y > x) y = y  x; }H3 2bbbb , LB  c $D P LB  c $D P   0%R  0initially: x = a; y = b;^ 2 bbb  p  0-R@ ``  Presult: x = GCD(a, b); y = GCD(a, b); ) 2bb bbb ) x   0;R ,$ 0 ,G1: (y = y') (x y x = x') (GCD(x, y) = GCD(x', y')) A1: (x = x') (y x y = y') (GCD(x, y) = GCD(x', y')) G2 = A1; A2 = G1  2d3l3 d33d33d33d33d3l3 d33d33d33d33d3l3d3l3d3l3d3l3t#    +    ' H  0޽h ? 33g& TIMING |40.1___PPT10v.PpCQ+ocD' M= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* +8+0+ 0 +    0qjL0  (   x   c $dyR `}  R :   0\{R  dwhile(x <> y){ if (x > y) x = x  y; }83 2bbb , J   0R   dwhile(x <> y){ if (y > x) y = y  x; }H3 2bbbb , LB   c $D P LB   c $D P    0lR  0initially: x = a; y = b;^ 2 bbb  p   0ԎR@ ``  Presult: x = GCD(a, b); y = GCD(a, b); ) 2bb bbb )    0HR p: WHard to certify using CSL without rewriting the program and adding auxiliary variables.4X 2Wg3d3 X H   0޽h ? 33g___PPT10i.PpCQ+D=' M= @B +$   0 @,$(  ,r , S ȥR `}  R r , S R ` R H , 0޽h ? 33g80___PPT10.pPE   0xL0 u m $ (  $x $ c $R `}  R   $ 0R  Hx = alloc(1); [x] = 2; if ( odd([x]) ){ //should never reach here }I 2Ib&  & LB $ c $D@ @ @ LB $ c $Dp p @   $ 0hR 0  Iy = alloc(1); [y] = 3; if ( even([y]) ){ //should never reach here }J 2Jb&  &  $ 0R  G1: ??? A1: ??? v 2d3l3d3l3d3`   $ 0R @ G2: ??? A1: ??? v 2d3l3d3l3d3`  *z P@  $ p,$D 0  $ 0RY@ { emp }2 2d33&     $ 0R P7 { emp }2 2d33&     $ 0R Y @ R*  23   z P@  $ ,$D 0 $ 0XRY@ t{ x _ }2 2d33   $ 0R P7 t{ y _ }2 2d33   $ 0$R Y @ R*  23  H $ 0޽h ? 33g`( TIMING |7|8.5(___PPT10.P +"D' M= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =-o6Bdissolve*<3<* $D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =-o6Bdissolve*<3<* $+~   0   0(t (  (x ( c $d `}  d   ( 0,d  Hx = alloc(1); [x] = 2; if ( odd([x]) ){ //should never reach here }I 2Ib&  & LB ( c $D@ @ @ LB ( c $Dp p @   ( 0VR 0  Iy = alloc(1); [y] = 3; if ( even([y]) ){ //should never reach here }J 2Jb&  & 8z  @ (  @,$D 0T ( 0 d  G1: emp A1: emp v 2d3l3 d3l3d3`@     R  ( 0<&d @ G2: emp A1: emp v 2d3l3d3l3d3`@       ( 0/dp -{ (emp, emp) }J 2`d3`@       ( 06d @g -{ (emp, emp }J 2`d3`@       ( 0>d ` "-{ (emp, x _) }J 2`d3`&      ( 0Dd  "-{ (emp, y _) }J 2`d3`&    &F   (  n ( 0"`Tn ( 0"`An ( 0"`  On ( 0"` F H ( 0޽h ? 33gB:___PPT10.P +ħD' M= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D1' =4@BB BB%(D' =-o6Bdissolve*<3<*(D' =1:Bhidden*o3>+B#style.visibility<*(%(D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-6B'blinds(horizontal)*<3<*(+}   0 @$(  r  S |Yd `}  d r  S @Zd ` d H  0޽h ? 33g___PPT10i.emЂf+D=' M= @B +r 0 @(  X  C    d  S ؛d 0  d $r1, & , rn disjoint`&    H  0޽h ? 3380___PPT10.?Qnx] @Ty.{ QqvB2\ѴX994M@<55Ь4-L+ O0YEw W+;}?f5ywf߮'7lu?~FX0o+#n?@İ @V dO4|@悙Ÿ&}cE3b6Neg`ߺ5awwrcZ804 #ຨm-qg߸\ ۷ v,dk(|4\\[Ifr^ wR߾Dmb010!>1$ `bact1z`x`^>=c }0b`Ðb0101c 1c00c־ 9### ##6cF1ĈňÈ`<Ɠ0&bڗO9ttG/YLK_S}IfMezcA:ZB~]+8>}c_)~R߬]t﵎3>:{m[ǩ Ij:!;}:7/=sGc&lV%D'mơ㜀p:YOhR2TzX}qkqe>=աnӧbc"EEoz<䉱qcBF uٮi;Ϻ1CC#B:k>Žmd9#>ƂHN{Z\M&-sfgO ⣔~2:~JRnrm8@<_c5[ĩᜰ G8g9c(<a!.rbr c :ڢ xd eMT9H]O]EmQsgRqPp\uۛj(k%Io' F2LM%&y1ew渕u\Id8N'V;N >qI4:z>NIFw:t [kFGZkԭZi /  _ ?үٟ,46^SNlZ{koutpZǏZtpuqW%fgkl[c۹K[D49#-;&>f vTwP)+#q#([\c$eraر] Ǖ4[NGۨcƋ:||_m=B{opɞɞɞɞT^^ɞIϿKGܝ+ep{5w&t?FJ9_6ͺTa&> n7z&Ξ5(%%Ůٵ] 3=ksf£gv-tZx̮Ι] ٵ9k3:gv-ܸw-|{%﮿\kmᇓwo۾ˮfwwBnߜ57/en}Z݋ߜظq[վo'/ksvv-N̮G_:gv-h=hn-__\o wZ-ʍ۽%[v-tpj7ߟٵ9Z7-7ksp̮݉ٵ] 3=ksf£gv-tjpؓ=IzIk֬iiiy%3?AžɞɞɞɞdrI't~CHkb1<ӗ %ɤ{z|s9.EO.'o\$[2<6ǹ2]]'3"ll.N]]]]DvW7$~"zIQW'J"fI+@&w \5%,D$K"$%%.%DE%c7C^Ŋߎ#WHŖK~1a{$Mǹ<0T3^.ųUoC%D>dLvx@L`39L7&ċĊEO\>$ f;xotE׸' coPČ-AE4\P6Sί)ArQd%) TP.P T'v c9]^K(YqvKjF/ԯїwW7Ui"bav(o%ϿTXlE5~}b*eg$tmKnް\۰*u>Ud%/ =9]g 1K.^AO 9Wsϯz,~Nxԏ()SM+ IU=4F|1 (X~LzQ&hcL,:b?c >vͨUմ.5 _$]@jC+?l;鼀{4a]z>{PFxLJ'B/v{WWHq{aiCY͖+NaVkJ MUA|s݁B~s5>lA ;UѬoM,_WN}e3Ws$h>ݶ\a9lsԮ\GGoLM A흶,$DND{.lnKdg_}cM>ԗ;bO𙁧8#\9bS3BKӥI{zsܟny &Su)pXQѵmFvo/~ךG7ɐTzy^Eǒ&vu.{5DQP۴ *L'eDL5!/jjPto~~2B<%,:Q3v `Fl=ೃ95r8wΈݗ5SQy>u G +Qv10(W\Y2[c׳ !>i"ȽFv$~x"qKwH]8j~~Q<>db@z9}ÉE<ʃN\xoh:ziQ9c*-0}ÌmA|H!x]_֖i\j_jzmg=%~{^F\[1vdUh+r*~%U@/GoVdпO~D5w2q)-k_8tjM?M*·jەn*UQJʵaoV4;WMG?U/3iǚq; t:DB=!@@Am=Gɩuo4Lt 7bQBe_ϴ^l]Lr  $5a\#$zJz={|/C(HitT/ &Kۻ G~AKo+++δdm>̵p5NT+2Ȝ2̩ '@WZ6Q:߅D4|j~)C>}U{c6=ymRi4L ,3KI+଴E \+Oȋ`|cyIPvQ0e2Q9M9_ZY*J7h<+)41$1>Q8#qq⫉o'SĚ+,f(5IB 1'۬tY}; 3k`ŚODBVȅ&ū?.4&&8LmXY 3 ̍ŔO7GX0]Lsq]y4QTSR `pd<ίi&} QAF @ %Q7ෆRwA`EK%` X.#cWLGXnjt=KHϟBOGIIəj%UsÜJ#Mn:;?):]t yHMhs53)N4)"0$eJ#un~zJR&2s|tJ?09JR2!Xl bccБ =x,<3y~2o%<)^ OxH7~DȘ'4Aƒ=\"/Bw\u.WD)+\"`.8f@!p"B KBL ),(Yh>Ӳ'OT!*$MӽR $)ithr5(MdfҖ\u:W;]XH%BNOFIRdyzi ,S_6=Czl:z. 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