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K7 1e/ph^/&0mΎ(>Vhpb芃5XhB2FSB3!H fC2NTD_(5Vz`K5Ĭ#UY L|;b$cOQ&3'$2:('}$'eHJQPb Sn!LxGi,UpktUY5e3#Q`)Jʲ!|R ]cf?T5"O<-MqB_yķ/KZjzE6y^%l` gaU]?Еnڱ0 U3x<t 6?0GUkx7yBȅF匶`G号(fP۠[sc8d ;2K&+_Vxor੓=bq&VPm!ѦBLجʣO+#?ԋJEbPp#dFs52 @&3 9SKgRxԘԩ$6k`:5k;Fv/ T`mpdE u@2 #7 S\D4 , 8J4A+ ]zb?NkJS "v:pPգfiR|E";HJ23!(9c!t\5GVǐ0#ɉ70`O/zQ]cĄ*Ait? }9#{G r*uf:? Q?; qMEI__+(A@%9r$ڊ -)30fUWJ+R2 "@">)e9d>d0:A 0vp p TEXPOINTINIT.USEAMSFONTSTrue,EMBEDFONTSTrue.USEBOLDAMS False(DEFAULTDISPLAYSOURCE\documentclass{slides}\pagestyle{empty} \begin{document} \end{document}  TEX2PS~latex $(base).tex; dvips -D $(res) -E -o $(base).ps $(base).dviH&EXTERNALEDITCOMMANDnotepad %D$GHOSTSCRIPTCOMMANDgswin32c2DEFAULTFONTSIZE108DEFAULTBITMAPpngmono2DEFAULTBLEND False>$DEFAULTTRANSPARENT FalseZ@DEFAULTWORKAROUNDTRANSPARENCYBUG False:"DEFAULTRESOLUTION1200.DEFAULTWIDTH3540DEFAULTHEIGHT200:(DEFAULTMAGNIFICATION2<___PPT10 26___PPT9{LA0?> 2/11/2004 MSR TalkO =&w<Precise Interprocedural Analysis using Random Interpretation==$&2Sumit Gulwani George Necula UC-Berkeley ,3%    dRandom Interpretation*= Random Testing + Abstract Interpretation Almost as simple as random testing but better soundness guarantees. Almost as sound as abstract interpretation but more precise, efficient, and simple.Z,          fExample OutlineFramework for intraprocedural random interpretation Advantages Investigate all analyses using one framework Design and proof of new analyses will be simpler A generic algorithm for interprocedural analysisV4 ^24 ^2  , OutlineFramework for intraprocedural random interpretation Affine join function Eval function Example A generic algorithm for interprocedural analysis@4+24+2>,+ eRandom Interpretation frameworkLGoal: Detect equivalences of expressions. Generic Algorithm: Choose random values for input variables. Execute assignments. Using Eval function to evaluate expressions. Execute both branches of conditionals and combine the program states at join points. Using Affine Join function. Compare values of expressions to decide equality. =0Z2?0Z2-0Z2U0Z20Z220Z2Z&? U3Affine Join functionUsed for combining program states at join points. fw: State State ! State Let r = fw(r1,r2). Then, r(y) =def wr1(y) + (1-w)r2(y) M>2   -Properties of Affine Joino Affine join preserves common linear relationships e.g. a+b=5. It does not introduce false relationships w.h.p.8on ,8. Eval function Eval: Expression State ! Value Used for executing expressions Defined in terms of Poly: Expression ! Polynomial Poly is abstraction specific Eval(e,r) = Evaluation of Poly(e) using r and random choices for non-program variables Poly must satisfy: Correctness: Poly(e1) = Poly(e2) iff e1 = e2 Linearity: Poly(e) is linear in program variables. n!nmcK+ -        +Zm#Example of Poly functionLinear Arithmetic (POPL 2003) Expression e := y | e1 e2 | ce Poly(e) = e Uninterpreted Functions (POPL 2004) Expression e := y | F(e) Poly(y) = y Poly(F(e)) = a Poly(e) + b2$E%  = , y   OutlineFramework for intraprocedural random interpretation Affine join function Eval function Example A generic algorithm for interprocedural analysis Random summary (Idea #1) Issue of freshness (Idea #2) Error probability and complexity Experimentsd4+2c4+1c>,+m  i  k!Idea #2: Generate fresh summaries #Generating 2 random summaries for P #Generating 2 random summaries for Q ;!Loops and Fixed point computationIn presence of loops (in procedures and call-graphs), fixed point computation is required. The number of iterations required to reach fixed point is kv(2kI+1) + 1 kv: # of visible variables kI: # of input variables <     , Error Probability and Complexity(Time Complexity = nkVkI2t Error probability = 1/qt-m n: size of program kV, kI: # of visible and input variables t: # of random summaries q: size of set from which random values are chosen m: kI kV (generic bound) kI + kV (for linear arithmetic) 4 (for unary uninterpreted functions)8      t    DPQs0   Related WorkIntraprocedural random interpretation Linear arithmetic (POPL 03) Uninterpreted functions (POPL 04) Interprocedural dataflow analysis (POPL 95, TCS 96) Sagiv, Reps, Horwitz Cons: simpler properties, e.g. liveness, linear constants Pro: better computational complexity Interprocedural linear arithmetic (POPL 04) Muller-Olm, Seidl Cons: O(k2) times slower Pro: works for non-linear relationships too &>5u,X&>5t,   <  3 % 9$G! Related WorkIntraprocedural random interpretation Linear arithmetic (POPL 03) Uninterpreted functions (POPL 04) Interprocedural dataflow analysis (POPL 95, TCS 96) Sagiv, Reps, Horwitz Cons: simpler properties, e.g. liveness, linear constants Pro: better computational complexity Interprocedural linear arithmetic (POPL 04) Muller-Olm, Seidl Cons: O(k2) times slower Pro: works for non-linear relationships too &>5u,X&"->%,   <  3 % 9$G Experiments  ConclusionRandomization buys efficiency, simplicity at cost of probabilistic soundness. Combining randomized techniques with symbolic techniques is powerful./8x"P  ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@ z?" dd@  " @ ` n?" dd@   @@``PR    @ ` ` p>>f-K0 8(    6o "0  T Click to edit Master title style! !$  0Lr "@  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     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Abstract interpretation analyzes statements once but uses complicated operations. Random interpretation simply executes program once (and captures effect of all paths).01 2  z  p00   L ,$D 0B  HDjJ?"0@NNN?N p0B   HDjJ?"0@NNN?N p00 ! 0@ @True 0 2 " < ?"6@ NNN?NTD =* 0 2   # <ħ ?"6@ NNN?NRB  =* 0 2  H  0޽h ?@                           ̙33 J TIMING.|25.2|2.1|5.5|13.5|10.8~ ___PPT10^ .|0u+TdD2 ' = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* C%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* D%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(+  -K0 p V(   x  c $L0     S H` 0<$@ 0  H  0޽h ? ̙33& TIMING |15.8J___PPT10*.W$+2D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =4@BBBB%(D' =,54*3>!Bstyle.color='`B@BPB<* 4D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* 4?%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* ?l%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* l%(+  -K0  6(   ~  s *p0   x  c $` 0  H  0޽h ? ̙33___PPT10i.W$+D=' = @B + -K0  6(   ~  s *0   x  c $ 00  H  0޽h ? ̙33___PPT10i.@+D=' = @B +o @--K0 X P  `  (  ` ~ ` s *0   x ` c $   ` 6 ` ,$D 0 r2: [a=4, b=1]H0   ` 6 0 ,$@ 0 r1: [a=2, b=3]H0  Ll `    ` `  ,$D 0 `  6C"?p  Ia := 2; b := 3;0 2 `  6C"? `   Ia := 4; b := 1;0 2T `  c $(  T  ` B c $9  Z2  `  s * @   ` 6 ` ,$D 0 $= f7(r1,r2): [a=72 + (1-7)4, b=73 +(1-7)1] i.e. [a=-10, b=15]^/ r' r *H ` 0޽h ?/@` ` ` ` ` `  ̙33 6 TIMING|54.7|7.1|7.9a ___PPT10A .k+ZG.Dm ' = @B D( ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* ` %(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*` %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*` %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* ` %(++0+` : ++0+` : ++0+ ` : +  0-K0  "(     6\:P ` zr2: [a=4, b=1]B0    6:0  zr1: [a=2, b=3]B0  ~  s *!:0  :   c $':  : " 0Pp  0"C"?@ f Ia := 2; b := 3;0 2  0-:C"?0 V Ia := 4; b := 1;0 2F  S f( F  @ S V9 R2  s * @ f  6`4:0p`  = f7(r1,r2): [a=72 + (1-7)4, b=73 +(1-7)1] i.e. [a=-10, b=15]:/ r' r *H  0޽h ?/@     ̙33___PPT10i.v+D=' = @B + -K0 \ *(  \ x \ c $I:0  : r \ S XJ:p  : H \ 0޽h ? ̙33___PPT10i.k+D=' = @B + -K0 h <(  h ~ h s *|Q:0  : ~ h s *PR:@ : H h 0޽h ? ̙33___PPT10i.k+D=' = @B +20-K0 (Q (    # 6l^: ` d4Example: Random Interpretation for Linear Arithmetic55 , 6_: P,$@ 0 I i=3, a=0, b=3 0  - 6e:| x ,$D 0 ?i=3 0  . 0h:C"?@  Ia := 0; b := i;0 2 / 0(l:C"?  Ka := i-2; b := 2;0 2 0 04p:C"?   l2c := b  a; d := i  2b;0 2 1 0t:C"?Pvv "assert (c+d = 0); assert (c = a+i)#0 2#, 2 6lx: p@,$D 0 Ji=3, a=-4, b=7 0  3 6|: ,$D  0 ei=3, a=-4, b=7 c=23, d=-23 .0 f 4 0p:C"?  `  l2c := 2a + b; d := b  2i;0 2 5 6؅:` & ,$D 0 I i=3, a=1, b=2 0  6 6: @P ,$D 0 Ti=3, a=-4, b=7 c=-1, d=1 0 F 7 S  WF 8 @ S  WL 9 @ c $ X2 : 0 F ; @ S  2 L < c $ PRB = s *Dp  > 6X: p ,$@ 0 Wi=3, a=-4, b=7 c=11, d=-11 0  ? 0:D  AFalse 0 2 @ 0:$ 1  AFalse 0 2R2 A s *P F B S   X C 0 A d D <G1HI1  ^ E 6GVHjIV A X F 0 . d G <G-Hr*I-  ^ H 6GHI . 2  I 0:&| F,$ 0 bw1 = 5@0 2  J 0:& ` 0 F ,$  0 bw2 = 2@0 2 F  p00 K  ~ PnB L HDjJ?"0@NNN?N p0B M  HDjJ?"0@NNN?N p00 N 0:2 + @True 0 2 O 0Ԭ:" /  @True 0 2 P <: ?"6@ NNN?N2   =* 0 2   Q < : ?"6@ NNN?N< $  =* 0 2  H  0޽h ?@ . A 7  / A 8  A F 9 0 : B 4 : ; : 1 < C . D C / E F 0 G F 4 H  33T TIMING8|35.7|2.2|3.|3.6|3.5|1.5|3.5{___PPT10[."(+of@D7' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*- %(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*, %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5 %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*I %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2 %(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*> %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6 %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*J %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3 %(++0+, : ++0+- : ++0+2 : ++0+3 : ++0+5 : ++0+6 : ++0+> : ++0+I : ++0+J : + &-K0  6(   ~  s *:0  : x  c $7"0 : H  0޽h ? ̙33___PPT10i.W$+D=' = @B +-K0 910)+ (     6:0P I i=3, a=0, b=3 0   6:|  ?i=3 0   0:C"?P0 Ia := 0; b := i;0 2  0:C"?  Ka := i-2; b := 2;0 2  0:C"?   l2c := b  a; d := i  2b;0 2  0@:C"?P v "assert (c+d = 0); assert (c = a+i)#0 2#,  6p: @ Ji=3, a=-4, b=7 0   6l:(  ei=3, a=-4, b=7 c=23, d=-23 .0 f  0:C"? p  l2c := 2a + b; d := b  2i;0 2  6:`6  I i=3, a=1, b=2 0   6N 0 P P  Ti=3, a=-4, b=7 c=-1, d=1 0 F  S WF  @ S  WL  @ c $X2  0  F  @ S  B L  c $ PRB  s *Dp  6N 0  Wi=3, a=-4, b=7 c=11, d=-11 0   0 NT  AFalse 0 2  0p:$0 1  AFalse 0 2R2  s *PF  S   X  0Qd  <G1HI1 ^  6GVHjIV Q X  0> d  <G-Hr*I-  ^  6GHI >B   0N&F bw1 = 5@0 2   0N& p@F  bw2 = 2@0 2  ! 6(N ` ExampleF  p00 #  ~` nB $ HDjJ?"0@NNN?N p0B %  HDjJ?"0@NNN?N p00 & 0N B+ @True 0 2 ' 0"N" /  @True 0 2 ( <%N ?"6@ NNN?NB*  =* 0 2   ) <N ?"6@ NNN?NL4 =* 0 2   + H,N ?"6@ NNN?NP`  ,$ 0 o The second assertion is true in the context i=2. We need two new ideas to make the analysis interprocedural.&p 2o_H  0޽h ?@                               33___PPT10."(+7?iDO' = @B D ' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+ %(+8+0++ N +7-K0  (, ](     68NT ,$@  0 Ni=2 .0 f  6h?N```,$@ 0 Da=0, b=i 0    0?NC"? pF Ia := 0; b := i;0 2  0CNC"? 0 F Ka := i-2; b := 2;0 2  0GNC"?` Ppl  l2c := b  a; d := i  2b;0 2  0KNC"?x6 & "assert (c+d = 0); assert (c = a+i)#0 2#,  6ON:* ,$D 0 Ja=8-4i, b=5i-8 0   6SN H ,$D  0 l"a=8-4i, b=5i-8 c=21i-40, d=40-21i .#0 !f  0YNC"?` @ l  l2c := 2a + b; d := b  2i;0 2  6(]N@ `,$D 0 F a=i-2, b=2 0    6haN  ,$D 0 Ya=8-4i, b=5i-8 c=8-3i, d=3i-8 0 F  S F7F  @ S FY L  @ c $ ?HX2  00 ;k` F  @ S l d7 L  c $` SWRB  s *D"   6fN PP ,$@ 0 \ a=8-4i, b=5i-8 c=9i-16, d=16-9i !0 !  0kN@$ M AFalse 0 2  0 oNB  AFalse 0 2R2  s *0` F  S l `B7 X  0pd  <GCH HICHp ^  6G HI H X  0d  <G-HI-g`` ^  6GPH'IPg`  ! 0tN,,$ 0 bw1 = 5@0 2  " 0`zN  ,$ 0 bw2 = 2@0 2  # 60N `P0 :&Idea #1: Keep input variables symbolic} $ HpN ?"6@ NNN?N`   Do not choose random values for input variables (to later instantiate by any context). Resulting program state at the end is a random summary.  2 % 6N= ,$@  0 ]a=0, b=2 c=2, d=-2 .0 fB & <D?"0@NNN?N 0  ,$D  0 ' 0N@M @True 0 2 ) 0N~ @True 0 2 + <ȄN ?"6@ NNN?Nj =* 0 2   , <N ?"6@ NNN?N @  =* 0 2  H  0޽h ?@                              33>V TIMING:|15.7|2.|3.5|1.7|0.5|0.3|28.5___PPT10."(+`OD\' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*! %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*" %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(Df' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*% %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*& %(+0+0+ N ++0+ N ++0+ N ++0+ N ++0+ N ++0+ N ++0+ N ++0+! N ++0+" N ++0+% N +\+ 0-K0  ! t(   x  c $N0  N f  6tN ,$D 0 .u = 52 -7 = 3 v = 51 -7 = -2 w = 51 -7 = -2/0 EEEEEE E  6dN'% ,$D 0 Fx = 5i-7" 0  E  0N,$D 0 Ew = 5$0 2E  6N ,$D 0 Cx = 3"0 E  6N,$@ 0 Ex = i+1"0 E  0`NC"?y0 E x := i+1; 0 2 A  0PNC"?y  Cx := 3; 0 2A  0NC"? & "  E return x; 0 2 AF  S F  @ S o L  @ c $ R2  s *  <N 8$0( 2 A d  <GH+ISy^  6G7HxI7So yL  c $AC  0@N  =* 0 2A  0N E Procedure P 0 2   0 N  BInput: i 0 2   0NC"?[ 0 ~  qAssert (u = 3); Assert (v = w);60 200 <2 A  0NC"?0  t"u := P(2); v := P(1); w := P(1);6 0 200 22#AL  c $[ L  c $  0N E Procedure Q 0 2   HN ?"6@ NNN?Na ,$D 0 Plugging the same summary twice is unsound. Fresh summaries can be generated by random affine combination of few independent summaries! 2z  p00    8p ,$D 0B  HDjJ?"0@NNN?N p0B   HDjJ?"0@NNN?N p00  0Obo @True 0 2 ! 0OdP q AFalse 0 2H  0޽h ?                  ̙33Z TIMING>|24.9|1.5|2.2|11.5|32.2|0.7|43.___PPT10v.pK+LDj' = @B D%' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* ,%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* ,%(++0+ N ++0+ O ++0+ O ++0+ O +S0 0&-K0 7x (  x ~ x s *,O0  O  x HO ?"6@ NNN?Na  ,$D 0 . 2 !x 0tO0@ E Procedure P 0 2  "x 6!O ,$D 0 K x=[5i-7,7-2i]"0 E #x 0 &OmD,$D 0 Hw=[5,-2]$ 0 2 E $x 6)Or H,$D 0 G x = [3,3]" 0  E %x 6.O`nx,$@ 0 I x=[i+1,i+1]" 0  E &x 02OC"?JNp E x := i+1; 0 2 A 'x 05OC"?JnP p Cx := 3; 0 2A (x 09OC"?U }{  E return x; 0 2 AF )x S pxF *x @ S pL +x @ c $U R2 ,x s *t  -x <?Of 8$0( 2 A d .x <GyHyIy$J^ /x 6G^MHRI^M$fJL 0x c $ 1x 0BO\  =* 0 2A 2x 0FOT BInput: i 0 2  3x 08JO# @True 0 2 4x 0(NOF $ AFalse 0 2 5x HRO ?"6@ NNN?N  ,$D 0 Bx = f7(5i-7,7-2i) = 47i-91 x = f6(5i-7,7-2i) = 40i-77 x = f2(5i-7,7-2i) = 19i-35 x = f0(5i-7,7-2i) = 7-2i x = f5(5i-7,7-2i) = 33i-63 x = f1(5i-7,7-2i) = 5i-7f 2 7x <fO ?"6@ NNN?N `&,$ 0 jProcedure Q calls P 3 times. Hence, generating 2 random summaries for Q requires 23 fresh summaries of P.8k0 2R      H x 0޽h ?o &x ,x )x  'x ,x *x  ,x (x +x  -x &x .x  -x 'x /x -x 0x  ̙33X TIMING<|18.4|1.5|5.1|18.7|21.1|5.|1.9___PPT10.pK+BDt' = @B D/' = @BA?%,( < +O%,( < +D^' =%(D' =%(D' =>@BBBBPB%(/%(D' =1:Bvisible*o3>+B#style.visibility<*x %(D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%x %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$x %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#x %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"x %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*7x %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5x %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5x :%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5x :U%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5x Un%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5x n%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5x %(++0+"x O ++0+$x O ++0+%x O ++0+7x O + d-K0    "| B (  | ~ | s *O0  O  | 6O P ,$D 0 XZu = [472-91, 402-77] =[3,3] v = [191-35, 7-21] =[-16,5] w = [331-63, 51-7] =[-30,-2][0 EEEEEE EEEEEEE | 0؍OC"?m p  qAssert (u = 3); Assert (v = w);60 200 <2 A | 0OC"?  t"u := P(2); v := P(1); w := P(1);6 0 200 22#AL | @ c $ m L | c $   | 0DOpjP E Procedure Q 0 2 F  p00  |   b B  | HDjJ?"0@NNN?N p0B  |  HDjJ?"0@NNN?N p00z "| HO ?"6@ NNN?N.  Bx = f7(5i-7,7-2i) = 47i-91 x = f6(5i-7,7-2i) = 40i-77 x = f2(5i-7,7-2i) = 19i-35 x = f0(5i-7,7-2i) = 7-2i x = f5(5i-7,7-2i) = 33i-63 x = f1(5i-7,7-2i) = 5i-7f 2H | 0޽h ?/@| | | | |  ̙33L6 TIMING|15.8|0.3|0.2___PPT10.pK+YD' = @B Du' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*| %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*| <%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*| <[%(+ -K0 p 6(   ~  s *O0  O x  c $ĸO`0 O H  0޽h ? ̙33___PPT10i.v+D=' = @B + 6-K0 @ 0(   x  c $dO0  O x  c $8O@P`  O H  0޽h ? ̙33___PPT10i.)+D=' = @B + py-K0 ` *(   x  c $O0  O r  S xO@ O H  0޽h ? ̙33___PPT10i. .+D=' = @B + Pr-K0  6(   ~  s *O0  O x  c $pO@ O H  0޽h ? ̙33___PPT10i. .+D=' = @B +Y c-K0 NNTt mN(  t ~ t s *O  O )#t 0P  t  #"&     P0   t  B4 ?"6@ NNN?N  I2 @`  t  BO ?"6@ NNN?Nt  K525 @`  t  BDQ ?"6@ NNN?Ndt  J49 @` t  B, Q ?"6@ NNN?Nd  J8K @`" t  BQ ?"6@ NNN?N0  jgzip @` t  B$Q ?"6@ NNN?N  J34 @` t  B$Q ?"6@ NNN?Nt K392 @` t  B4-Q ?"6@ NNN?Ndt J53 @` t  B6Q ?"6@ NNN?Nd K23K @`  t  B'Q ?"6@ NNN?N0 hli @` t  BGQ ?"6@ NNN?N  I4 @` t  B@QQ ?"6@ NNN?Nt K825 @` t  BYQ ?"6@ NNN?Ndt J31 @` t  BcQ ?"6@ NNN?Nd K28K @`# t  BkQ ?"6@ NNN?N0 kijpeg @` t  BPuQ ?"6@ NNN?Np  J47 @` t  B}Q ?"6@ NNN?Ntp L1700 @` t  BQ ?"6@ NNN?Ndpt J63 @`  t  BQ ?"6@ NNN?Npd K29K @` !t  B\Q ?"6@ NNN?N0p Ngo @` #t  BhQ ?"6@ NNN?NP p PTime @`! $t  BQ ?"6@ NNN?NtPp iVar @`! %t  BQ ?"6@ NNN?NdPtp iInp @` &t  BQ ?"6@ NNN?NPdp PLine @`" 't  BQ ?"6@ NNN?N0Pp jProg @`B (t  <o ?"0@NNN?N0P PB )t  61 ?"0@NNN?N0p pB *t  61 ?"0@NNN?N0 B +t  61 ?"0@NNN?N0 B ,t  61 ?"0@NNN?N0 B .t  <o ?"0@NNN?N0 B /t  <o ?"0@NNN?N0P0 B 0t  61 ?"0@NNN?NP B 1t  61 ?"0@NNN?NdPd B 2t  61 ?"0@NNN?NtPt B 3t  61 ?"0@NNN?NP B 5t  <o ?"0@NNN?N P  6t <Q ?"6@ NNN?NDl P oRandom Inter (this paper): 0 2 00 22  ^ 7t <Q ?"6@ NNN?NF I,$@ 0 Random Intra (POPL 2003)L 0 2 00 22    ` 8t <Q ?"6@ NNN?N@PL,$@ 0 Det Inter (TCS 96): 0 2 00 22  v  P  t  #"&     P  ,$D 0 t  BQ ?"6@ NNN?Nv  K756 @` ?t  BDS ?"6@ NNN?N v  K160 @` @t  B, S ?"6@ NNN?Nv  J24 @` At  BS ?"6@ NNN?N v  J34 @` Bt  BS ?"6@ NNN?Nv p K107 @` Ct  B#S ?"6@ NNN?N pv  K170 @`  Dt  B-S ?"6@ NNN?Nv Pp SSpeedup @`h Et  B|8S ?"6@ NNN?N Pv p  D(Var)P @`B Ft  <o ?"0@NNN?N PPB Gt  61 ?"0@NNN?N ppB Ht  61 ?"0@NNN?N B It  61 ?"0@NNN?N B Jt  61 ?"0@NNN?N B Lt  <o ?"0@NNN?N  B Mt  <o ?"0@NNN?N P B Nt  61 ?"0@NNN?Nv Pv B Ot  <o ?"0@NNN?NP a PP  t  #"&     PP ,$D 0 St  B'S ?"6@ NNN?N'P  K2.0 @` Tt  BMS ?"6@ NNN?N'  I6 @` Ut  BVS ?"6@ NNN?N'P K1.3 @` Vt  B_S ?"6@ NNN?N' J20 @` Wt  BhS ?"6@ NNN?N'P K2.3 @` Xt  BqS ?"6@ NNN?N' I3 @` Yt  BzS ?"6@ NNN?N'pP K1.9 @` Zt  BS ?"6@ NNN?Np' J17 @`  [t  BȌS ?"6@ NNN?N'PPp SSpeedup @`V \t  BS ?"6@ NNN?NP'p  D(Inp)> @`B ]t  <o ?"0@NNN?NPPPB ^t  61 ?"0@NNN?NpPpB _t  61 ?"0@NNN?NPB `t  61 ?"0@NNN?NPB at  61 ?"0@NNN?NPB ct  <o ?"0@NNN?N P B dt  <o ?"0@NNN?NP B et  61 ?"0@NNN?N'P' B ft  <o ?"0@NNN?NPPP  t <S ?"6@ NNN?Ni   Inp: # of input variables that were constants Var: # of local variable that were constants D(Var): # of fewer local variable constants discovered Random Inter discovers 10-70% more facts; Random Intra is faster by 10-500 times; Det Inter is faster by 2 times. .1 2f10 <2s0 2\   7  s  H+-H t 0޽h ? ̙33\ 0 TIMING|27.5|32.5 ___PPT10 .`t S+إ,D` ' = @B D ' = @BA?%,( < +O%,( < +D%' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*7t %(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t %(D%' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8t %(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t %(+p+0+7t S ++0+8t S + -K0 P 0(   x  c $ S0  S x  c $S0 S H  0޽h ? ̙33___PPT10i.o Wc+D=' = @B + 0 0 +(   d  c $w     s *R ZF    H  0~dm.F ? ̙33P 0  `(   X  C w   X  S YX ZF  X bFor example, we can specify how to extend any analysis in this framework to an interprocedural analysis. 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