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A lattice L is a logical lattice over theory T if DL = finite conjunctions of atomic facts over T E L E iff E )T E Let L1 and L2 be logical lattices over T1 and T2 resp. Then logical product of L1 and L2 is L1*L2, where DL1*L2 = finite conjunctions of atomic facts over T1 [ T2 E L1*L2 E iff E )T1 [ T2 E and AlienTerms(E ) Terms(E)mHi\91 </  . b<{1 Outline Logical product combination of lattices Abstract interpreter for logical product lattice Join operator Existential quantification operator Correctness and Complexity J)1 N*1NAbstract Interpreter for L1*L2 Outline Logical product combination of lattices Abstract interpreter for logical product lattice Join operator Existential quantification operator Correctness and Complexity TZ @Z@ )Background: Combining Decision Procedures  Join OperatorIf E = JoinL(E1,E2), then E is the least upper bound of E1 and E2 in lattice L Examples: Joinla(z=0 y=10, z=5 y=5) = z+y=10 0z 5 Joinuf(z=a y=F(a), z=b y=F(b)) = y=F(z) Joinla*uf(z=a-1 y=F(a), z=b-1 y=F(b)) = ? Z  'Z     Join Operator,If E = JoinL(E1,E2), then E is the least upper bound of E1 and E2 in lattice L Examples: Joinla(z=0 y=10, z=5 y=5) = z+y=10 0z 5 Joinuf(z=a y=F(a), z=b y=F(b)) = y=F(z) Joinla*uf(z=a-1 y=F(a), z=b-1 y=F(b)) = y=F(1+z) We next show how to construct JoinL1*L2 using JoinL1 and JoinL2.ZA  ' "  Z    P  Outline Logical product combination of lattices Abstract interpreter for logical product lattice Join operator Existential quantification operator Correctness and Complexity tZ$ Z$ #Existential Quantification OperatorIf E = EQL(E ,V), then E is the least (i.e., most precise) element in lattice L such that: E L E Vars(E) V = ; Examples: EQla(xa ay, {a}) = x y EQuf(x=F(a) y=F2(a), {a}) = y=F(x) EQla*uf(aby z=c+1 a=F2(b) c=F(b), {a,b,c}) = ?  [ zO    c#Existential Quantification OperatorZIf E = EQL(E ,V), then E is the least (i.e., most precise) element in lattice L such that: E L E Vars(E) V = ; Examples: EQla(xa ay, {a}) = x y EQuf(x=F(a) y=F2(a), {a}) = y=F(x) EQla*uf(aby z=c+1 a=F2(b) c=F(b), {a,b,c}) = F(z-1)y We can construct EQL1*L2 using EQL1 and EQL2. l[ /O     c>  Outline Logical product combination of lattices Abstract interpreter for logical product lattice Join operator Existential Quantification operator Correctness and Complexity jZ2 Z2  Correctness Our algorithms for JoinL1*L2 and EQL1*L2 are sound. They are complete when the underlying theories T1 and T2 are convex, stably infinite, and disjoint. Proof of correctness is non-trivial. ~ zComputational Complexity> Complexity of JoinL1*L2 and EQL1*L2 is worst-case quadratic in complexity of JoinL1, JoinL2, EQL1, EQL2. Steps required for fixed-point computation DL(E) = max # of elements in a chain above E in lattice L DL1 L2(E) DL1(E1) + DL2(E2) + |AlienTerms(E)| where E1 and E2 are purified and saturated components of E../8        - ?Conclusion and Future WorkDefined combination L1*L2 of two lattices L1 and L2. This logical product is more precise than reduced product. Described abstract interpretation operators for L1*L2 in terms of corresponding operators for L1 and L2. Lends modularity to design & implementation of abstract interpreters. Future Work: Handle non-convex theories (eg. arrays) more precisely. Handle non-atomic facts involving negation & disjunction. Perform experiments. 5;iF 5;iF    Jk/px6P-   0` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@z?" dd@  " @ ` n?" dd@   @@``PR    @ ` ` p>>fL0 8(    6H  "@   T Click to edit Master title style! !$  0(  "0   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     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S  T r`r` @    x*  V++VV  TD r`r`  @   z*  V++VVH  0~dm.F ? ̙3380___PPT10.47L P 0(    NlH r`r` )    ^*  V++VV  NLr`r`  )   `*  V++VV  T|~ r`r` @    ^*  V++VV  TŤ r`r`  @   `*  V++VVH  0~dm.F ? ̙3380___PPT10.47U 0HL0   (   x  c $8˛ `       0̛  "6 ?T    R  s *\; ; \  0\w  C"?wT P  FAshish Tiwari SRI  0ϛ  C"? f  oRADPLV  0޽h ?"` ̙33y___PPT10Y+D='  = @B +  0NL0 3(  r  S |. @     0/ s)  Oa1 := a1+1; a2 := a2+2; b1 := F(b1); b2 := F(b2); c1 := F(2c1-c2); c2 := F(c2);0 2800 22  0+B#style.visibility<*;m%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*m%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DL' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Xz%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*z%(+  0aL0 4F(  x  c $Tե @     c $ۥ 0   " (0PpH  0޽h ? ̙33___PPT10i.o`g+D='  = @B +H/  0];L0 `3!%0 (  r  S  @     6P 9  g Join Node, 0   8X  6X C"?  Cx := g; 0 2R  s *C C R  s *mC C    6 n h NE .0 2    6h   KE.0 2    <   YConditional Node ,0     <l  = 0 R  s *\Gj  BGS|H IS| d  <G1HI1  < ^ Kp.0    B G 4 0   6   @True 0 2  6  AFalse 0 2  6 z\Tt ME00 2   6   gE1 H0 2  (  6%  LE2.0 2   6* E  WAssignment Node,0    6L/ qk LE2.0 2 2  B3 ]@p 4 0 R  s *XX  67 x ME00 2   6< v7Op LE1.0 2 RB  s *DE`RB   s *D UghB ! 6D?"0@NNN?N@bgB " 6D?"0@NNN?NFN N A # <A  ?"6@ NNN?NZ,$D 0 cKE = JoinL1*L2(E1,E2) We show how to get JoinL1*L2 from JoinL1 and JoinL2.L0 2                        0 $ <$Q  ?"6@ NNN?N 8J ,$@ 0 RjE1 = MeetL1*L2(E, p) E2 = E MeetL1*L2(E,E ) = E E 60 2                      % <.I ;+QCD'  = @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<*#L%(DL' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%)%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%)=%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%=l%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$6%(+  0m0L0 4F(  x  c $4 @     c $X} 0   " (0PpH  0޽h ? ̙33___PPT10i.o`g+D='  = @B +M  0L0 `.x (  x r x S  @    x <<  ?"6@ NNN?N$ ,$@ 0 (a1=2y2-y1 y1 4y3 a2 y1 = y2 y1 = a2 0 F20 22           x <$  ?"6@ NNN?N  ,$@  0 +a2=F(a1) y1=F(y1) y2=F(F(y1)) y1 = a1 Z 0 200 F2 00 22         x <ҵ  ?"6@ NNN?N Q1y1 4y3 F(2y2-y1) y1=F(y1) y2=F(F(y1))2           x @ 6?"0@NNN?N l ,$@ 0  x 6?"0@NNN?Nl ,$@ 0( x <4ڵ  ?"6@ NNN?ND ,$ 0 J Purification" 0 2  gl hN R x h ,$@ 09 x  B  ?"6@ NNN?NN H y1=y2`0 2        B x  BDo?"0@NNN?NhR Rgl r' 7  x er u ,$@ 09 x  B  ?"6@ NNN?N' !  y1=a1`0 2        B x  BDo?"0@NNN?Nr7 7 oz hN R x   o ,$@ 03 x  <  ?"6@ NNN?NN H y1=a2`0 2        B x  BDo?"0@NNN?NhR RB %x <D?"0@NNN?Nf 33 ,$D 0R &x <  ?"6@ NNN?N  ,$ 0 ty1 = 4y3P 0 2      & 'x <  ?"6@ NNN?N2  , ,$ 0 H Saturation" 0 2   )x <  ?"6@ NNN?N"? ty1 = 4y3P 0 2      8 '! ,x '!B *x  <D?"0@NNN?N'z!z +x  <   ?"6@ NNN?N2 ??"0 2  G .x <P  ?"6@ NNN?N$% =This classic algorithm was given by Nelson and Oppen in 1979.">0 2>  / H x 0޽h ?/@x x x x x x  ̙332P TIMING4|27.5|28.4|39|6.2|4.1|24.1l2___PPT10L2.Р+O@D1'  = @B D0' = @BA?%,( < +O%,( < +Dl' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* x %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* x %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x !%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x !)%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x "%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*x "*%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*x %(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*x !%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*x !)%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*x "%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*x "*%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*x %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*x "%(DY' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'x %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*x %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*x !%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*x %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*x "*%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*x %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*x !)%(D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&x %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%x %(++0+x 0 ++0+x 0 ++0+x 0 ++0+&x 0 ++0+'x 0 +  0L0 p3P(  r  S F @     S G 0<$D 0   H  0޽h ? ̙33& TIMING |75.2___PPT10n.Q3F+YDB'  = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+  0$L0  50(  x  c $d @   x  c $d 0   H  0޽h ? ̙33___PPT10i.Q3F+D='  = @B +  0L0  ++2)=hR((  h (h B Ԕ?"6@ NNN?N=x h c $4k @    h <l  ?"6@ NNN?NT,$@ 0 y=F(a) a=ha,bi0 200 22          , h <x  ?"6@ NNN?N4,$@ 0  z=a-1 a=ha,bi0 200 22            h <X  ?"6@ NNN?NQ,$@ 0 y=F(b) b=ha,bi0 200 22            ,  h <  ?"6@ NNN?N ,$@  0 z=b-1 b=ha,bi 0 2 00 22            5 h <\  ?"6@ NNN?Nf z=a-1 y=F(a) >0 2       4 h <  ?"6@ NNN?N"- " z=b-1 y=F(b)>0 2        h@ 6?"0@NNN?N,$@ 0 h 6?"0@NNN?N,$@ 0 h@ 6?"0@NNN?Na,$@  0 h 6?"0@NNN?N,$@ 0i h <p  ?"6@ NNN?Nf mf ,$@ 0  ha,bi=1+zL 0 2         h <؞  ?"6@ NNN?Nq =q ,$@! 0  y=F(ha,bi)Z 0 2          L h <ܨ f?"6@ NNN?Nk ,$@ 0 nJoinla20 2    h@ 6?"0@NNN?Nka ,$@ 0 h@ 6?"0@NNN?N 35f ,$@ 0L  h < f?"6@ NNN?NqLq ,$@ 0 nJoinuf20 2    !h 6?"0@NNN?NL,$@ 0 "h@ 6?"0@NNN?N,$@ 0 #h@ 6?"0@NNN?Nq q ,$@  0M $h < f?"6@ NNN?NL ~ L,$@- 0 oEQuf*la20 2    %h <p  ?"6@ NNN?N< 8  Fy=F(1+z)" 0 2   &h 6?"0@NNN?Nf 3 ,$@* 0 'h@ 6?"0@NNN?Nq  ,$@+ 0 ,h 6?"0@NNN?N ,$@ 02 -h N<  ?"6@ NNN?Nt- 8$0( 2  2 .h <?"6@ NNN?N ,$@1 0 /h 6?"0@NNN?N L ,$@, 02 1h <?"6@ NNN?N2 2h <?"6@ NNN?N 3h@ 6?"0@NNN?N 4h 6?"0@NNN?N" 5h@ 6?"0@NNN?NL ,$@0 0 6h <pŸ  ?"6@ NNN?NI I,$@. 0  { ha,bi }Z 0 2           8h@ 6?"0@NNN?N ,$@/ 0# 9h <˸  ?"6@ NNN?N]M  y Joinuf+la: 0 2      :h 6Ѹ  1G ,Combining Join Operators2 +B#style.visibility<*h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* h%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* h%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* h%(D ' =%(DI ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h %(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*h %(D ' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h %(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*h %(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* h%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h %(D' =%(DV' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$h%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* 0 2      2 ( NN  ?"6@ NNN?Nt- 8$0( 2  2 ( <?"6@ NNN?N52 ( <?"6@ NNN?N1 ( 6?"0@NNN?N (@ 6?"0@NNN?N ( 6?"0@NNN?N.or,$@ 0 ( <S  ?"6@ NNN?N7 , ,$@  0 &c a z-1 a aF(z-1)  0 2 00 22            ! ( <]  ?"6@ NNN?N}N wEQuf+la:0 2     ( 6lb  G B.Combining Existential Quantification Operators2 ( <?"6@ NNN?Nn ( 6?"0@NNN?NnnMXz   ( ,  &,$@! 0B (B BDjJ?"0@NNN?N  ( <8f  ?"6@ NNN?N<  jc a z-1@0 2     \z ]D >   (  ,$@" 0B !(B BDjJ?"0@NNN?N]    "( <k  ?"6@ NNN?ND  >  na aF(z-1)@ 0 2     K #( <q f?"6@ NNN?N,$@ 0 mDefla20 2   K $( <|v f?"6@ NNN?N  ,$@ 0 mDefuf20 2    %(@ 6?"0@NNN?N-f O,$D  0J &( <| f?"6@ NNN?N M ,$@ 0 lEQla20 2   " '( NGYHVIR?"0@NNN?NEMe ,$D 0 (( 6?"0@NNN?N67 ,$@  0 )(@ 6?"0@NNN?N M4e ,$@ 0 *( 6?"0@NNN?N 4 i ,$@ 0" +(@ HGQHI8?"0@NNN?NI i ,$D 0 ,(@ 6?"0@NNN?N  ,$@ 0F -( <܃  ?"6@ NNN?N  ,$@ 0 ha = F(c)" 0 2   .( 6?"0@NNN?N L Q ,$@ 0 /( 6?"0@NNN?N (  0(@ 6?"0@NNN?N oQ . ,$@ 0 1( 6?"0@NNN?N o. ,$@ 0 2( NZG5H{I5?"0@NNN?N U  ,$D 02 3( 6?"6@ NNN?Nf ,$@# 02 4( 6?"6@ NNN?Ng,$D$ 0 5( 6?"0@NNN?Nu,$@ 0 6( 6?"0@NNN?Nj ,$@  0 7( 6?"0@NNN?Nu,$@ 0H ( 0޽h ?(( ((( (( ( ((3(((((((((((((( ((%( (&('( (((( (&()( ((*(((+(&( (,((-(.( ((/(-((0( ((1(((2(4(#(5(3($(6((4(7( ̙33>N TIMING2|7.8|12.8|38.9|9.6|4|12.9=___PPT10=.sЪ+~D;'  = @B Dm;' = @BA?%,( < +O%,( < +D' =%(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<* (%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D" ' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*7(%(D' =4@BBBB%(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<* (%(D' =4@BBBB%(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<*%(%(Dq' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'(%(Dd' =%(D ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+(%(Dq' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0(%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2(%(D" ' =%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$(%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4(%(++0+(0 ++0+(0 ++0+ (0 ++0+ (0 ++0+(0 ++0+(0 ++0+#(0 ++0+$(0 +  0L0 64F(  4x 4 c $(( @    4 c $x) 0   " (0PpH 4 0޽h ? ̙33___PPT10i.o`g+D='  = @B +  0@=L0 `4$(  r  S $M @   r  S M z0   H  0޽h ? ̙33___PPT10i.m +D='  = @B +  0[hL0  680(  8x 8 c $T @   x 8 c $U 0   H 8 0޽h ? ̙33___PPT10i.H%M+D='  = @B +  0L0 P4n(  r  S \ @     s *4]  "6 ?    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L 2 L7D.-՜.+,0    6On-screen Show Times New RomanComic Sans MSMonotype CorsivaArialcmsy10 MT ExtraDefault Design Combining Abstract Interpreters MotivationOutlineLogical Product of LatticesOutlineAbstract Interpreter for L1*L2Outline*Background: Combining Decision ProceduresJoin OperatorJoin Operator Outline$Existential Quantification Operator$Existential Quantification Operator Outline CorrectnessComputational ComplexityConclusion and Future Work  Fonts UsedDesign Template Slide Titles%_ Sumit GulwaniSumit Gulwani  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)Current UserSummaryInformation(PowerPoint Document(DocumentSummaryInformation8