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O  =+9Join Algorithms for the Theory of Uninterpreted Functions::"  tSumit Gulwani Ashish Tiwari George Necula UC-Berkeley SRI UC-Berkeleyu8=>>Definition: Join in theory T E = JoinT(E1,E2) iff E1 )T E and E2 )T E If (E1 )T g) and (E2 )T g), then E )T g E1, E2, E: conjunction of ground facts in theory T g: ground fact in theory T E is the strongest conjunction of ground facts that is implied by both E1 and E2 in theory TD<" 4" y    f ff f-ffH Example of JoinsLE: Linear Arithmetic with Equality JoinLE(x=1 y=4, x=3 y=2) = x+y=5 LI: Linear Arithmetic with Inequalities JoinLI(x=1 y=4, x=3 y=2) = x+y=5 1x3 UF: Uninterpreted Functions JoinUF(x=a y=F(a), x=b y=F(b)) = y=F(x) 8$))11, 1 '  (0  :Motivation: Program Analysis using Abstract Interpretation;; ,Join for Uninterpreted Functions is not easy  Join(F(a)=a F(b)=b G(a)=G(b), a=b) = GFi(a)=GFi(b) The result of join is not finitely representable using standard data-structures like EDAGs   a) %$Relatively Complete Join: Definition[Recall, Join(E1,E2): strongest conjunction of ground facts g s.t. E1 )T g and E2 )T g RCJoin(E1,E2,K): strongest conjunction of ground facts g s.t. E1 )T g and E2 )T g and Terms(g) 2 K Example E1: F(a)=a F(b)=b G(a)=G(b) E2: a=b K: { GF(a),GF(b) } RCJoin(E1,E2,K): GF(a) = GF(b)\L  O   =n;  #Relatively Complete Join: Algorithm RCJoin(E1,E2,K): Let D1=EDAG(E1) and D2=EDAG(E2) Extend D1 and D2 to represent K Congruence close D1 and D2 Let D=product construction of D1 and D2 Output D|          "       Step 1: Constructing EDAGs:E1: F(a)=a F(b)=b G(a)=G(b) E2: a=b K: { GF(a),GF(b) };   b  Step 2: Extending EDAGs:E1: F(a)=a F(b)=b G(a)=G(b) E2: a=b K: { GF(a),GF(b) };   b Step 3: Congruence Closure:E1: F(a)=a F(b)=b G(a)=G(b) E2: a=b K: { GF(a),GF(b) };   b (Step 4: Product Construction (Intuition):E1: F(a)=a F(b)=b G(a)=G(b) E2: a=b K: { GF(a),GF(b) };   b (Step 4: Product Construction (Algorithm):E1: F(a)=a F(b)=b G(a)=G(b) E2: a=b K: { GF(a),GF(b) };   b  .Future Work: Join Algorithm for other theoriesFor example, theory of commutative functions (CF) Useful in modeling floating point operations More challenging than uninterpreted functions (UF) E1: x=a y=b E2: x=b y=a JoinUF(E1,E2) = true JoinCF(E1,E2) = F(C[a],C[b]) = F(C[b], C[a]) \2`n2`Pu a &Future Work: Combining Join AlgorithmsFor example, theory of linear arithmetic and uninterpreted functions (LA+UF) E1: x=a y=b E2: x=b y=a JoinUF(E1,E2) = true JoinLA(E1,E2) = x+y=a+b JoinLA+UF(E1,E2) = F(x+c)+F(y+c) = F(a+c)+F(b+c) .& .P  b- V   .Future Work: Context-sensitive Join Algorithms1Join(E1,E2) E = Join(E1E, E2E) Useful in interprocedural analysis This is a representation issue. Representing result of join using conjunction of ground facts is not context-sensitive. E1: x=a y=F(a) E2: x=b y=F(b) JoinUF(E1,E2) a=b = y=F(x) a=b JoinUF(E1 a=b,E2 a=b) = y=F(x) x=a=b #DYq DYb-%  Conclusion^Join Algorithms are useful in program analysis. They are generalization of decision procedure. JoinT(E, g) = g iff E )T g E: conjunction of ground facts in theory T g: ground fact in theory T We showed a relatively complete join algorithm for uninterpreted functions. Join algorithms open up several interesting problems.~`|pf fff fTf>p  C/P(   0` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@z?" dd@  " @ ` n?" dd@   @@``PR    @ ` ` p>> \T(    6|  0@  T Click to edit Master title style! !  0h ` `  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     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*  @  Ib, 2nz J pJ  T    ,$D 0T T c $ T T c $  T 06S"`?J :  IF, 2 T 0;S"`?J p:  IG, 2 T 0(@S"`?Z pJ  IG, 2 T 0DS"`?Z J  IF, 2T T c $ T T c $  /T 0I  9 Add extra nodes to EDAGs s.t. terms in K are represented: 2:, AT <N  `p   GT 0Op P}  k D1 = EDAG(E1)B 2   HT 0+B#style.visibility<*T%(+  0L0  #C`(  `x ` c $H| 0@   F P P  `   @  ` 0oS"`?P @@  IF, 2 ` 0S"`?=* -P  Ia, 2T ` c $@ *  ` 0S"`?0 @  IG, 2T `B c $@ *   ` 0S"`?@ 0@  IG, 2  ` 0@S"`?* P  Ib, 2T  ` c $@ *   ` 0ĊS"`? @  IF, 2T  `B c $@ }* F P   `   rb ` BGHIwoP = lb `B <GgHCKI o  Z ` s *o  @ R ` s *o *  ` 0菠S"`?  @  Ia, 2 ` 0 S"`? *  @  Ib, 2:F J pJ  `    T ` c $ T ` c $  ` 0(S"`?J :  IF, 2 ` 0lS"`?J p:  IG, 2 ` 0S"`?Z pJ  IG, 2 ` 0ȦS"`?Z J  IF, 2T ` c $ T ` c $  ` s *o * ,$D 0 ` s *o * ,$D 0 /` 0t P { F(n) = F(m) if n=m 2,  A` <   `p   B` 0tp P}  k D1 = EDAG(E1)B 2   C` 08p N }  k D2 = EDAG(E2)B 2   H ` 0޽h ?````````` ` ` ` ` ` ` ` ``` ````` ``` ``` ``` ``` `````` 33* TIMING|11|1.7___PPT10.+>>D~' = @B D9' = @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<*`%(+;  0L0 &%2Fd#(  dx d c $| 0@  | F P P  d   @  d 0#zS"`?P @@  IF, 2 d 0%zS"`?=* -P  Ia, 2T d c $@ *  d 0h|S"`?0 @  IG, 2T dB c $@ *   d 0|S"`?@ 0@  IG, 2  d 0|S"`?* P  Ib, 2T  d c $@ *   d 0|S"`? @  IF, 2T  dB c $@ }* F P   d   rb d BGHIwoP = lb dB <GgHCKI o  Z d s *o  @ R d s *o *  d 0,|S"`?  @  Ia, 2 d 0|S"`? *  @  Ib, 2:F J pJ  d    T d c $ T d c $  d 0l|S"`?J :  IF, 2 d 0D|S"`?J p:  IG, 2 d 0 |S"`?Z pJ  IG, 2 d 0|S"`?Z J  IF, 2T d c $ T d c $ R d s *o * R d s *o *  z P"P  "d  [ ,$D 0 #d 0DxC"?0 "J B  J30, 2 $d 0@xC"?" r  G4* 2 %d 0xC"?P   G1* 2 &d 00xC"?0r P  G3* 2 'd 0xC"?Pi   G2* 2 (d 0xC"?` `  76 2 )d 0TxC"?`   G5* 2 *d 0|xC"?0 q @  J20, 2 +d 0xC"?0 0 G P  J10, 2 ,d 0$xC"?P 0 pP  J40, 2 -d 0 )xC"?P @  J50, 2 .d 0-xC"?P +pK  J60, 2 Ad </x  `p x  Bd 02xp P}  k D1 = EDAG(E1)B 2   Cd 0 :xp N }  k D2 = EDAG(E2)B 2   \ Dd 0P@x `0 ^,$D 0 C1: {a, Fa, F2(a), & } C4: {b, Fb, F2(b), & } C6: {G(a), GF(a), & G(b), GF(b), & } 20<2,022 3t  Ed 0Ox 0 @`,$D 0 +1C10: {a, b} C20: {F(a), F(b)} C30: {GF(a), GF(b)}2022 P g Fd 0[x  ,$D  0 C6 C30: { GF(a), GF(b)}j022, H d 0޽h ?ddddddddd d d d d d d d ddd ddddd ddd ddd ddd ddd dddddd 33*\ TIMING@|26|5.1|50.1|3.1|13.9|6.1|0|25.1___PPT10.+xDr' = @B D-' = @BA?%,( < +O%,( < +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<*Dd%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Dd,%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Dd,@%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Dd@X%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ed %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ed %(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ed2%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Fd%(+Y  0L0 44BD1(  x  c $͠ 0@   F P P     @   0ϠS"`?P @@  IF, 2  0hԠS"`?=* -P  Ia, 2T  c $@ *   0\٠S"`?0 @  IG, 2T B c $@ *    0ݠS"`?@ 0@  IG, 2   0S"`?* P  Ib, 2T   c $@ *    0S"`? 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33d 6 TIMING|40.6|4.4|5.5 ___PPT10 .AR+D ' = @B D ' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(+Z  05(L0 l^(  lr l S  0@    l S ` P<$@ 0   l ^Atxp_fig g xp SOURCE\documentclass{slides}\pagestyle{empty} \begin{document} $$\bigwedge\limits_{\mbox{c}}$$ \end{document} 6EXTERNALNAMEtxp_fig$ BLEND False0TRANSPARENT False,KEEPFILES False.DEBUGPAUSE False,RESOLUTION1200*TIMEOUT (none)&BOXWIDTH348(BOXHEIGHT200"BOXFONT10(BOXWRAP FalseL2WORKAROUNDTRANSPARENCYBUG FalseD*ALLOWFONTSUBSTITUTION False6BITMAPFORMATpngmono&ORIGWIDTH186PICTUREFILESIZE1833H l 0޽h ? 33 > TIMING"|33.3|2.2|2.7|5.5 ___PPT10 .^q+ED ' = @B D= ' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*lO]%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*l]k%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*lk%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*l%(+   0(L0 pP(  pr p S  0@    p S ` `<$@ 0  H p 0޽h ? 33 2 TIMING|64.7|7|7.7 ___PPT10r .^ *y+EDF ' = @B D ' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p1%(+  0 L0 t$(  tr t S * 0@   r t S +` `  H t 0޽h ? 33___PPT10i.`+D=' = @B +  0 |(  |X | C x   x | S  xx 0  x  H | 0޽h ? 3380___PPT10.b rd`R~5)0 o'EN 1Oh+'0 `h    ,4<Join Algorithms for the Theory of Uninterpreted FunctionsSumit GulwaniSumit Gulwani279Microsoft PowerPoint@@ϐ2@ 'G g  O  y--$xx--'--%--'@"Arial-. :2 ! "Join Algorithms for the Theory of ."System<-@"Arial-. 2 ), Uninterpreted.-@"Arial-. 2 )W Functionse.-@"Arial-. 33'2 FSumit Gulwani .-@"Arial-. 332 FAAshish.-@"Arial-. 332 FRTiwari.-@"Arial-. 332 FlGeorge y.-@"Arial-. 332 FNecula.-@"Arial-.  2 NUC.-@"Arial-.  2 N-.-@"Arial-. O2 N0Berkeley SRI UC.-@"Arial-.  2 Nw-.-@"Arial-. 2 NyBerkeley.-՜.+,0T     On-screen ShowUC Berkeley-EECSBK Arialcmsy10Comic Sans MSSymbolDefault Design:Join Algorithms for the Theory of Uninterpreted FunctionsDefinition: Join in theory TExample of Joins;Motivation: Program Analysis using Abstract Interpretation-Join for Uninterpreted Functions is not easy%Relatively Complete Join: Definition$Relatively Complete Join: AlgorithmStep 1: Constructing EDAGsStep 2: Extending EDAGsStep 3: Congruence Closure)Step 4: Product Construction (Intuition))Step 4: Product Construction (Algorithm)/Future Work: Join Algorithm for other theories'Future Work: Combining Join Algorithms/Future Work: Context-sensitive Join Algorithms Conclusion  Fonts UsedDesign Template Slide Titles%_' 0Sumit GulwaniSumit Gulwani  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&()*+,-.012345689:;<=>ERoot EntrydO)Pictures8Current User7SummaryInformation('PowerPoint Document(KDocumentSummaryInformation8/