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A substitution s1 is more general than s2 if there exists s such that s1 = s(s2). A substitution s is a unifier for an equality e1=e2 if e1[s(y)/y] = e2[s(y)/y]. Example Consider the equality F(u) + F(v) = F(a) + F(b). {ua, vb} is a unifier for it and so is {u1, a1, vb}. The former unifier is more general than the latter. RZZZ :   3#8+8FUnification Terminology Continued & A set of unifiers {s1,& ,sk} for e1=e2 is complete if for all unifiers s of e1=e2, 9 i s.t. si is more general than s. wVUnification Type of Theories"Unitary: All equalities e1=e2 have a complete set of unifiers that is singleton. Finitary: All equalities e1=e2 have a complete set of unifiers whose cardinality is finite. Bitary: There exists an equality e1=e2 whose complete set of unifiers has 2 unifiers of the form y z1 and y z2 #5?=,RUnExamples of Bitary Theories  +Bitary: There exists an equality e1=e2 whose complete set of unifiers has 2 unifiers of the form y z1 and y z2 Commutative Functions F(F(y,y),F(z1,z2)) = F(F(y,z1),F(y,z2)) Combination of Linear Arithmetic + Uninterpreted Functions F(F(y)+F(y)) + F(F(z1)+F(z2)) = F(F(y)+F(z1)) + F(y)+F(z2)) Rt(=<=# T 6  3Summary of ResultsOutlineyUnification type of theory Assertion checking algorithm (unitary/finitary theories) coNP-hardness (bitary theories) V : ":" >C   3Connection between Assertion Checking & Unification An assertion e1 = e2 holds at a program point p iff the assertion Unif(e1=e2) holds at p. ] 1)Assertion Checking AlgorithmBackward analysis strengthened with Unification Perform weakest precondition computation. At each step replace the formula y by Unif(y), which is a stronger and simpler formula. Termination (reach fixpoint across loops)? Yes, because of unifier computations. PTIME for unitary theories (no disequality guards). Bounded for finitary theories.t0+y0K -+yPBU  /Advantage of Backward Analysis with Unification00Handling equality guards OutlineyUnification type of theory Assertion checking algorithm (unitary/finitary theories) coNP-hardness (bitary theories) jV  ! V  >C   /Reducing Unsatisfiability to Assertion Checking Py: boolean 3-SAT instance with m clauses IsUnsatisfiable(y) { for j=1 to m cj := F; for i=1 to k do if (*) 8 j s.t. var i occurs positively in clause j, cj := T; else 8 j s.t. var i occurs negatively in clause j, cj := T; Assert (c1=F c2=F & cm=F); } )'2..  6""*Encoding disjunction The check c1=F c2=F can be encoded by some appropriate assertion e1=e2 in a bitary theory. The above trick can be recursively applied to construct an assertion that encodes c1=F c2=F & cm=F^i 1iOs ConclusionComplexity of assertion checking depends on the unification type of the theory of program expressions: Unitary (PTIME), Bitary (coNP-hard), Finitary (Decidable) The assertion checking algorithm is based on backward analysis strengthened with unification. For some infinite-height abstract domains, a (goal-driven) backward analysis is more efficient than forward analysis. Use of unification is yet another non-trivial use of theorem proving in program analysis.N >x= Proof of Termination9At each program point, the proof obligation has the form: DiscussionMThe assertion checking algorithm is based on backward analysis strengthened with unification. Backward Analysis vs. Forward Analysis Use of Theorem Proving in Program Analysis Combining (Forward) Abstract Interpreters [PLDI 06]: using an extension of Nelson-Oppen combination method. This paper: Backward Analysis using Unification_S- SK& E/px  P-  0` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@z?" dd@  " @ ` n?" dd@   @@``PR    @ ` ` p>>f^N0 >(    6 "@  T Click to edit Master title style! !$  0e "0  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0e "` e @*   0p e "< l  e H*/17   0 e "  e B* ~B  Hp?"@@H  0޽h ? ̙33y___PPT10Y+D=' = @B + Default Design00 @(    N ir`r` )  i x*  V++VV  Nir`r`  ) i z*  V++VVd  c $ ?w  i4  N4ir`r`  ZF i RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  Tdir`r` @  i x*  V++VV  Tir`r`  @ i z*  V++VVH  0~dm.F ? ̙3380___PPT10.47L P 0(    NX^ir`r` )  i ^*  V++VV  NDr`r`  ) i `*  V++VV  Twir`r` @  i ^*  V++VV  TXzir`r`  @ i `*  V++VVH  0~dm.F ? ̙3380___PPT10.47UC0:^N0 RJ  (   x  c $a`  a    0a "6 ?=  a R  s *\; ; \V  0޽h ?"` ̙33y___PPT10Y+D=' = @B +  0^N0  3&r(  r  S ' I HH  0޽h ?       !#$ ̙33___PPT10i.^|+D=' = @B + 0W^N0 56,( (  ,r , S @a=@  a r , S a a  , <da &^ N Assignment( 0    , 6aa C"? g [ y := e80 2R , s * R , s *   , <fa  |Non-det Conditional*0   , <aa  = 0 R , s * j , BGS|H IS|A d , <G1HI1A +  , < a u  K*.0  ^ , 6  , 6daFt  >True0 2 , 6)j+D=' = @B + 0ǐ^N0 p4$(  r  S ,ʾ@   r  S ʾ0  H  0޽h ? ̙33___PPT10i.n+D=' = @B + 0^N0 6P$(  Pr P S ־@   r P S ־d0  H P 0޽h ? ̙33___PPT10i.P =ȭ+D=' = @B +F  0Q^N0 E = 6 h (  hr h S ?@   r h S 0@r<  t h BA ?"6@ NNN?NZ   y = si(y)f0 20 0  h BJ ?"6@ NNN?NGtA Ai=1"0 2   h BM ?"6@ NNN?NE? ?k"0 2   h BxR ?"6@ NNN?N' ! ?y"0 2  6  h 0U 6 WDB &In each successive loop iteration, above formula becomes stronger. We prove this cannot happen indefinitely: Assign the following measure to the above formula { # of conjuncts representing unifier si | i=1 to k } Show this measure decreases in some well-founded ordering.nP2P9P;Pm 2& ;H h 0޽h ? ̙33___PPT10i.PN;'+D=' = @B + 0f^N0 8$(  r  S H6@   r  S  70  H  0޽h ? ̙33___PPT10i.8`+D=' = @B +0 0 4(   d  c $w   a  s * Sumit Gulwani.-@BComic Sans MS-. 02 A*Microsoft Research, Redmond.-@BComic Sans MS-. 2 N? Ashish Tiwari.-@BComic Sans MS-.  2 TKSRI.-՜.+,0H    On-screen Show~ Times New RomanComic Sans MScmsy10ArialSymbolcmmi10 WingdingsDefault DesignAssertion Checking UnifiedExample+Abstract Program Model / Problem StatementSummary of ResultsOutlineUnification Terminology$Unification Terminology Continued Unification Type of TheoriesExamples of Bitary TheoriesSummary of ResultsOutline4Connection between Assertion Checking & UnificationAssertion Checking Algorithm0Advantage of Backward Analysis with UnificationHandling equality guardsOutline0Reducing Unsatisfiability to Assertion CheckingEncoding disjunction ConclusionProof of Termination Discussion  Fonts UsedDesign Template Slide Titles*_y~Microsoft EmployeeMicrosoft Employee  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?ABCDEFGIJKLMNOQRSTUVW\Root EntrydO)Current UserPSummaryInformation(@PowerPoint Document(~DocumentSummaryInformation8H