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I}OPMs0 3+(     0e0e?4Slide Image Placeholder 1      L9 0e0e?(Notes Placeholder 2 0   t"At past POPLs, we've seen fairly powerful domains for reasoning using LA+UFS and the like. When applied to this example, those domains give poor results. ... The problem arises because they can only refer to a bounded number of locations (variables for example). To state a fact about every array element, we need quantifiers." Say loop unrolled for explanatory purposes.u u u   l@5 ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPM:0 0(     0e0e?4Slide Image Placeholder 1   a   (T 0e0e?(Notes Placeholder 2 0   ;"There are many other places where quantifiers are useful."< < <   l4g ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPM0 ]UP(     0e0e?4Slide Image Placeholder 1       x 0e0e?(Notes Placeholder 2 0   "Let's look at the quantifier from the first slide. ... We want to take advantage of all the accumulated knowledge and effort from AI over the past 30 years."     l ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPM 0 ?7p(     0e0e?4Slide Image Placeholder 1       0e0e?(Notes Placeholder 2 0   Things to say: (1) Implication is ideal order, but we are incomplete; stuff like induction is not handled. (2) Second rule is analogous to function subtyping (since B in a negative position). (3) We use environment in doing subtyping rule. Say I'll assume in rest of presentation that we use one base domain. Also, all examples are with one quantifier, but in general we support many.     ll ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPM !0 p(     0e0e?4Slide Image Placeholder 1   G   Ȕ 0e0e?(Notes Placeholder 2 0   w!Issue: This sort of assumes 1 < n" " "   l, ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPM 10 p(     0e0e?4Slide Image Placeholder 1   G   p 0e0e?(Notes Placeholder 2 0   w!Issue: This sort of assumes 1 < n" " "   l ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPM 20 p(     0e0e?4Slide Image Placeholder 1   G    0e0e?(Notes Placeholder 2 0   w!Issue: This sort of assumes 1 < n" " "   lp ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPM.0 G(     0e0e?4Slide Image Placeholder 1      t 0e0e?(Notes Placeholder 2 0   N     l ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPM#$0 s(     0e0e?4Slide Image Placeholder 1    J     0e0e?(Notes Placeholder 2 0    z$Mention other transfer funcs (paper)% % %   l ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPM,&0 0|(     0e0e?4Slide Image Placeholder 1    S   H& 0e0e?(Notes Placeholder 2 0    -Can think of templates as defn of RHS domain.. . .   l ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPM7'0 P(     0e0e?4Slide Image Placeholder 1    ^   = 0e0e?(Notes Placeholder 2 0    8Slide from example with joins to show where we're going.9 9 9   l0G ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPM 40 pp(     0e0e?4Slide Image Placeholder 1    G   S 0e0e?(Notes Placeholder 2 0    w!Issue: This sort of assumes 1 < n" " "   l4V ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPM)0 (     0e0e?4Slide Image Placeholder 1   C    ]C 0e0e?(Notes Placeholder 2 0  C  %Mention near end at that it's really interesting! "One of the main contributions of this paper is to identify under-approx as the core of the quantification problem. It's used in all our transfer functions."     loC ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPMq*0 1)(     0e0e?4Slide Image Placeholder 1   C    lC 0e0e?(Notes Placeholder 2 0  C  rMention that we use the same strategy (guess and refine) for the other under-approxes we do that aren't discussed.s s s   lC ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPM730 (     0e0e?4Slide Image Placeholder 1   C ^   ,C 0e0e?(Notes Placeholder 2 0  C  8Slide from example with joins to show where we're going.9 9 9   lC ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPM+0 o(     0e0e?4Slide Image Placeholder 1   C F   C 0e0e?(Notes Placeholder 2 0  C  v Mention results are preliminary.! ! !   lC ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPM,0  U(     0e0e?4Slide Image Placeholder 1   C ,   C 0e0e?(Notes Placeholder 2 0  C  \Mention it's optional because we can generate one. Say this is one example of domain construction. Talk about related work here! "What differentiates is the parameterizability of our approach. It allows us to generate more expressive invariants than other work."     lC ?6Slide Number Placeholder 3_   f*   H  0޽h ? I}OPMr`51 oR@oV:$hXuteH Y f S +  ȯ m  D l k 1   N1Oh+'0 `h    ,4<Automatically Discovering Universally Quantified InvariantsEECSSumit Gulwani1019Microsoft Office PowerPoint@ޗa@%65H@aJhLGPg  9  y--$xx--'@"Calibri-. 92 /!Lifting Abstract Interpreters to ."System9-@"Calibri-. .2 ;Quantified Logical Domains.-@"Calibri-. "2 K2Sumit Gulwani, MSR.-@"Calibri-. $2 V4Bill McCloskey, UCB.-@"Calibri-. "2 `7Ashish Tiwari, SRI.-@"Calibri-.  2 s1{.-՜.+,0     uOn-screen Show UC Berkeleyt y CalibriArialSymbolcmsy10 Office Theme<Lifting Abstract Interpreters to Quantified Logical DomainsMotivating ExampleHow Are Quantifiers Useful?What Do Quantifiers Look Like?Universally Quantified DomainTransfer Function ExampleTransfer Function ExampleTransfer Function ExampleTransfer Function ExampleOutlineQuantifier IntroductionOutlineTransfer Function ExampleJoining QuantifiersJoining QuantifiersUnder-Approximation ExampleOutline Experiments&Quantified Domain Construction Works!  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