Abstract Interpretation and the Heap

Transcription

1 Absrac Ierpreai ad he Heap Cmpuer Sciece ad Arificial Ielligece Labrary MIT Wih slides ad eamples by Mly Sagiv. Used wih permissi. Nv 18, 2015 Nvember 18,

2 Recap: Cllecig Semaics Cmpue fr each prgram he rue se f saes ha ca ccur a each pi 2

3 Eample: Cllecig Ierpreai empy reur = =mallc(..); e=; = NULL T F 3

4 Eample: Absrac Ierpreai empy reur = =mallc(..); e=; = NULL T F 4

5 Ccree Ierpreai A slighly differe view f he sae - Ev: Var Values - Oe map per field - Field: Lc Values - Values = Lc Ams Eample - Ev =[ 30, p 79] - Fields: e = 30 40, 40 50, 50 79, val=[30 1, 40 2, 50 3, 79 4, 90 5] p 5

6 The TVLA Apprach Represe he sre wih lgical predicaes - The d absraci hese predicaes - A apprach buildig absracis isead f a sigle e Lcais Idividuals Prgram variables Uary predicaes Fields Biary predicaes Eample - U = {u1, u2, u3, u4, u5} - = {u1}, p = {u3} - = {<u1, u2>, <u2, u3>, <u3, u4>, <u4, u5>} u1 u2 u3 u4 u5 p 6

7 Impra ai Trasiive clsure f a biary predicae (u, v) - u, v u = v ( w. u, w w, v ) - + u, v ( w. u, w w, v ) 7

8 Ccree Ierpreai Rules Sae: - : predicae fr variable. - : predicae fr e field s, = (, ) - = ull = v. v = 0 - = mallc() = v. v = IsNew(v) - = y = v. v = y(v) - = y. e = v. v = w. y w (w, v) -. e = y = v w. v, w = v v, w ( v y w ) 8

9 Saig prgram prperies pis a acyclic lis - v w. v v, w + (w, v) The heap reverses he lis pied a by i - v w r. v v, w ( w, r r, w ) 9

10 Caical Absraci Cver lgical srucures f ubuded size i buded size Guaraees ha umber f lgical srucures i every prgram is fiie Every firs-rder frmula ca be cservaively ierpreed Same idea we eplred las ime, bu revisied i Three Valued Lgic 10

11 Kleee Three-Valued Lgic 1: 0: 1/2: A True False Ukw ji semi-laice: 0 1 = 1/2 1/2 Ifrmai rder Lgical rder 11

12 Blea Cecives [Kleee] 0 1/ /2 0 1/2 1/ / / /2 1 1/2 1/2 1/

13 Key idea Predicaes describig prgram sae are w predicaes i 3-Valued Lgic - Le U be he se f idividuals i he ccree dmai (peially ifiie) - Le U be he se f idividuals i he absrac dmai (fiie) - Le f: U U - The a predicae p B ver U ca be absraced p S ver U as fllws p S u 1, u k = p B u 1,, u k f u 1 = u 1,, f u k = u k } - Sice U is buded, p S ca be represeed wih a able 13

14 Caical Absraci = NULL; while ( ) d { } = mallc(); e=; = (u1,u2)=1 (u1,u3)=0 (u2,u3)=1 (u3,u3)=0 u1 u1 s(u1,u23)=1/2 (u23,u23)=1/2 u2 u2,3 u3 14

15 Big Idea Yu ca icrease precisi by rackig addiial predicaes 15

16 Cycliciy predicae c[]() = v 1,v 2 : (v 1 ) * (v 1,v 2 ) + (v 2, v 2 ) c[]()=0 u 1 u 2 u c[]()=0 u1 u 2.. Frm he absrac graph ale we ca ell here are cycles, bu he predicae ells us his is he case. 16

17 Cycliciy predicae c[]() = v 1,v 2 : (v 1 ) * (v 1,v 2 ) + (v 2, v 2 ) c[]()=1 u 1 u 2 u c[]()=1 u1 u

18 Heap Sharig predicae is(v) = v 1,v 2 : (v 1,v) (v 2,v) v 1 v 2 is(v)=0 is(v)=0 is(v)=0 u 1 u 2 u u1 u 2.. is(v)=0 is(v)=0 18

19 Heap Sharig predicae is(v) = v 1,v 2 : (v 1,v) (v 2,v) v 1 v 2 is(v)=0 is(v)=1 is(v)=0 u 1 u 2 u u1 u2 u 3.. is(v)=0 is(v)=1 is(v)=0 19

20 Reachabiliy predicae [](v1, v2) = * (v1,v2) [] [] [] u 1 u 2 [] [] [] u [] u1 u 2.. [] [] 20

21 Addiial Isrumeai predicaes c iorder(v) reachable-frm-variable-(v) fb (v) = v 1 : f(v, v 1 ) b(v 1, v) ree(v) dag(v) = v 1 : (v, v 1 ) dle(v,v 1 ) 21

22 Isrumeai (Summary) Refies he absraci Adds glbal ivarias Bu requires updae-frmulas (geeraed aumaically i TVLA2) 22

23 Parial Ccreizai (fcus) Helpful i makig rasfer fucis mre precise Epad a absrac heap i a clleci f mre ccree 23

24 Parial Ccreizai Based Trasfrmer (s=tp ) Tp r Tp r Tp Absrac Semaics Tp s r Tp s r Tp Tp r Tp r Tp r Tp Tp r Tp r Tp r Tp Parial Ccreizai s (v) = v1: Tp(v1) (v1,v) Caical Absraci s Absrac Semaics Tp r Tp s r Tp Tp r Tp r Tp Tp r Tp r Tp r Tp 24

25 MIT OpeCurseWare hp://cw.mi.edu Fudameals f Prgram Aalysis Fall 2015 Fr ifrmai abu ciig hese maerials r ur Terms f Use, visi: hp://cw.mi.edu/erms.