1. 1 Automatic Predicate Abstraction of C Programs Parts of the slides are from http://research.microsoft.com/slam/presentations/pldi2001.ppt

2. 2 Predicate abstraction by C2BP program P Boolean program BP(P,E) C2BP predicates E

3. 3 C2BP: Predicate Abstraction for C Programs Given P : a C program E = {e 1,...,e n } : set of C boolean expressions over the variables in P –no side effects, no procedure calls Produces a boolean program B same control-flow structure as P only vars are 3-valued booleans {{e 1 },...,{e n }} properties true of B are true of P

4. 4 Assumptions Operates on intermediate representation of C language –CFG : if-then-else statements and gotos –expressions : side-effect free, no short-circuit evaluation, no multiple dereferences(e.g. **p) –Function call occurs at the top-most level z=x+f(y);  w=f(y); z=x+w; –Return statement is in the following form: return r; (where r is a local variable) Uses a logical memory model –for all i: p+i = p (where p is a pointer)

5. 5 Weak/Strong predicates p is weaker than q iff q ) p For example, –‘T’ is weaker than ‘F’ since ‘F’ ) ‘T’ –x < 5 is weaker than x < 4 since x < 4 ) x < 5 ‘Weak’ means ‘easy to be satisfied’ –‘T’ is the weakest predicate. ‘Strong’ means ‘hard to be satisfied’ –‘F’ is the strongest predicate.

6. 6 Precondition p 2 P(s, q) iff –8 M 2 States : M 2 [[p]] ) ([[s]] M) 2 [[q]] (where [[p]] denotes the states that satisfy p and [[s]] denotes the transfer function) P(s, q) = the set of predicates whose truth before s entails the truth of q after s terminates Example: –(y<3) 2 P(x=y+1, x<20) –(y<19) 2 P(x=y+1, x<20) It is easy to see that if p,p’ 2 P(s,q) then p Ç p’ 2 P(s,q) –i.e, P(s,q) is closed under Ç –Note that p Ç p’ is weaker than p and p’

7. 7 Weakest Precondition WP(s, q) = –the weakest predicate p such that p 2 P(s, q) –equivalently, Ç (P(s,q)) For assignment statement, –WP(x = e, q) = q[e/x] Example: –WP(x = y+1, x<20) = (x<20)[(y+1)/x] = (y+1)<20 = (y<19) –(y<3) 2 P(x = y+1, x<20) but (y<3) ) (y<19)

8. 8 Predicate strengthening Example –Given only E={ x<19, y==7, y==z, z<8}, –WP(x = y+1, x<19) = y<18 -> we don’t have (y<18) as a predicate! panic? –Find the weakest boolean expression consisting of E that implies (y<18)! How? –S E (p) = Ç { e | e is a boolean expression over E, e ) p } –But by disjunctive normal form theorem… –c 1 Æ … Æ c n is a cube over E iff (c i = e i or c i = : e i ) and e i 2 E –S E (p) = Ç { e | e is a cube over E, e ) p } –S E (y<18) = (y==7) Ç (y==z Æ z<8)

9. 9 Strengthening & Weakening p S E (p) W E (p)

10. 10 Translating Assignment Statements In general, –“WP(s, e i ) is true before s” implies “predicate e i is true after s” –“WP(s, : e i ) is true before s” implies “predicate e i is false after s “ But in some cases, WP(s, e i ) is not expressible using E The assignment statement “x=e;” is translated to –{e i } = {S E (WP(x=e, e i ))} ? true : {S E (WP(x=e, : e i ))} ? false : *; for each e i 2 E

11. 11 Translation example Translating “x = y+1;” –Given E={ x<19, y==7, y==z, z<8}, –S E (WP(x=y+1, x<19)) = S E (y<18) = (y==7) Ç (y==z Æ z<8) –S E (WP(x=y+1, : (x<19))) = S E (WP(x=y+1, x ¸ 20)) = S E (y ¸ 19) = false –For {x<19}, {x<19} = ( {y==7} || ( {y==z} && {z<8} ) ) ? true : false ? false : *; –Similar for {y==7}, {y==z}, {z<8}

12. 12 Handling Pointers Statement : *p = y; predicates E = { x==5 } Weakest Precondition: –WP(*p=y, x==5) = (x==5)[y/*p] = (x==5)??? –What if *p and x alias? Correct Weakest Precondition: –WP(*p=y, x==5) = (p==&x Æ y==5) Ç (p!=&x Æ x==5) Uses Das’s pointer analysis to prune infeasible alias scenarios

13. 13 Translating conditional statements (1/2) if (e) { … } else { … } is translated to if (*) { assume(e); … } else { assume( : e); … } But what if ‘e’ is not expressible in terms of the predicates in E? Then find a condition weaker than ‘e’ such that it is expressible in terms of E

14. 14 Predicate weakening W E (p) = the strongest predicate among those which are expressible in terms of ‘E’ and are implied by ‘p’ W E (p) = : (S E ( : p))

15. 15 Translating conditional statements (2/2) if (e) { … } else { … } is translated to if (*) { assume(W E (e)); … } else { assume(W E ( : e)); … } Example –E = { x>3, x==10 } –if(x>5) { … } else { … }  –if(*) { assume(x>3); … } else { assume(!(x==10)); … }

16. 16 Handling Procedures Procedures abstracted in two passes: –a signature is produced for each procedure in isolation –procedure calls are abstracted using the callees’ signatures only

17. 17 Global/Local predicates Each predicate in E is annotated as being either global or local to a particular procedure e i 2 E is … –global iff e i refers to global variables only –local to procedure R iff e i refers to a local variable of R (e i cannot refer to local variables of another procedure due to scoping)

18. 18 Formal/Return predicates e i 2 E is … –a formal predicate of procedure R iff e i is a local predicate of R and doesn’t refer to any local variables of R –a return predicate of procedure R iff one of the following conditions is true: (1) e i refers to return variable ‘r’ of R and doesn’t refer to any local variable except ‘r’ (2) e i is a formal predicate and e i refers to a global var or to a pointer dereference of formal parameter of R

19. 19 Example program int bar(int *q,int y,int z) { int m1,m2; m1 = y+1; return m1; } void foo(int *p, int x) { int r; r=bar(p,x,*p); } bar { y>=0; // formal *q<=y; // formal, return y==m1; // return } foo { *p<=0; // formal, return x==0; // formal r==0 // local }

20. 20 Translating procedure “bar” int bar(int *q,int y,int z) { int m1,m2; m1 = y+1; return m1; } bar({y>=0}, {*q<=y}) { bool {y==m1}; {y==m1} = false ? true : !{y==m1} ? false : *; } bar { y>=0; // formal *q<=y; // formal, return y==m1; // return }

21. 21 Translating procedure “foo” “foo” invokes “bar” but only the signature of “bar” is needed to handle the procedure call Translated form of “foo”: foo({*p<=0}, {x==0}) { bool {r==0}; … }

22. 22 Signature of procedure “bar” int bar(int *q,int y,int z) { int m1,m2; m1 = y+1; return m1; } bar { y>=0; // formal *q<=y; // formal, return y==m1; // return } Signature of “bar”: int bar(int *q,int y,int z); Takes (y>=0), (*q<=y) as formal predicates Returns (*q<=y), (y==m1) as return predicates Return var = m1

23. 23 Resolving/Strengthening formal predicates r=bar(p,x,*p); (in procedure foo) Formal predicates of “bar” (y>=0)[p/q, x/y, *p/z] = (x>=0) –S foo (x>=0) = (x==0) –S foo ( : (x>=0)) = false (*q<=y)[p/q, x/y, *p/z] = (*p<=x) –S foo (*p<=x) = (*p<=0) Æ (x==0) –S foo ( : (*p<=x)) = : (*p<=0) Æ (x==0) Signature of “bar”: int bar(int *q,int y,int z); Takes (y>=0), (*q<=y) Returns (*q<=y), (y==m1) Return var = m1 Predicates of “foo”: *p<=0; x==0; r==0

24. 24 Translated procedure invocation prm1 = {x==0} ? true : false ? false : *; prm2 = {*p<=0}&&{x==0} ? true : !{*p<=0}&&{x==0} ? false : *; r1,r2 = bar(prm1,prm2); “bar” returns the resulting value of predicates (*q<=y) and (y==m1) Signature of “bar”: int bar(int *q,int y,int z); Takes (y>=0), (*q<=y) Returns (*q<=y), (y==m1) Return var = m1 Predicates of “foo”: *p<=0; x==0; r==0

25. 25 Resolving return predicates r=bar(p,x,*p); (in procedure foo) Return predicates of “bar” (*q<=y)[r/m1, p/q, x/y, *p/z] = (*p<=x) (y==m1)[r/m1, p/q, x/y, *p/z] = (x==r) Let R = { (*p<=x), (x==r) } Predicates of “foo” to be updated –(*p<=0) (“p” is passed to “bar”) –(r==0) (“r” is assigned by proc call) Let U foo = { (*p<=x), (r==0) } Signature of “bar”: int bar(int *q,int y,int z); Takes (y>=0), (*q<=y) Returns (*q<=y), (y==m1) Return var = m1 Predicates of “foo”: *p<=0; x==0; r==0

26. 26 How to update caller’s predicates r=bar(p,x,*p); (in procedure foo) R = { (*p<=x), (x==r) } U foo = { (*p<=0), (r==0) } Let T = R [ (E foo \ U foo ) = { (*p<=x), (x==r), (x==0) } Strengthening U foo over T –S T (*p<=0) = (*p<=x) Æ (x==0) –S T ( : (*p<=0)) = : (*p<=x) Æ (x==0) –S T (r==0) = (x==r) Æ (x==0) –S T ( : (r==0)) = : (x==r) Æ (x==0) Signature of “bar”: int bar(int *q,int y,int z); Takes (y>=0), (*q<=y) Returns (*q<=y), (y==m1) Return var = m1 Predicates of “foo”: *p<=0; x==0; r==0

27. 27 Translated procedure return …; r1,r2 = bar(prm1,prm2); r1,r2 represents {*p<=x}, {x==r} U foo strengthened over T –S T (*p<=0) = (*p<=x) Æ (x==0) –S T ( : (*p<=0)) = : (*p<=x) Æ (x==0) –S T (r==0) = (x==r) Æ (x==0) –S T ( : (r==0)) = : (x==r) Æ (x==0) Updating predicates of U foo –{*p<=0} = r1&&{x==0} : true ? !r1&&{x==0} : false ? *; –{r==0} = r2&&{x==0} : true ? !r2&&{x==0} : false ? *; Signature of “bar”: int bar(int *q,int y,int z); Takes (y>=0), (*q<=y) Returns (*q<=y), (y==m1) Return var = m1 Predicates of “foo”: *p<=0; x==0; r==0

28. 28 Finally… foo({*p<=0}, {x==0}) { bool {r==0}; bool prm1, prm2, r1, r2; prm1 = {x==0} ? true : false ? false : *; prm2 = {*p<=0}&&{x==0} ? true : !{*p<=0}&&{x==0} ? false : *; r1,r2 = bar(prm1,prm2); {*p<=0} = r1&&(x==0) : true ? !r1&&(x==0) : false ? *; {r==0} = r2&&(x==0) : true ? !r2&&(x==0) : false ? *; } Signature of “bar”: int bar(int *q,int y,int z); Takes (y>=0), (*q<=y) Returns (*q<=y), (y==m1) Return var = m1 Predicates of “foo”: *p<=0; x==0; r==0

29. 29 Formal Properties of C2BP soundness –B has a superset of the feasible paths in P –If {e i } is true (false) at some point on a path in B, then e i is true (false) at that point along a corresponding path in P complexity –linear in size of program –exponential in number of predicates

30. 30 Conclusions Contributions –C2BP handles the constructs of C pointers, recursion, structs, casts modular abstraction of procedures –Provably sound Defects –Doesn’t use physical memory model