1. Checking correctness properties of object-oriented programs K. Rustan M. Leino Microsoft Research, Redmond, WA Lecture 3 EEF summer school on Specification, Refinement, and Verification 23 Aug 2002, Turku, Finland

2. Review of methods class T { method m(this, x, y) requires Pre modifies w ensures Post } class T { method m(this, x, y) requires Pre modifies w ensures Post } class U <: T { mimpl m(this, x, y) is Body } class U <: T { mimpl m(this, x, y) is Body }

3. Review of methods Turku trivia This Side The Other Side class T { method m(this, x, y) requires Pre modifies w ensures Post } class T { method m(this, x, y) requires Pre modifies w ensures Post } class U <: T { mimpl m(this, x, y) is Body } class U <: T { mimpl m(this, x, y) is Body }

4. this self Review of methods class T { method m(this, x, y) requires Pre modifies w ensures Post } class T { method m(this, x, y) requires Pre modifies w ensures Post } class U <: T { mimpl m(this, x, y) is Body } class U <: T { mimpl m(this, x, y) is Body }

5. class Counter { method inc(this) … method dec(this) … … } class SimpleCounter <: Counter { field x: int mimpl inc(this) is this.x := this.x + 1 mimpl dec(this) is this.x := this.x – 1 … } class OpCounter <: Counter { field i: int field d: int mimpl inc(this) is this.i := this.i + 1 mimpl dec(this) is this.d := this.d + 1 … } Subtyping What should be the specifications of methods inc and dec?

6. Abstract variables class Counter { abstract field n: int method inc(this) requires true modifies this.n ensures this.n = this.n 0 + 1 method dec(this) requires true modifies this.n ensures this.n = this.n 0 - 1 … } class SimpleCounter <: Counter { field x: int rep this.n this.x mimpl inc(this) is this.x := this.x + 1 mimpl dec(this) is this.x := this.x – 1 … } class OpCounter <: Counter { field i: int field d: int rep this.n this.i – this.d mimpl inc(this) is this.i := this.i + 1 mimpl dec(this) is this.d := this.d + 1 … } Two questions: How do we reason about the abstract expressions? What do modifies clauses with abstract variables mean?

7. Shape of verification condition UnivBackPred /\ BackPred(Program) ==>VC(MethodImpl)

8. 2-dimensional heap $ denotes current value of heap $ denotes current value of heap (o,f) denotes a heap location (o,f) denotes a heap location tr[ o.f ] = get($, o, f) tr[ o.f ] = get($, o, f) tr[ o.f 0 ] = get($ 0, o, f) tr[ o.f 0 ] = get($ 0, o, f) get($, o, f) = get($, o, f) = select($, o, f)if f is concrete absfun(o, f)($)if f is abstract

9. Meaning of rep class T { rep this.a E } gives rise to the following program-specific axiom: ( this ::this null /\ typeof(this) absfun(this, a) = (λ $ :: tr[ E ])) class T { rep this.a E } gives rise to the following program-specific axiom: ( this ::this null /\ typeof(this) absfun(this, a) = (λ $ :: tr[ E ])) Example: class OpCounter { rep this.n this.i – this.d } gives rise to: ( this ::this null /\ typeof(this) absfun(this, n) = (λ $ :: get($,this,i) – get($,this,d))) Example: class OpCounter { rep this.n this.i – this.d } gives rise to: ( this ::this null /\ typeof(this) absfun(this, n) = (λ $ :: get($,this,i) – get($,this,d)))

10. Proving postconditions class Counter { abstract field n: int method inc(this) requires true modifies this.n ensures this.n = this.n 0 + 1 method dec(this) requires true modifies this.n ensures this.n = this.n 0 - 1 … } class SimpleCounter <: Counter { field x: int rep this.n this.x mimpl inc(this) is this.x := this.x + 1 mimpl dec(this) is this.x := this.x – 1 … } class OpCounter <: Counter { field i: int field d: int rep this.n this.i – this.d mimpl inc(this) is this.i := this.i + 1 mimpl dec(this) is this.d := this.d + 1 … }

11. Modifies clauses class Counter { abstract field n: int method inc(this) requires true modifies this.n ensures this.n = this.n 0 + 1 method dec(this) requires true modifies this.n ensures this.n = this.n 0 - 1 … } class SimpleCounter <: Counter { field x: int rep this.n this.x mimpl inc(this) is this.x := this.x + 1 mimpl dec(this) is this.x := this.x – 1 … } class OpCounter <: Counter { field i: int field d: int rep this.n this.i – this.d mimpl inc(this) is this.i := this.i + 1 mimpl dec(this) is this.d := this.d + 1 … }

12. Dependency declarations class Counter { abstract field n: int method inc(this) requires true modifies this.n ensures this.n = this.n 0 + 1 method dec(this) requires true modifies this.n ensures this.n = this.n 0 - 1 … } class SimpleCounter <: Counter { field x: int in n rep this.n this.x mimpl inc(this) is this.x := this.x + 1 mimpl dec(this) is this.x := this.x – 1 … } class OpCounter <: Counter { field i: int in n field d: int in n rep this.n this.i – this.d mimpl inc(this) is this.i := this.i + 1 mimpl dec(this) is this.d := this.d + 1 … }

13. Dependency relation Relation (o,a) –S (p,b) states that in heap S, the value at location (o,a) may depend on the value at location (p,b) Relation (o,a) –S (p,b) states that in heap S, the value at location (o,a) may depend on the value at location (p,b) –S is a reflexive, transitive, and antisymmetric –S is a reflexive, transitive, and antisymmetric For every concrete field f: ( o, S, S, E, p, x :: S = store(S, o, f, E) ==> get(S, p, x) = get(S, p, x) \/(p,x) –S (o,f)) For every concrete field f: ( o, S, S, E, p, x :: S = store(S, o, f, E) ==> get(S, p, x) = get(S, p, x) \/(p,x) –S (o,f))

14. Dependency axioms Declaration field f in a Declaration field f in a allows this.a to be a function of this.f, and allows this.a to be a function of this.f, and gives rise to the following program-specific axiom: gives rise to the following program-specific axiom: ( o, S :: (o,a) –S (o,f)) ( o, S :: (o,a) –S (o,f))

15. More dependencies class BigOpCounter <: Counter { field i: BigNumber maps x into n field d: BigNumber maps x into n rep this.n this.i.x – this.d.x mimpl inc(this) is this.i.add(1) mimpl dec(this) is this.d.add(1) … } class BigNumber { abstract field x: Z method add(this, v) requires true modifies this.x ensures this.x = this.x 0 + v … }

16. More dependency axioms Declaration field f maps x into a Declaration field f maps x into a allows this.a to be a function of this.f.x, and allows this.a to be a function of this.f.x, and gives rise to the following program-specific axiom: gives rise to the following program-specific axiom: ( o, S :: (o,a) –S (get(S,o,f), x)) ( o, S :: (o,a) –S (get(S,o,f), x))

17. Meaning of modifies modifies this.a, … means: modifies $ ensures ( o, f ::get($,o,f) = get($ 0,o,f) \/(this,a) –$ 0 (o,f) \/(o,f) –$ 0 (this,a) \/…) modifies this.a, … means: modifies $ ensures ( o, f ::get($,o,f) = get($ 0,o,f) \/(this,a) –$ 0 (o,f) \/(o,f) –$ 0 (this,a) \/…)

18. Negative dependency information For any concrete field f: ( o, X, S, p :: (o,f) –S (p,X) o=p /\ f=X) For any concrete field f: ( o, X, S, p :: (o,f) –S (p,X) o=p /\ f=X) Each declaration field f in a, b, c gives rise to the following program-specific axiom: ( o, X, S, p :: (o,X) –S (p,f) ==> (o=p /\ X=f) \/(o,X) –S (p,a) \/(o,X) –S (p,b) \/(o,X) –S (p,c)) Each declaration field f in a, b, c gives rise to the following program-specific axiom: ( o, X, S, p :: (o,X) –S (p,f) ==> (o=p /\ X=f) \/(o,X) –S (p,a) \/(o,X) –S (p,b) \/(o,X) –S (p,c)) Similar axioms are added for maps into Similar axioms are added for maps into

19. Proving modifies clauses class Counter { abstract field n: int method inc(this) requires true modifies this.n ensures this.n = this.n 0 + 1 method dec(this) requires true modifies this.n ensures this.n = this.n 0 - 1 … } class SimpleCounter <: Counter { field x: int rep this.n this.x mimpl inc(this) is this.x := this.x + 1 mimpl dec(this) is this.x := this.x – 1 … } class OpCounter <: Counter { field i: int field d: int rep this.n this.i – this.d mimpl inc(this) is this.i := this.i + 1 mimpl dec(this) is this.d := this.d + 1 … }

20. Alias confinement It seems that to support maps into soundly, one needs some form of alias confinement (For details, see Using data groups to specify and check side effects, Leino, Poetzsch-Heffter, Zhou, PLDI02.) It seems that to support maps into soundly, one needs some form of alias confinement (For details, see Using data groups to specify and check side effects, Leino, Poetzsch-Heffter, Zhou, PLDI02.)

21. Exercise Prove the following program correct: Prove the following program correct: class Counter { abstract field n: int method inc(this) requires true modifies this.n ensures this.n = this.n 0 + 1 method adjust(this, v) requires 0 v modifies this.n ensures this.n = this.n 0 + v mimpl adjust(this, v) is if n=0 then skip else this.inc(); this.adjust(n-1) end }

22. Example: valid/state paradigm class T { abstract field valid: bool abstract field state: unit method init(this, x, y, z) requires P(x,y,z) modifies this.valid, this.state ensures this.valid method operation0(this, x, y, z) requires this.valid /\ R0(x,y,z) modifies this.state ensures true method operation1(this, x, y, z) requires this.valid /\ R1(x,y,z) modifies this.state ensures true … method close(this) requires this.valid modifies this.valid, this.state ensures true field f: int in valid, state field g: int in valid, state field h: int in state rep this.valid this.f < this.g … }

23. Example: visitor class Container { field x: int field y: int method put(this, z) requires true modifies this.x, this.y ensures (this.x = z /\ this.y = this.y 0 ) \/ (this.x = this.x 0 /\ this.y = z) method pick(this) returns (z) requires true modifies ε ensures z = this.x \/ z = this.y method map(this, visitor) … mimpl map(this, visitor) is visitor.apply(this.x); visitor.apply(this.y) } class Visitor { method apply(this, z) … }

24. Example: visitor class Container { field x: int field y: int … method map(this, visitor) requires visitor.valid modifies visitor.state ensures true mimpl map(this, visitor) is visitor.apply(this.x); visitor.apply(this.y) } class Visitor { abstract field valid: bool abstract field state: unit method apply(this, z) requires this.valid modifies this.state ensures true }

25. Information hiding and modular verification Instead of proving Verif(Method, Program): Instead of proving Verif(Method, Program): UnivBackPred /\ BackPred(Program) ==> VC(MethodImpl) prove Verif(Method, Module): UnivBackPred /\ BackPred(Module) ==> VC(MethodImpl) where MethodImpl Module Program

26. Modular soundness Well-formed(Module) /\ Verif(Method, Module) ==> Verif(Method, Program) Well-formed(Module) /\ Verif(Method, Module) ==> Verif(Method, Program)

27. Summary Method specifications apply unchanged to all method overrides Method specifications apply unchanged to all method overrides Data abstraction (abstract variables, abstraction dependencies, rep functions) gives subclasses the ability to operate differently Data abstraction (abstract variables, abstraction dependencies, rep functions) gives subclasses the ability to operate differently Information hiding poses restrictions on formal system for data abstraction, likely including alias confinement Information hiding poses restrictions on formal system for data abstraction, likely including alias confinement

28. References C.A.R. Hoare. Proof of correctness of data representations. In Acta Informatica 1(4), pp. 271-281, Springer, 1972. C.A.R. Hoare. Proof of correctness of data representations. In Acta Informatica 1(4), pp. 271-281, Springer, 1972. K. Rustan M. Leino. Toward Reliable Modular Programs. PhD thesis, California Institute of Technology. Technical Report Caltech- CS-TR-95-03, Caltech, 1995. K. Rustan M. Leino. Toward Reliable Modular Programs. PhD thesis, California Institute of Technology. Technical Report Caltech- CS-TR-95-03, Caltech, 1995. K. Rustan M. Leino and Greg Nelson. Data abstraction and information hiding. Research Report 160, Compaq SRC, Nov. 2000. To appear in TOPLAS. K. Rustan M. Leino and Greg Nelson. Data abstraction and information hiding. Research Report 160, Compaq SRC, Nov. 2000. To appear in TOPLAS. Peter Müller. Modular Specification and Verification of Object- Oriented Programs. PhD thesis, FernUniversität Hagen. Volume 2262 of LNCS, Springer, 2002. Peter Müller. Modular Specification and Verification of Object- Oriented Programs. PhD thesis, FernUniversität Hagen. Volume 2262 of LNCS, Springer, 2002. K. Rustan M. Leino. Data groups: Specifying the modification of extended state. In OOPSLA 98, pp. 144-153, ACM, 1998. K. Rustan M. Leino. Data groups: Specifying the modification of extended state. In OOPSLA 98, pp. 144-153, ACM, 1998. K. Rustan M. Leino, Arnd Poetzsch-Heffter, and Yunhong Zhou. Using data groups to specify and check side effects. In PLDI 02, SIGPLAN Notices 37(5), pp. 246-257, ACM, May 2002. K. Rustan M. Leino, Arnd Poetzsch-Heffter, and Yunhong Zhou. Using data groups to specify and check side effects. In PLDI 02, SIGPLAN Notices 37(5), pp. 246-257, ACM, May 2002.