Technologies for finding errorsTechnologies for finding errorsin object-oriented softwarein object-oriented software

Technologies for finding errorsTechnologies for finding errorsin object-oriented softwarein object-oriented software

K. Rustan M. LeinoMicrosoft Research, Redmond, WA

Lecture 1Summer school on Formal Models of Software2 Sep 2003, Tunis, Tunisia

Review: Tool architectureSource programSource program

Verification conditionVerification condition

Counterexample contextCounterexample context

Warning messagesWarning messages

Automatic theorem proverAutomatic theorem prover

Post processorPost processor

Sugared commandSugared command

Primitive commandPrimitive command

Passive commandPassive commandTra

nsl

ato

rTra

nsl

ato

rFocus today

Commands and their possible outcomes• Normal termination

– terminates normally in some state

• Erroneous termination– goes wrong, crashes the computer

• Non-termination– diverges, fails to terminates, results in infinite

recursion

• Miraculous termination– fails to start, blocks

(partial/miraculous commands)

you breach

contract,demon wins

demon breaches contract,you win!

Commands

C ::= w := E| assert P| assume P| var w in C end| C0 ; C1

| C0 [] C1

Semantics• Hoare logic

– {P} C {R} says that if command C is started in (a state satisfying) P, then:• C does not go wrong, and• if C terminates normally, then it

terminates in (a state satisfying) R

• Weakest preconditions– for a given C and R, the weakest P

satisfying {P} C {R}– written wp(C, R) or simply C.R

Command semantics—assignment

• evaluate E and change value of w to E• (w := E).R ≡ R[w := E]

• (x := x + 1).(x ≦ 10)≡ x+1 ≦ 10≡ x < 10

• (x := 15).(x ≦ 10)≡ 15 ≦ 10≡ false

• (y := x + 3*y).(x ≦ 10)≡ x ≦ 10

• (x,y := y,x).(x < y)≡ y < x

replace w by Ein R

Command semantics—assert

• if P holds, do nothing, else go wrong• (assert P).R ≡ P ∧ R

• (assert x < 10).(0 ≦ x)≡ 0 ≦ x < 10

• (assert x = y*y).(0 ≦ x)≡ x = y*y ∧ 0 ≦ x≡ x = y*y

• (assert false).(x ≦ 10)≡ false

logical AND,conjunction

Command semantics—assume

logicalimplication

logical NOT,negation

logical OR,disjunction

• if P holds, do nothing, else block

• (assume P).R ≡ ￢ P ∨ R≡ P ⇒ R

• (assume x < 10).(0 ≦ x)≡ 10 ≦ x ∨ 0 ≦ x≡ 0 ≦ x

• (assume x = y*y).(0 ≦ x)≡ x = y*y ⇒ 0 ≦ x≡ true

• (assume false).(x ≦ 10)≡ true

• introduce w with an arbitrary initial value,then do C

• (var w in C end).R ≡ (∀w ･ C.R)

• (var y in y := x end).(0 ≦ x)≡ (∀y ･ (y := x).(0 ≦ x))≡ (∀y ･ 0 ≦ x)≡ 0 ≦ x

• (var y in x := y end).(0 ≦ x)≡ (∀y ･ (x := y).(0 ≦ x))≡ (∀y ･ 0 ≦ y)≡ false

Command semantics—local variable

provided w does not occur free in R

• do C0, then C1

• (C0 ; C1).R ≡ C0.(C1.R)

• (x := x+1 ; assert x ≦ y).(0 < x)≡ (x := x+1).( (assert x ≦ y).(0 < x) )≡ (x := x+1).(0 < x ≦ y)

≡ 0 < x+1 ≦ y≡ 0 ≦ x < y

• (assume 0 ≦ y+z ; x := y).(0 ≦ x)≡ (assume 0 ≦ y+z).( (x:=y).(0 ≦

x) )≡ (assume 0 ≦ y+z).(0 ≦ y)≡ 0 ≦ y+z ⇒ 0 ≦ y≡ -y ≦ z ⇒ -y ≦ 0≡ 0 ≦ z

Command semantics—sequential composition

• do either C0 or C1 (the demon chooses which)

• (C0 [] C1).R ≡ C0.R ∧ C1.R

• (x := x+1 [] x := x + 2).(x ≦ 10)≡ (x := x+1). (x ≦ 10) ∧ (x := x+2).(x ≦ 10)≡ x ≦ 9 ∧ x ≦ 8

≡ x ≦ 8• (assume false [] x := y).(0 ≦ x)

≡ (assume false).(0 ≦ x) ∧ (x:=y).(0 ≦ x) ≡ true ∧ 0 ≦ y≡ 0 ≦ y

Command semantics—choice composition

Convenient shorthands• skip = assert true = assume true• wrong = assert false• magic = assume false• P C = assume P; C• if P then C0 else C1 end

= P C0 [] ￢ P C1

• havoc w = var w’in w := w’end

Change such that• change w such that P

= havoc w ; assume P

• change x such that y = x+1≡ havoc x ; assume y = x+1≡ x := y-1

• change x such that y < x≡ x := y+1 [] x := y+2 [] …

• change x such that x = x+1≡ havoc x ; assume false≡ magic

• change r such that r*r = y≡ y < 0 magic [] 0 ≦ y r := √y [] 0 ≦ y r := -√y

Specification statementw:[P, Q]

= requires P modifies w ensures Q= assert P ;

var w0 inw0 := w ;change w such that Q

end• x:[true, x0=x+1]

≡ x := x-1• r:[0 ≦ y, r*r = y]

≡ assert 0 ≦ y ; (r := √y [] r := -√y)• x:[0 ≦ x, x2 ≦ x0 < (x+1)2] ≡ ?• x,y,z,n:[1≦x≦y∧1≦z∧2≦n, xn+yn=zn] ≡ ?

Variables with internal structure: maps

• x := a[i] = x := select(a, i)• a[i] := E = a := store(a, i, E)

where

(∀m,i,j,v ･ i ≠j ⇒

select(store(m, i, v), i) = v ∧ select(store(m, i, v), j) = select(m, j))

Example: maps(a[5] := 12 ; a[7] := 14 ; x := a[5]).(x=12)

=(a[5] := 12 ; a[7] := 14).(select(a, 5) = 12)=(a[5] := 12).(select(store(a, 7, 14), 5) = 12)=select(store(store(a, 5, 12), 7, 14), 5) = 12= { select/store axiom, since 7 ≠ 5 }

select(store(a, 5, 12), 5) = 12= { select/store axiom, since 5 = 5 }

12 = 12=true

RefinementB ⊆ C

=(∀R ･ B.R ⇒ C.R )

• change x such that y < x ⊆ x := y+4• assert x < 10 ⊆ skip• skip ⊆ assume x < 10• wrong ⊆ C• C ⊆ magic

command Bis refined bycommand C

• C is “better” than B

• “anyone who requests B would be happy with C”

Compositions are monotonic with respect to refinement• if B ⊆ C then:

– var w in B end ⊆ var w in C end– A;B ⊆ A;C– B;D ⊆ C;D– A [] B ⊆ A [] C

• var x in ... change x such that y < x ... end⊆ var x in ... x := y+4 ... end

Commands form a lattice• Commands form a semi-lattice under

ordering ⊆, with meet operation [], top element magic, and bottom element wrong

• A lattice theorem: B ⊆ C0 ∧ B ⊆ C1 ≡ B ⊆ C0 [] C1

• Corollary: C0 [] C1 ⊆ C0

Example application of lattice theoremLet B = x:[true, x = |x0| ]. Then:• B ⊆ assume 0 ≦ x = C0

• B ⊆ assume x ≦ 0 ; x := -x = C1

• B ⊆ assume x = -3 ; x := 3 = C2

• B ⊆ magic = C3

Therefore:B ⊆ C0 [] C1 [] C2 [] C3

Procedures• proc P(x,y,z) returns (r,s,t) spec S• call to P:

a,b,c := P(E0, E1, E2) =

var x,y,z,r,s,t inx := E0 ; y := E1 ; z := E2 ;S ;a,b,c := r,s,t

end

Example: procedure• proc Add(x) returns (r)

spec requires 0 ≦ xmodifies kensures k = k0+x ∧ r = k0

• a := Add(k+25)= var x,r in

x := k+25 ;k:[0 ≦ x, k = k0+x ∧ r = k0] ;a := r

end

Procedure implementations• proc P(x,y,z) returns (r,s,t) spec S

• impl P(x,y,z) returns (r,s,t) is CProof obligation: S ⊆ C

• Let C0, ..., Cm-1 be the declared implementations of P. Then, the language implementation of a call to P can replace S by:

C0 [] ... [] Cm-1

Exercise• Redefine (in terms of the commands we've

seen) the specification statement so that the postcondition mentions x,x’ instead of x0,x

• Example:– old form:

x:[0 ≦ x, x*x ≦ x0 < (x+1)*(x+1)]– new form:

x:[0 ≦ x, x’*x’≦ x < (x’+1)*(x’+1)]

ExerciseDefine

while {inv J} B do w: S end where:– B is the loop guard– S is the loop body– J is the loop invariant– w is the list of assignment targets in S

in terms of the commands we've seen.

Loop (answer to exercise)

while {inv J} B do w: S end=

assert J ;change w such that J ;if B then

S ; assert J ; magicelse

skipend

Summary• Language is built up from 6 primitive

commands• Semantics can be given by weakest

preconditions• Partial (miraculous) commands are

important and very useful• select/store handle “map” variables• Procedures are names for specifications• Procedure implementations are hints for

compiler