Program verification: flowchart programs

Book: chapter 7

History

Verification of flowchart programs: Floyd, 1967 Hoare’s logic: Hoare, 1969 Linear Temporal Logic: Pnueli, Krueger, 1977 Model Checking: Clarke & Emerson, 1981

Program Verification

Predicate (first order) logic. Partial correctness, Total correctness Flowchart programs Invariants, annotated programs Well founded ordering (for

termination) Hoare’s logic

Predicate (first order logic)

Variables, functions, predicates

Terms

Formulas (assertions)

Signature

Variables: v1, x, y18Each variable represents a value of some given

domain (int, real, string, …). Function symbols: f(_,_), g2(_), h(_,_,_).Each function has an arity (number of

paramenters), a domain for each parameter, and a range.

f:int*int->int (e.g., addition), g:real->real (e.g., square root)

A constant is a predicate with arity 0. Relation symbols: R(_,_), Q(_).Each relation has an arity, and a domain for each

parameter.R : real*real (e.g., greater than).Q : int (e.g., is a prime).

Terms

Terms are objects that have values. Each variable is a term. Applying a function with arity n to n

terms results in a new term.Examples: v1, 5.0, f(v1,5.0),

g2(f(v1,5.0))

More familiar notation: sqr(v1+5.0)

Formulas

Applying predicates to terms results in a formula.

R(v1,5.0), Q(x)More familiar notation: v1>5.0 One can combine formulas with the

boolean operators (and, or, not, implies).

R(v1,5.0)->Q(x)x>1 -> x*x>x One can apply existentail and universal

quantification to formulas.x Q(X) x1 R(x1,5.0) X Y R(x,y)

A model, A proofs

A model gives a meaning (semantics) to a first order formula: A relation for each relation symbol. A function for each function symbol. A value for each variable.

An important concept in first order logic is that of a proof. We assume the ability to prove that a formula holds for a given model.

Example proof rule (MP) :

Flowchart programs

Input variables: X=x1,x2,…,xlProgram variables: Y=y1,y2,…,ymOutput variables: Z=z1,z2,…,zn

start

haltY=f(X)

Z=h(X,Y)

Assignments and tests

Y=g(X,Y) t(X,Y)FT

start

halt

(y1,y2)=(0,x1)

y2>=x2

(y1,y2)=(y1+1,y2-x2) (z1,z2)=(y1,y2)

Initial condition

Initial condition: the values for the input variables for which the program must work.

x1>=0 /\ x2>0

FT

start

halt

(y1,y2)=(0,x1)

y2>=x2

(y1,y2)=(y1+1,y2-x2) (z1,z2)=(y1,y2)

The input-output claim

The relation between the values of the input and the output variables at termination.

x1=z1*x2+z2 /\ 0<=z2<x2

FT

Partial correctness, Termination, Total correctness Patial correctness: if the initial

condition holds and the program terminates then the input-output claim holds.

Termination: if the initial condition holds, the program terminates.

Total correctness: if the initial condition holds, the program terminates and the input-output claim holds.

start

halt

(y1,y2)=(0,x1)

y2>=x2

(y1,y2)=(y1+1,y2-x2) (z1,z2)=(y1,y2)

Subtle point:

The program ispartially correct

withrespect tox1>=0/\x2>=0and totally correctwith respect tox1>=0/\x2>0

T F

start

halt

(y1,y2)=(0,x1)

y2>=x2

(y1,y2)=(y1+1,y2-x2) (z1,z2)=(y1,y2)

Annotating a scheme

Assign an assertion for each pair of nodes. The assertion expresses the relation between the variable when the program counter is located between these nodes.

A

B

C D

E

FT

Annotating a scheme with invariants

A): x1>=0 /\ x2>=0B): x1=y1*x2+y2 /\

y2>=0C): x1=y1*x2+y2 /\

y2>=0 /\ y2>=x2D):x1=y1*x2+y2 /\

y2>=0 /\ y2<x2E):x1=z1*x2+z2 /\ 0<=z2<x2Notice: (A) is the initial

condition, is the input-output condition.

start

halt

(y1,y2)=(0,x1)

y2>=x2

(y1,y2)=(y1+1,y2-x2) (z1,z2)=(y1,y2)

A

B

C D

E

FT

Verification conditions: assignment

A) B) [Y\g(X,Y)]

A): x1>=0 /\ x2>=0

B): x1=y1*x2+y2 /\ y2>=0

B) [Y\g(X,Y)] =x1=0*x2+x1 /\

x1>=0

(y1,y2)=(0,x1)

A

B

A

B

(y1,y2)=(0,x1)

Y=g(X,Y)

(y1,y2)=(y1+1,y2-x2)

Second assignment

C): x1=y1*x2+y2 /\ y2>=0 /\ y2>=x2

B): x1=y1*x2+y2 /\ y2>=0

B)[Y\g(X,Y]: x1=(y1+1)*x2+y2-x2 /\ y2-x2>=0

C

B

(z1,z2)=(y1,y2)

Third assignment

D):x1=y1*x2+y2 /\ y2>=0 /\ y2<x2

E):x1=z1*x2+z2 /\ 0<=z2<x2

E)[Z\g(X,Y]: x1=y1*x2+y2 /\ 0<=y2<x2

E

D

Verification conditions: tests

B) /\ t(X,Y) C)B) /\¬t(X,Y) D)

B): x1=y1*x2+y2 /\y2>=0

C): x1=y1*x2+y2 /\ y2>=0 /\ y2>=x2

D):x1=y1*x2+y2 /\ y2>=0 /\ y2<x2

y2>=x2

B

C

D

B

C

Dt(X,Y)

FT

FT

Exercise: prove partial correctness

Initial condition: x>=0

Input-output claim:

z=x!

start

halt

(y1,y2)=(0,1)

y1=x

(y1,y2)=(y1+1,(y1+1)*y2) z=y2

TF

Annotating a scheme

Assign an assertion for each pair of nodes. The assertion expresses the relation between the variable when the program counter is located between these nodes.

start

halt

(y1,y2)=(0,x1)

y2>=x2

(y1,y2)=(y1+1,y2-x2) (z1,z2)=(y1,y2)

A

B

C D

E

falsetrue

Annotating a scheme with invariants

A): x1>=0 /\ x2>=0B): x1=y1*x2+y2 /\

y2>=0C): x1=y1*x2+y2 /\

y2>=0 /\ y2>=x2D):x1=y1*x2+y2 /\

y2>=0 /\ y2<x2E):x1=z1*x2+z2 /\ 0<=z2<x2Notice: (A) is the initial

condition, Eis the input-output condition.

start

halt

(y1,y2)=(0,x1)

y2>=x2

(y1,y2)=(y1+1,y2-x2) (z1,z2)=(y1,y2)

A

B

C D

E

falsetrue

Verification conditions: assignment

A) B) [Y\g(X,Y)]

A): x1>=0 /\ x2>=0B): x1=y1*x2+y2 /\

y2>=0

B) [Y\g(X,Y)] =x1=0*x2+x1 /\

x1>=0

A

B

(y1,y2)=(0,x1)

(y1,y2)=(0,x1)

A

B

Y=g(X,Y)

Assignment condition

(y1,y2)=(0,x1)

A

B

y1=2

y1=x1

2=x1

Another way to understand condition

(y1,y2)=(0,x1)

A

B

y1=2

y1=x1

Use two versions of variables: before assignment and after. E.g., y1 and y1’, respectively.

postcondition: y1’=x1assignment: y1’=2precondition: 2=x1

2=x1

Assignment condition

(y1,y2)=(0,x1)

A

B

y1=y1+5

y1=10

y1=5

Assignment condition

(y1,y2)=(0,x1)

A

B

y1=y1+5

y1=10

y1=5Postcondition: y1’=10

Assignment: y1’=y1+5

Precondition: y1+5=10, I.e., y1=5

Verification conditions: assignment

B): x1=y1’*x2+y2’ /\ y2’ >=0

Assignment: y1’=0 /\ y2’=x1

B) [Y\g(X,Y)] =x1=0*x2+x1 /\ x1>=0(or simply x1>=0)

A

B

(y1,y2)=(0,x1)

A): x1>=0 /\ x2>=0

Second assignment

Precondition:B): x1=y1*x2+y2 /\

y2>=0

Assignment:y1’=y1+1/\y2’=y2-x2

Postcondition:B)[Y\g(X,Y)]:

x1=(y1+1)*x2+y2-x2 /\ y2-x2>=0

(y1,y2)=(y1+1,y2-x2)

C

B

(y1,y2)=(y1+1,y2-x2)

Second assignment

C): x1=y1*x2+y2 /\ y2>=0 /\ y2>=x2

B): x1=y1*x2+y2 /\ y2>=0

B)[Y\g(X,Y)]: x1=(y1+1)*x2+y2-x2 /\ y2-x2>=0

C

B

(z1,z2)=(y1,y2)

Third assignment

D):x1=y1*x2+y2 /\ y2>=0 /\ y2<x2

E):x1=z1*x2+z2 /\ 0<=z2<x2

E)[Z\g(X,Y]: x1=y1*x2+y2 /\ 0<=y2<x2

E

D

Verification conditions: tests

B) /\ t(X,Y)) C)(B) /\ ¬t(X,Y)) D)

B): x1=y1*x2+y2 /\ y2>=0

C): x1=y1*x2+y2 /\ y2>=0 /\ y2>=x2

D):x1=y1*x2+y2 /\ y2>=0 /\ y2<x2

y2>=x2

B

C

D

B

C

Dt(X,Y)

falsetrue

falsetrue

Exercize: prove partial correctness

Initial condition: x>=0

Input-output claim: z=x!

start

halt

(y1,y2)=(0,1)

y1=x

(y1,y2)=(y1+1,(y2+1)y2) z=y2

truefalse

What have we achieved?

For each statement S that appears between points X and Y we showed that if the control is in X when (X) holds and S is executed, then (Y) holds.

Initially, we know that (A) holds. The above two conditions can be combined

into an induction on the number of statements that were executed: If after n steps we are at point X, then (X)

holds.

Another example

(A) : x>=0

(F) : z^2<=x<(z+1)^2

z is the biggest numberthat is not greaterthan sqrt x.

start

(y1,y2,y3)=(0,0,1)

A

halt

y2>x

(y1,y3)=(y1+1,y3+2) z=y1

B

C

D

F

truefalse

E

y2=y2+y3

Some insight

1+3+5+…+(2n+1)=(n+1)^2

y2 accumulates theabove sum, untilit is bigger than x.

y3 ranges over oddnumbers 1,3,5,…

y1 is n-1.

start

(y1,y2,y3)=(0,0,1)

A

halt

y2>x

(y1,y3)=(y1+1,y3+2) z=y1

B

C

D

F

truefalse

E

y2=y2+y3

Invariants

It is sufficient to have one invariant for every loop(cycle in the program’sgraph).

We will have(C)=y1^2<=x /\ y2=(y1+1)^2 /\ y3=2*y1+1

start

(y1,y2,y3)=(0,0,1)

A

halt

y2>x

(y1,y3)=(y1+1,y3+2) z=y1

B

C

D

F

truefalse

E

y2=y2+y3

Obtaining (B)

By backwards substitution in (C).

(C)=y1^2<=x /\ y2=(y1+1)^2 /\ y3=2*y1+1

(B)=y1^2<=x /\ y2+y3=(y1+1)^2 /\ y3=2*y1+1

start

(y1,y2,y3)=(0,0,1)

A

halt

y2>x

(y1,y3)=(y1+1,y3+2) z=y1

B

C

D

F

truefalse

E

y2=y2+y3

Check assignment condition

(A)=x>=0(B)=y1^2<=x /\ y2+y3=(y1+1)^2 /\ y3=2*y1+1(B) relativized is 0^2<=x /\ 0+1=(0+1)^2 /\ 1=2*0+1Simplified: x>=0

start

(y1,y2,y3)=(0,0,1)

A

halt

y2>x

(y1,y3)=(y1+1,y3+2) z=y1

B

C

D

F

truefalse

E

y2=y2+y3

Obtaining (D)

By backwards substitution in

(B).

(B)=y1^2<=x /\ y2+y3=(y1+1)^2 /\ y3=2*y1+1

(D)=(y1+1)^2<=x /\ y2+y3+2=(y1+2)^2 /\ y3+2=2*(y1+1)+1

start

(y1,y2,y3)=(0,0,1)

A

halt

y2>x

(y1,y3)=(y1+1,y3+2) z=y1

B

C

D

F

truefalse

E

y2=y2+y3

Checking

(C)=y1^2<=x /\ y2=(y1+1)^2 /\ y3=2*y1+1

(C)/\y2<=x) (D)

(D)=(y1+1)^2<=x /\ y2+y3+2=(y1+2)^2 /\ y3+2=2*(y1+1)+1

start

(y1,y2,y3)=(0,0,1)

A

halt

y2>x

(y1,y3)=(y1+1,y3+2) z=y1

B

C

D

F

truefalse

E

y2=y2+y3

y1^2<=x /\

y2=(y1+1)^2 /\ y3=2*y1+1 /\y2<=x (y1+1)^2<=x /\

y2+y3+2=(y1+2)^2 /\

y3+2=2*(y1+1)+1y1^2<=x /\ y2=(y1+1)^2 /\ y3=2*y1+1 /\y2<=x (y1+1)^2<=x /\ y2+y3+2=(y1+2)^2

/\ y3+2=2*(y1+1)+1

y1^2<=x /\

y2=(y1+1)^2 /\

y3=2*y1+1 /\y2<=x (y1+1)^2<=x /\

y2+y3+2=(y1+2)^2 /\

y3+2=2*(y1+1)+1

Not finished!

Still needs to:

Calculate (E) bysubstituting backwardsfrom (F).

Check that(C)/\y2>x(E)

start

(y1,y2,y3)=(0,0,1)

A

halt

y2>x

(y1,y3)=(y1+1,y3+2) z=y1

B

C

D

F

truefalse

E

y2=y2+y3

Proving termination

Well-founded sets

Partially ordered set (W,<): If a<b and b<c then a<c (transitivity). If a<b then not b<a (asymmetry). Not a<a (irreflexivity).

Well-founded set (W,<): Partially ordered. No infinite decreasing chain a1>a2>a3>…

Examples for well founded sets Natural numbers with the bigger than

relation. Finite sets with the set inclusion relation. Strings with the substring relation. Tuples with alphabetic order:

(a1,b1)>(a2,b2) iff a1>a2 or [a1=a2 and b1>b2].

(a1,b1,c1)>(a2,b2,c2) iff a1>a2 or [a1=a2 and b1>b2] or [a1=a2 and b1=b2 and c1>c2].

Why does the program terminate

y2 starts as x1. Each time the loop is

executed, y2 is decremented.

y2 is natural number The loop cannot be

entered again when y2<x2.

start

halt

(y1,y2)=(y1+1,y2-x2) (z1,z2)=(y1,y2)

(y1,y2)=(0,x1)

A

B

D

E

falsey2>=x2

C

true

Proving termination

Choose a well-founded set (W,<). Attach a function u(N) to each

point N. Annotate the flowchart with

invariants, and prove their consistency conditions.

Prove that (N) (u(N) in W).

How not to stay in a loop?

Show that u(M)>=u(N).

At least once in each loop, show that u(M)>u(N).

S

M

N

TN

M

How not to stay in a loop?

For stmt: (M)(u(M)>=u(N)’rel)

For test (true side):((M)/\test)(u(M)>=u(N))

For test (false side):((M)/\

¬test)(u(M)>=u(L))

stmt

M

N

test

N

M

true

L

false

What did we achieve?

There are finitely many control points. The value of the function u cannot

increase. If we return to the same control point,

the value of u must decrease (its a loop!).

The value of u can decrease only a finite number of times.

Why does the program terminate

u(A)=x1u(B)=y2u(C)=y2u(D)=y2u(E)=z2

W: naturals> : greater than

start

halt

(y1,y2)=(y1+1,y2-x2) (z1,z2)=(y1,y2)

(y1,y2)=(0,x1)

A

B

D

E

falsey2>=x2

C

true

Recall partial correctness annotation

A): x1>=0 /\ x2>=0B): x1=y1*x2+y2 /\

y2>=0C): x1=y1*x2+y2 /\

y2>=0 /\ y2>=x2D):x1=y1*x2+y2 /\

y2>=0 /\ y2<x2E):x1=z1*x2+z2 /\ 0<=z2<x2

start

halt

(y1,y2)=(0,x1)

y2>=x2

(y1,y2)=(y1+1,y2-x2) (z1,z2)=(y1,y2)

A

B

C D

E

falsetrue

Strengthen for termination

A): x1>=0 /\ x2>0B): x1=y1*x2+y2 /\

y2>=0/\x2>0C): x1=y1*x2+y2 /\

y2>=0 /\ y2>=x2/\x2>0

D):x1=y1*x2+y2 /\ y2>=0 /\ y2<x2/\x2>0

E):x1=z1*x2+z2 /\ 0<=z2<x2This proves that u(M) is

natural for each point M.

start

halt

(y1,y2)=(0,x1)

y2>=x2

(y1,y2)=(y1+1,y2-x2) (z1,z2)=(y1,y2)

A

B

C D

E

falsetrue

We shall show:

u(A)=x1u(B)=y2u(C)=y2u(D)=y2u(E)=z2u(A)>=u(B)u(B)>=u(C)u(C)>u(B)u(B)>=u(D)u(D)>=u(E)

start

halt

(y1,y2)=(y1+1,y2-x2) (z1,z2)=(y1,y2)

(y1,y2)=(0,x1)

A

B

D

E

falsey2>=x2

C

true

Proving decrement

C): x1=y1*x2+y2 /\ y2>=0 /\ y2>=x2/\x2>0

u(C)=y2u(B)=y2u(B)’rel=y2-x2

C) y2>y2-x2(notice that C) x2>0)

start

halt

(y1,y2)=(0,x1)

y2>=x2

(y1,y2)=(y1+1,y2-x2) (z1,z2)=(y1,y2)

A

B

C D

E

falsetrue

Integer square prog.

(C)=y1^2<=x /\ y2=(y1+1)^2 /\ y3=2*y1+1

(B)=y1^2<=x /\ y2+y3=(y1+1)^2 /\y3=2*y1+1

start

(y1,y2,y3)=(0,0,1)

A

halt

y2>x

(y1,y3)=(y1+1,y3+2) z=y1

B

C

D

F

truefalse

E

y2=y2+y3

u(A)=x+1u(B)=x-y2+1u(C)=max(0,x-y2)u(D)=x-y2+1u(E)=u(F)=0u(A)>=u(B)u(B)>u(C)u(C)>=u(D)u(D)>=u(B)Need some invariants,i.e., y2<=x/\y3>0at points B and D,and y3>0 at point C.

start

(y1,y2,y3)=(0,0,1)

A

halt

y2>x

(y1,y3)=(y1+1,y3+2) z=y1

B

C

D

F

truefalse

E

y2=y2+y3