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Axiomatic Semantics and Program Verification

David Evans

http://www.cs.virginia.edu/~evans

It is easier to write an incorrect program than understand a correct one.

Alan Perlis

CS655: Programming Languages

University of Virginia

Computer Science

Operational Semantics

- Map to execution of a virtual machine
- Depends on informal understanding of machine
- Were able to prove that all programs in a language without loops terminate
- Awkward notion of equivalence
- Hard to prove properties about all possible executions of a program – need to simulate execution

CS 655: Lecture 19

Static Semantics

- Can prove properties about simple properties (type checking) easily
- Cannot take into account any dynamic properties
- Proofs must assume type of reference does not change throughout execution

CS 655: Lecture 19

Axiomatic Semantics

- Reason about programs using axioms (mathematical rules about program text fragments)
- Depends on informal (almost formal) understanding of logic
- Better than depending on informal understanding of Virtual Machine (Operational Semantics)
- Allows reasoning about all possible executions
- Can prove more interesting properties about programs than static semantics
- Can deal with control flow, dynamic properties, etc.

CS 655: Lecture 19

Floyd-Hoare Rules

pre-condition

post-condition

P { code fragment } Q

Partial correctness:

For all execution states which satisfy P, if the code fragment terminates, the resulting execution state satisfies Q.

Total correctness:

For all execution states which satisfy P, the code fragment terminates and the resulting execution state satisfies Q. (Sometimes people write: P [ code fragment ] Q.)

CS 655: Lecture 19

A simple example

{ true } while true do x := 1 { 2 + 2 = 5 }

Partial correctness:

Yes! Since code doesn’t terminate,

any post-condition is satisfied.

Total correctness:

No! Since code doesn’t terminate,

no total correctness post-condition

could be satisfied.

CS 655: Lecture 19

A Toy Language: Algorel

Program ::= Statement

Statement ::=

Variable :=Expression

| Statement ; Statement

| while Pred doStatementend

Expression ::= Variable | IntLiteral

| Expression + Expression

| Expression*Expression

Pred ::= true | false | Expression <= Expression

CS 655: Lecture 19

Assignment Axiom

P[e/x]

{x:=e side-effect-free(e) }

P

P is true after x:=e, iff P with e substituted for x was true before the assignment.

CS 655: Lecture 19

Assignment Example

wp{ x := x + 1 } x = 3

P[e/x] {x:=e sef(e) }P

wp = (x = 3)[x + 1/x]

wp = ((x + 1)= 3)

wp = (x = 2)

CS 655: Lecture 19

Weakest Preconditions

P { S } Q

- Given Q and S, what is the weakest P such that P { S } Q

x = 2 { x := x + 1 } x = 3

- Is there a stronger precondition?
- Is there a weaker precondition?
- Is there always a weakest precondition for any S and Q?

CS 655: Lecture 19

If Axiom

side-effect-free (b)

- (P b { s1 } Q)
- (P b { s2 } Q)

P { if b then s1 else s2 } Q

CS 655: Lecture 19

If Example

side-effect-free (x < 3)

- (P x < 3 {x := x + 1} x 3)
- (P (x < 3) {x := x – 1} x 3)

P { if(x < 3) then x := x + 1 else x := x – 1} x 3

weakest-precondition: P = x 3 x 2

CS 655: Lecture 19

An Algorel Program Fragment

% Pre-condition: ?

while n <= x do

n := n + 1;

result := result * n;

end

% Post-condition: result = x!

CS 655: Lecture 19

Goal: find weakest pre-condition

P & x0 = x

{

while n <= x do

n := n + 1;

result := result * n;

end

}

result = x0!

Elevator speech

CS 655: Lecture 19

Rule for while

P Inv

Inv { Pred } Inv

Inv & Pred { Statement } Inv

(Inv &~Pred) Q

whilePreddoStatementend terminates

P { while Preddo Statement end} Q

CS 655: Lecture 19

What can go wrong?

- Invariant too weak
- Can’t prove (Inv &~Pred) Q
- Invariant too strong
- Can’t prove Inv & Pred { Statement } Inv
- Can’t prove P Inv

(for sufficiently weak P)

CS 655: Lecture 19

Summary

- Once you have the right invariant, proving partial correctness is tedious but easy
- Computers can do this quickly
- Picking the right invariant is hard and not well understood
- Computers can do this slowly in special circumstances, often need help (from programmer or language designer)
- Next time: Proof-Carrying Code
- Can quickly and automatically prove interesting properties (type safety, memory safety, etc.) about arbitrary code if you are given the right invariants

CS 655: Lecture 19

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