Program verification -- 096229

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Program verification -- 096229

Ofer Strichman

Room [email protected] hours: after class

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Agenda for the first class

Up to slide 36: what is this course about ?

Slide 37: why not take this course Slide 38-39: why take this course Slide 40: requirements

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The goal – reliable software systems

Importance of reliable software – almost needless to mention.

A nice collection of famous bugs:http://en.wikipedia.org/wiki/Software_bugs

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Here are some famous ones: ESA Ariane 5 Flight 501 self-destruction 40

seconds after takeoff (June 4, 1996). A conversion from 64-bit floating point to 16 bit

integer with a value larger than possible with Arian 4. The overflow caused a hardware trap.

The 2003 North America blackout was triggered by a local outage that went undetetected.

A race condition in General Electric’s monitoring software prevented an alarm.

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Here are some famous ones: The MIM-104 Patriot bug, which resulted in the

deaths of 28 Americans in Dharan, Saudi Arabia (February 25, 1991).

The radar classifies detections according to a trajectory model it builds in real time. (“in x mSec the missile should be in position y”)

Due to rounding of the clock values, it accumulates inaccuracies. After several hours this inaccuracy is critical.

The Pentium bug Incorrect floating-point division. Cost Intel ~ $400,000,000

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Not unusual to have more than 50% of resources allocated to testing

Testing and verification are (becoming) the bottleneck of development

bEgInSlIdE

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Remarks by Bill Gates17th Annual ACM Conference on Object-Oriented

Programming, Seattle, Washington, November 8, 2002

“… When you look at a big commercial software

company like Microsoft, there's actually as much

testing that goes in as development. We have as

many testers as we have developers. Testers

basically test all the time, and developers basically

are involved in the testing process about half the

time…

bEgInSlIdE

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“… We've probably changed the industry we're in.

We're not in the software industry; we're in the

testing industry, and writing the software is the

thing that keeps us busy doing all that testing.”

“…The test cases are unbelievably expensive; in

fact, there's more lines of code in the test harness

than there is in the program itself. Often that's a

ratio of about three to one.”

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Quality dilemma: quality / features / time More efficient methods for test and

verification are always needed No ‘silver-bullet’ when it comes to testing. Formal verification is on the rise...

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Formal methods

Formal = based on rigorous mathematical logic concepts. It is ‘machine-readable’, and hence can be

used by a verification algorithm. Once we formally specify what we expect

from the program, we can try to prove that the program satisfies the specification.

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Local assertions: at this point in the program always x > y. How is it different than writing ‘assert ( x > y);’ ?

A more global property: The array bound is never exceeded

Temporal properties:If a request is sent, acknowledgement is eventually received

Specification can be: Informal, textual, visual...

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... or it can be formal

(1· x · 5 U x=7) Æ G (x¸ 0)

(Translated from: The value of x will be between 1 and 5, until at some point it will become 7, and it will always (“Globally”) be positive.)

1· x · 5

x=7

x¸ 0

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A system can be general and messy...

struct node *copy_node(struct node *f) {

static int count = 0;

struct node *temp;

count ++;

if (f == NULL) return NULL;

if ((temp = (struct node *) malloc (sizeof (struct node))) == NULL)

fprintf(stderr,"Out of memory");

temp -> tag = f -> tag;

temp -> name = strdup(f -> name);

temp -> left = copy_node(f -> left);

temp -> right = copy_node(f -> right);

temp -> cnf_id = f -> cnf_id;

return temp;

}

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...or its underlying algorithm can be modeled

E.g., as a finite-state machine

s1 s3s2pull

release

release

extended malfunction

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The program can be tested...

Try inputs x1 = 2, x2 = 25, str = ‘llk’ x1 = 0, x2 = 31, str = ‘aaa’ ...

... As good as the set of input test-cases ... and as good as the reference system

Manual inspection Regression testing Specification-based testing

For simple, non-critical code, this is the easiest and most immediate option.

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...or can be formally verified

Given the formal specification G(:malfunction) // always not

malfunctioning

And the system model

Run a model-checker program to formally verify that the system satisfies the specification.

s1 s3s2pull

release

releaseextended malfunction

bEgInSlIdE

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“We also went back and say, OK, what is the state-of-the-art in terms of being able to prove programs, prove that a program behaves in a certain way? This is the kind of problem that has been out there for many decades...

...When I dropped out of Harvard, this was an interesting problem that was being worked on, and actually at the time I thought, well, if I go and start a company it will be too bad, I'll miss some of the big breakthroughs in proving software programs because I'll be off writing payroll checks. “

bEgInSlIdE

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...and it turns out I didn't miss all that much in terms of rapid progress which now has become, I'd say, in a sense, a very urgent problem. And although a full general solution to that is in some ways unachievable, for many very interesting properties this idea of proof has come a long way in helping us quite substantially just in the last year.

We call the system that does this kind of proof -- a model-checking system.

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Model-checking

Input: A finite state machine M A temporal property P P

Output: Does M satisfy P ?

M

P = “never in state 11”

11

0001 10

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Model-checking

Nondeterminism is used to model systems with inputs

Model-checking is exponential in the number of inputs

M

11

00 1001 Minputs

outp

uts=

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Model-checking

The model M defines a set of behaviors (traces) L(M)

The model P defines a set of allowed behaviors L(P)

M

11

00 1001

P

Never “11”

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The model checking problem: L(M) µ L(P)

Q: What if this problem is too hard in practice?

Model-checking as language-inclusion

L(M)

L(P)error

?

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Program verification

Bill: Prove that the program is correct !

MS-employee: let me quote Carl Popper’s answer about verification vs. falsification in science. Science does not progress by finding eternal

truths... ... only by refuting previous theories.

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Bill: Ok, So just prove that the program satisfies the specification !

MS-employee : ok ... but even if I prove that the system

satisfies the specification, there can still be an error in the proof process or the proof tool that I use...

... and there is no guarantee that the specification fully specifies what we consider as ‘correct’.

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Bill: Ok, fine, so just increase the probability that the program is correct !

MS-employee : Well, boss, if you want this process to be

automatic, please provide me a model of the program, and a formal (machine-readable) specification...

... Such that checking the model with respect to the specification is a decidable problem.

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Bill: Will it be better than testing ? MS-employee: when it is possible, it is

much better.

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Testing / Formal Verification

A very crude dichotomy:

In practice:Many types of testing, Many types of formal verification.

Testing

Correct with respect to the set of test inputs, and reference system

Easy to perform

Dynamic

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Testing / Formal Verification

TestingFormal Verification

Correct with respect to the set of test inputs, and reference system

Correct with respect to all inputs, with respect to a formal specification

Easy to performDecidability problems, Computational problems ,

DynamicStatic

A very crude dichotomy:

In practice:Many types of testing, Many types of formal verification.

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Testing + formal-verification

Bill: is there a mid-way ? MS-employee: Yes: Specify formally, verify

informally

This is called run-time verification... The idea is to write a small program that

monitors the tested program It checks during run-time that the tested

program satisfies a given (Formal) specification.

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Where do we start?

Bill: ok, where do we start ? MS-employee: with some basics of Logics Bill: Why Logic ? MS-employee: I will show you an example.

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Reminder: some Boolean connectives

Æ and Ç or : not ! implies $ equal

A propositional formula: :(a Ç (b ! (c Æ : a)))

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Program transformation

if(!a && !b) h();

else if(!a) g();

else f();

if(a) f();

else if(b) g();

else h();

*

if(a) f();

else {

if(!b) h();

else g(); }

(

How can we check that these two versions are equivalent?

+

if(!a) {

if(!b) h();

else g(); }

else f();

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Code transformation example – cont’d

Represent procedures as independent Boolean variablesoriginal := optimized :=

if :a Æ b then h if a then f

else if :a then g else if b then g

else f else h

Transform if-then-else chains into Boolean formulas

transform(if x then y else z) ≡ (x Æ y) Ç (:x Æ z)

Check equivalence of boolean formulaetransform(original) , transform(optimized)

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Code transformation example – cont’d

original ≡ if :a Æ :b then h else if :a then g else h

≡ (:a Æ :b) Æ h Ç :(:a Æ :b) Æ if :a then g else f

≡ (:a Æ :b) Æ h Ç :(:a Æ :b) Æ (:a Æ g Ç a Æ f )

optimized ≡ if a then f else if b then g else h

≡ (a Æ f ) Ç ( :a Æ if b then g else h )

≡ ( a Æ f ) Ç ( :a Æ (b Æ g Ç :bÆ h) )

Verification Condition : ((:a Æ :b) Æ h Ç :(:a Æ :b) Æ (:a Æ g Ç a Æ f )) $

(a Æ f ) Ç (:a Æ (b Æ g Ç :b Æ h))

The question: is valid ?

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Not just software...

ÆÆ

Ç

ÇÇ

Prove the equivalence of two circuits:

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So...

We will learn: A brief introduction to Propositional Logic

Incl. General logic jargon such as: validity, satisfiability, soundness, completeness,...

In a later stage in the course we will also learn algorithms for checking whether a given formula is valid/satisfiable

Temporal Logic, and its ability to formally specify systems

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So...

... and Specification-based testing (specifically: Run-

time verification) Other types of testing

... And also Modeling of systems Formal verification of systems against formal

specification with model-checking. More... if we have the time

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Why not take this course

In the end of the course you will NOT be able to automatically verify the big software you wrote in C

No one can, (completely) automatically. After your Ph.D, you might be able to do it with some

manual guidance.

On the bright side ... with certain restrictions, or compromises on what it means to ‘verify’, you might be able to verify even large C programs.

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Why take this course

There are some restricted type of programs that can be verified automatically.

Finite-state programs (e.g. protocols) and more. The ‘killer-application’: hardware circuits.

There is an industry behind it, incl. in Haifa. Developers of verification tools: IBM, Intel. Users of such tools: IBM, Intel, Melanox, Freescale, Verisity

(now Cadence), Galileo (now Marvel)...

Background in hardware is not necessary at all.

Most developers & users in this field do not have it.

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Why take this course (cont’d)

Very different than other courses you took...

Combines Theory (logic, automata-theory) Algorithms Acquaintance with type of programs and

systems you have probably not seen before.

Not difficult... Only assumes ‘mathematical maturity’,

not specific knowledge.

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Requirements

~5 homework assignments 2 exams, 3-questions each (1.5 hours).

Program verification -- 096229

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