Loop Analysis and Repair - PowerPoint Presentation

Slide1

Loop Analysis and Repair

Nafi Diallo

Computer Science

NJIT

Advisor: Dr. Ali MiliSlide2

Outline

1. Introduction

2. Progress

3. Prospects4. Conclusion

IntroductionResearch ProgressProposed workConclusion

Loop Analysis and Repair- RAMICS 2015

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Loop Analysis and Repair

Loop

Analysis

Convergence

Termination + Absence of AbortCorrectness/IncorrectnessLoop RepairDiagnose/Remove FaultsVerification

Loop Analysis and Repair- RAMICS 20153

1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide4

Definition

We consider a while loop w of the form :

denotes

the state

space of

and

represent the function of the loop

:

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide5

Definition

T is the relational vector defined by:

The loop semantics is defined by means of the reflexive transitive closure of 𝑇∩𝐵

:

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide6

Invariant Relation

Interpretation:

pairs

of states (

s,s') that are separated by an arbitrary number of iterationsExample:

𝐰𝐡𝐢𝐥𝐞(𝐤! = 𝐧){𝐤 = 𝐤 + 𝟏; 𝐟 = 𝐟∗ 𝐤;}An invariant relation is :

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide7

Invariant Relations and Invariant Assertions

All invariant assertions stem from invariant relations

Only a subset of invariant relations can be derived from invariant assertions

Invariant Relation

Invariant Assertion

Inductive Invariant Relation

Invariant Assertion

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3.

Prospects

4. Conclusion

R is an invariant relation

ν is a vector

A is an invariant assertionSlide8

Convergence:

Integrating Abort freedom with Termination

A general framework for Convergence

Theorem 1

We consider a while loop w of the form

on

space S, and

we let R be

an

invariant relation for w. Then:

Capturing aspects of abort freedom

Theorem

2

We consider a while loop w of the form

w: while (t) {b}

on space

,

and we let B′ be a superset of

If B′ satisfies the following conditions:

The

following relation

is transitive, for an arbitrary vector V .

( condition of concordance)

then

is

an invariant relation for w.

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide9

Abort-Freedom Invariant Relations

Logical form of Theorem 2

Applications

:

Array out of bounds

Illegal arithmetic operations

Arithmetic overflow

Illegal Pointer reference

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide10

Termination: Example

Abort condition: Illegal arithmetic operation

Termination condition

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide11

Termination: Example

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress3.

Prospects4. ConclusionSlide12

Correctness/Incorrectness

A necessary

condition

of correctness

Proposition: Let w be a while loop of the form

that terminates

for all

states in

S. Let

R be an invariant relation for w, and let V be a

specification

on S.

If w is

correct with respect to V then

.

Interpretation

: Incorrect if the invariant relation is incompatible with the specification

A

sufficient condition of correctnessProposition: Given a while loop w of the form

that terminates

for all

states in its space

, and given a specification

on

, if an invariant relation R of

w satisfies

the condition

then

w is correct with respect

to

U.

Interpretation

: Correct only if the invariant relation subsumes the specification.

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide13

Algorithm

for Verifying Loop Correctness

13

S = S and

Termination

(w)

[

CumR

= L

]

More-

inv

-relation(w)?

yes

R=get-inv-relation(w)

Loop

w

,

Space

S

,

Specification

spec

Necessary(R,spec)

?

Correct

OR

Incorrect

OR

Undecided

INPUT

OUTPUT

No

No

yes

[

CumR=CumR∩R

]

sufficient(CumR,spec)

?

yes

No

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide14

Relative Correctness

Definition 1

Given

a specification R and

a program P

defines

the competence domain

of P and denotes the set

of states on which 𝑷 obeys 𝑹

.

Definition 2

Given

a specification R and

two programs P and P’, deterministic

𝑷

’ more-correct than 𝑷 with respect to 𝑹:

𝑷’ has a larger competence domain than 𝑷.

Denoted:

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide15

Relative Correctness

Therefore

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide16

Impact of Relative Correctness on Testing

Impact

on

Test Data Generation

: vs.

Impact on Oracle Design:

: oracle for absolute correctness.

: oracle for relative correctness

.

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide17

Absolute Correctness and Relative Correctness

Loop Analysis and Repair- RAMICS 2015

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(Absolute) Correctness

Relative Correctness

Culminates

1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide18

A Formal Definition of fault

Definition 1

A feature in a program P is a

statement, condition, formula, or combination

thereofDefinition 2

Given Specification

, Program

, feature

in

:

A feature

is said to be a

fault

in

if and only if there exists a substitute

of

that would make

more-correct

.A pair of features

is said to be a

(monotonic) fault removal

of

if and only if program

obtained from

by substituting

for

is more-correct than

.

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide19

Loop Repair

Mutation Testing consists of :

Generating

Mutants

Testing Mutants against some sample test dataSelecting mutants that passRejecting mutants that failWe argueSelecting mutants that pass is wrong: a mutant may pass the test but still not be more-correct.Rejecting mutants that fail is also wrong: a mutant may fail and still be more-correct.Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide20

Proving Relative Correctness

Traditionally

:

static

verification techniques applicable only to correct programs;dynamic testing techniques used to expose/ diagnose faults in incorrect programs.Using relative correctness: We remove a fault from a program, check a set of conditions statically and locally, and conclude that the new program is more-correct than the old, all

Without testing (and its attending uncertainties),Without remorse (final determination: no going back to question wisdom of fault removal).Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide21

A Framework for Monotonic Fault Removal

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3. Prospects

4. ConclusionSlide22

Proving Relative Correctness

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3. Prospects

4. Conclusion

Theorem

Let R be a specification on space

and let

be a while loop of the form

which terminates for all

.

Let

be an invariant relation of

that is incompatible with

and let

be the largest invariant relation of such that Let

be the while loop that has as an invariant relation , terminates for all

and admits an invariant relation

that is compatible with

And satisfies the condition

Then

is strictly more correct that

 Slide23

Illustration

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3. Prospects

4. Conclusion

Specification

Invariant RelationsSlide24

Illustration

Loop Analysis and Repair- RAMICS 2015

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1. Introduction

2. Progress

3. Prospects

4. Conclusion

Compute constraints

Pick

, thus consider

to be involved in fault

Working by elimination, we choose mutant 2.

Testing for absolute correctness fails but testing for relative correctness succeeds

We repeat the process and end up with one incompatible invariant relation

Compute constraints and remove 2

nd

fault

 Slide25

Deployment

FxLoop

Analyzer (C++ based)

Client-server application

Thin clientServer(HTTP) has 2 componentsCCA compilerInvariant relation generator using semantic matchingDatabases of recognizers based on application domainComputing termination conditionTraditional senseIn combination with Abort-freedomCorrectness verification

Loop Analysis and Repair- RAMICS 201525

1. Introduction

2. Progress

3.

Prospects

4. ConclusionSlide26

Analytical Research

To further

explore the implications and applications of relative correctness,

To

derive techniques for proving relative correctness by static analysis of the source code.Loop Analysis and Repair- RAMICS 201526

1. Introduction2. Progress

3. Prospects

4. ConclusionSlide27

Thank you!

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