Covering Arrays –Mathematical, Engineering, and Scientific Perspectives

Alan Hartman

May 2006Fields Institute Workshop on Covering Arrays – Carleton University, Ottawa

Copyright 2006 © IBM Corporation

Outline

• Mathematics

• Engineering

• Science

Mathematics

• Definition

• My favorite constructions

Covering Array - Definition

• A covering array of strength t for k parameters over domains of size n with R test cases is:

• An Rxk matrix with entries in {0,1,…n-1}

• Every Rxt submatrix contains every ordered t-tuple over {0,1,…n-1} in at least one of its rows

• is the minimum R such that there exists a covering suite of R test cases with parameters t, k, n

• Equivalent to finding the maximum k such that there exists a covering suite of R test cases with parameters t, k, n

),,( nktCA

Combinatorial Constructions RkCA )2,,2(

Let be the smallest integer such thatR

2/

1

R

Rk

K 2-3 4 5-10 11-15 16-35 36-56 57-126

R 4 5 6 7 8 9 10

First row all 0

Subsequent columns are incidence vectors of

subsets of an set 2/R 1R

Combinatorial Constructions RkCA )2,,2(

2/

1

R

Rk

First row all 0

Subsequent columns are incidence vectors of

subsets of an set

00 in first row

11 since any pair of subsets intersect

01 and 10 since the subsets are the same size

Best possible by Sperner’s Lemma and

Erdös-Ko-Rado Theorem

2/R 1R

2/R

Recursion for t>2

Let - the Turan number - be the maximum

number of edges over all graphs with vertices

having no clique of size

),,()),(1(),,( 2 nktCAntTnktCA

Uses perfect covering suites (OAs) of strength 2

to square the number of columns

),( ntT

1nt

Example: t=3, n=2

The maximum number of edges over all graphs

with 3 vertices having no clique of size 3 is 2.

)2,,3(3)2,,3( 2 kCAkCA

A A A AA ...

kA1 kA2 kAkkA3 ...

A ...A 2A k-1A

The general construction

A1 A

2 A

3 … A

k

N

k

T(t,n)+1

k2

T(t,n)+1

B[i,j]

C[i , j]=AB[j,i]

k2

),,()),(1(),,( 2 nktCAntTnktCA

Engineering Aspects• Motivation

• Applications

• Input Coverage

• FSM Reduction

• Military

• Ease of Use

• Effectiveness

Applications

•Software input coverage test suites

•Software and hardware FSM model reduction

•Blind disoriented robots on a line

Testing - Motivation

• Testing is important

• Testing is expensive

• Poor testing is even more expensive

Downtime Hourly Cost• Brokerage operations $6,450,000

• Credit card authorization $2,600,000

• Ebay (1 outage 22 hours) $225,000

• Amazon.com $180,000

• Package shipping services $150,000

• Home shopping channel $113,000

• Catalog sales center $90,000

• Airline reservation center $89,000

• Cellular service activation $41,000

• On-line network fees $25,000

• ATM service fees $14,000

A Testing Problem

• Test an internet site:– Operating system: Win, Linux, Solaris

– Browser: Explorer, Netscape

– Printer: Epsom, HP, IBM

– Protocol: Token Ring, Ethernet

36 combinations (OS, Br, Prt, Pcl)

1 tester

3 days

A Covering Array Solution

OS Browser Printer Protocol

Windows Explorer Epsom TR

Windows Netscape HP Enet

Windows Explorer IBM Enet

Linux Netscape Epsom TR

Linux Explorer HP Enet

Linux Netscape IBM TR

Solaris Explorer Epsom Enet

Solaris Netscape HP TR

Solaris Explorer IBM Enet

Model Based Testing with FSMs

• An abstract model of SW

• Nodes are labeled by tuples of state variables

• Arcs are labeled by possible inputs to the SW

• Test cases are generated by judicious choices of paths through the state space

• Problem: State and Arc explosion

• Solution: Covering Arrays to sample the arcs at each state

Blind disoriented synchronized robots

•k robots start at points 1,2,…k on a line

•They cannot see each other

•They have no common sense of direction

•They all move at speed 1

•They have to rendezvous in a minimum number of steps

Blind disoriented synchronized robots•Stage 1: Identify the extreme robots

•Stage 2: Extreme robots proceed towards the centre.

•Optimize stage 1 by giving each robot a column in a binary covering suite with t=2

•Interpret 0 as a move ½ left then ½ right

•Interpret 1 as a move ½ right then ½ left

•After steps the extreme robots are known

)2,,2( kCA

Usability and Extensibility

• The Pairwise Website (www.pairwise.org) lists 20 tools for constructing covering arrays

•Key factors for usability:

•NO MATH

•Constraints! – but how

•Distinction between legal and illegal inputs

• 1. CATS (Constrained Array Test System) *) [Sherwood] Bell Labs.

• 2. OATS (Orthogonal Array Test System) *) [Phadke] AT&T

• 3. AETG Telecordia Web-based, commercial

• 4. IPO (PairTest) *) [Tai/Lei]

• 5. TConfig [Williams] Java-applet

• 6. TCG (Test Case Generator) *) NASA

• 7. AllPairs Satisfice Perl script, free, GPL

• 8. Pro-Test SigmaZone GUI, commercial

• 9. CTS (Combinatorial Test Services) IBM Free for non-commercial use

• 10. Jenny [Jenkins] Command-line, free, public-domain

• 11. ReduceArray2 STSC, U.S. Air Force Spreadsheet-based, free

• 12. TestCover Testcover.com Web-based, commercial

• 13. DDA *) [Colburn/Cohen/Turban]

• 14. Test Vector Generator GUI, free

• 15. OA1 k sharp technology

• 16. CTE-XL Daimler Chrystler GUI, free

• 17. AllPairs [McDowell] Command-line, free

• 18. Intelligent Test Case Handler (replaces CTS) IBM Free for non-commercial use

• 19. CaseMaker Díaz & Hilterscheid GUI, commercial

• 20. PICT Microsoft Command-line, free

WHITCH

•WHITCH is open and extensible

• 62 downloads – since January

• You get a free Covering Array GUI

• Why not use it to liven up your next grant application?

•www.alphaworks.ibm.com/tech/whitch

WHITCH – Interface

• GUI panel for defining Types

• GUI panel for defining Columns

• GUI panels for choosing algorithms, variations etc.

• Java interfaces for plugging in new ideas, algorithms, panels

Science

• Empirical Software Engineering

• What are the issues

But is it all worth the trouble?

• Schroeder, Bolaki, Gopu, 2004 pointed out that random test suites of the same size are equally as effective as covering arrays in detecting injected SW defects

• James Bach has been trumpeting the news in all the engineering forums

• Do we have an answer?

SBG Arguments

• Previous papers compared pairwise to “conventional testing”

• Correlated code coverage with fault detection

• Small applications (6-8 KLOC)

• Artificially seeded bugs

Empirical SW Testing

• Need further experiments to determine if and when covering arrays do help:

• Real software

• Real faults

• Real software testers