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Testing Event Sequences


SEQUENCE COVERING ARRAY LIBRARY

Note: a technical paper providing more depth than this introduction is available here.

Many testing problems involve sequences of operations. For example, an embedded system may accept multiple sensor inputs and generate output to several communication links and effectors such as machine controls. It is important to test combinations of connected components, but also to test the order in which they could be connected. There is an empirical basis for the use of sequence covering arrays. In reviews of various failure reports, when sequences of events were involved, the critical condition for triggering failures generally was whether or not a particular event had occurred prior to a second one, not necessarily if they were back to back. In other words, the report might say something like 'failure occurred when <event A> if B is already connected'. So it appeared that A didn't have to immediately follow B, but the fact that it followed B at some point after B had already occurred was sufficient to trigger a failure. Sequence covering arrays, as we have defined them, ensure that any t events will be tested in every possible order.

For example, we may have a factory automation system that uses certain devices interacting with a control program. For this problem we can define a sequence covering array, which is a set of tests that ensure all t-sequences of events have been tested. The t events in the sequence may be interleaved with others, but all t-way permutations will be tested. We want to test the following events:

Event Description
a connect air flow meter
b connect pressure guage
c connect satellite link
d connect pressure readout
e engage drive motor
f engage steering control


There are 6! = 720 possible sequences for these six events, and the system should respond correctly and safely no matter the order in which they occur. Operators may be instructed to use a particular order, but mistakes are inevitable, and should not result in injury to users or compromise the mission. We want to test this system as thoroughly as possible, but time and budget constraints do not allow for testing all possible sequences, so we will test all 3-event sequences. With six events, a, b, c, d, e, and f, one subset of three is {b, d, e}, which can be arranged in six permutations: <b d e>, <b e d>, <d b e>, <d e b>, <e b d>, <e d b>. A test that covers the permutation <d b e> is: <a d c f b e>; another is <a d c b e f>. With only 10 tests, we can test all 3-event sequences:

Test Sequence
1 a b c d e f
2 f e d c b a
3 d e f a b c
4 c b a f e d
5 b f a d c e
6 e c d a f b
7 a e f c b d
8 d b c f e a
9 c e a d b f
10 f b d a e c


(Keep in mind that this example is small for convenience. A real system may have, for example, 10 devices to connect, in which case the number of permutations is 10!, or 3,628,800 tests for exhaustive testing. In that case, a 3-sequence covering array with 14 tests would be a much more dramatic improvement.)

Definition. We define a sequence covering array, SCA(N, S, t) as an N x S matrix where entries are from a finite set S of s symbols, such that every t-length permutation of symbols from S occurs in at least one row; the t symbols in the permutation are not required to be adjacent.

Example 1. Consider the problem of testing four events, a, b, c, and d. There are 4! = 24 possible permutations of these four events, but we can test all 3-sequences of these events with only six tests:

  1 2 3 4
1 a d b c
2 b a c d
3 b d c a
4 c a b d
5 c d b a
6 d a c b


Example 2. Generating a 2-sequence covering array is trivial: list the events in some order for one test and in the reverse order for the second test.

  1 2 3 4
1 a b c d
2 d c b a


As shown in example 2, only two tests are needed to cover all 2-way permutations of symbols. Other values of t > 2 will require more. The example below covers all 3-way permutations for five events in eight tests. Arrays for other event set sizes can be found in the library table following.

  1 2 3 4 5
1 a b c d e
2 e d c b a
3 b a e d c
4 c e a b d
5 d e a b c
6 c d b a e
7 a e c d b
8 b d c e a


SEQUENCE COVERING ARRAY LIBRARY


The sequence covering arrays are provided in comma separated value form. Currently arrays are for 3-way interactions only. Longer permutations will be added in the future. For 2-way permutations, event sequences can simply be reversed, as shown in Example 2, so these arrays are not included in the library. Please note that these arrays were prepared with a quick and dirty greedy algorithm in Oct 09, and can probably be improved upon. We are working on this now. If you have an algorithm that generates smaller arrays, we would be happy to host them, citing your algorithm of course; feel free to email me at kuhn@nist.gov if interested.

3-way permutations 4-way permutations
Events Number  Tests Test file Number  Tests Test file
5 8 test3_5.csv 29
test4_5.csv
6 10 test3_6.csv 38 test4_6.csv
7 12 test3_7.csv 50 test4_7.csv
8 12 test3_8.csv 56 test4_8.csv
9 14 test3_9.csv 68 test4_9.csv
10 14 test3_10.csv 72 test4_10.csv
11 14 test3_11.csv 78
test4_11.csv
12 16 test3_12.csv 86
test4_12.csv
13 16 test3_13.csv 92
test4_13.csv
14 16 test3_14.csv 100
test4_14.csv
15 18 test3_15.csv 108
test4_15.csv
16 18 test3_16.csv 112
test4_16.csv
17 20 test3_17.csv 118
test4_17.csv
18 20 test3_18.csv 122
test4_18.csv
19 22 test3_19.csv 128
test4_19.csv
20 22 test3_20.csv 134
test4_20.csv
21 22 test3_21.csv 134
test4_21.csv
22 22 test3_22.csv 140
test4_22.csv
23 24 test3_23.csv 146
test4_23.csv
24 24 test3_24.csv 146
test4_24.csv
25 24 test3_25.csv 152
test4_25.csv
26 24 test3_26.csv 158
test4_26.csv
27 26 test3_27.csv 160
test4_27.csv
28 26 test3_28.csv 162
test4_28.csv
29 26 test3_29.csv 166
test4_29.csv
30 26 test3_30.csv 166
test4_30.csv
40 32 test3_40.csv 198
test4_40.csv
50 34 test3_50.csv 214
test4_50.csv
60 38 test3_60.csv 238
test4_60.csv
70 40 test3_70.csv
test4_70.csv
80 42 test3_80.csv
test4_80.csv
90 44 test3_90.csv
test4_90.csv
100 44 test.3_100csv
test.4_100csv