Locating arrays with error correcting ability Masakazu Jimbo joint work with Xiao-Nan Lu † Chubu University ‡ Tokyo University of Science Dedicated to Professor Helleseth’s 70-th birthday MMC Workshop, September 7, 2017. Masakazu Jimbo (Chubu Univ.) Locating arrays with error correcting ability Sept. 7 1 / 23
An example of interaction testing: Software testing ▶ Applications F = { f 1 , f 2 , . . . , f k } are installed in a PC. ▶ Each application has two states S= { 1, 2 } . f 1 f 2 · · · f k ▶ Some state of some application ( f , σ ), t 1 1 2 · · · 1 ( σ ∈ S ) or such a combination t 2 2 2 · · · 2 { ( f i , σ i ) , ( f j , σ j ) } may cause a “fault” in PC. . . . . ... . . . . ▶ A pair ( f , σ ) is called a 1-way interaction. A . . . . combination of pairs { ( f i , σ i ) , ( f j , σ j ) } is 2 1 · · · 2 t N called a 2-way interaction. ▶ We want to find such faulty interactions by designing a testing array of testing suits. Masakazu Jimbo (Chubu Univ.) Locating arrays with error correcting ability Sept. 7 2 / 23
An example of interaction testing: Software testing ▶ Applications F = { f 1 , f 2 , . . . , f k } are installed in a PC. ▶ Each application has two states S= { 1, 2 } . f 1 f 2 · · · f k ▶ Some state of some application ( f , σ ), t 1 1 2 · · · 1 ( σ ∈ S ) or such a combination t 2 2 2 · · · 2 { ( f i , σ i ) , ( f j , σ j ) } may cause a “fault” in PC. . . . . ... . . . . ▶ A pair ( f , σ ) is called a 1-way interaction. A . . . . combination of pairs { ( f i , σ i ) , ( f j , σ j ) } is 2 1 · · · 2 t N called a 2-way interaction. ▶ We want to find such faulty interactions by designing a testing array of testing suits. Masakazu Jimbo (Chubu Univ.) Locating arrays with error correcting ability Sept. 7 2 / 23
An example of interaction testing: Software testing ▶ Applications F = { f 1 , f 2 , . . . , f k } are installed in a PC. ▶ Each application has two states S= { 1, 2 } . f 1 f 2 · · · f k ▶ Some state of some application ( f , σ ), t 1 1 2 · · · 1 ( σ ∈ S ) or such a combination t 2 2 2 · · · 2 { ( f i , σ i ) , ( f j , σ j ) } may cause a “fault” in PC. . . . . ... . . . . ▶ A pair ( f , σ ) is called a 1-way interaction. A . . . . combination of pairs { ( f i , σ i ) , ( f j , σ j ) } is 2 1 · · · 2 t N called a 2-way interaction. ▶ We want to find such faulty interactions by designing a testing array of testing suits. Masakazu Jimbo (Chubu Univ.) Locating arrays with error correcting ability Sept. 7 2 / 23
An example of interaction testing: Software testing ▶ Applications F = { f 1 , f 2 , . . . , f k } are installed in a PC. ▶ Each application has two states S= { 1, 2 } . f 1 f 2 · · · f k ▶ Some state of some application ( f , σ ), t 1 1 2 · · · 1 ( σ ∈ S ) or such a combination t 2 2 2 · · · 2 { ( f i , σ i ) , ( f j , σ j ) } may cause a “fault” in PC. . . . . ... . . . . ▶ A pair ( f , σ ) is called a 1-way interaction. A . . . . combination of pairs { ( f i , σ i ) , ( f j , σ j ) } is 2 1 · · · 2 t N called a 2-way interaction. ▶ We want to find such faulty interactions by designing a testing array of testing suits. Masakazu Jimbo (Chubu Univ.) Locating arrays with error correcting ability Sept. 7 2 / 23
Interaction testing: Terminologies ▶ Let F = { f 1 , f 2 , . . . , f k } be the set of k factors . ▶ For each f ∈ F , let S = { 1 , 2 , . . . , s } be the set of possible levels or values . ▶ A t -way interaction is a choice of a set K of t factors, and a selection of a value σ f ∈ S for each factor f ∈ K . ( F ) { } T = ( f , σ f ) | f ∈ K with K ∈ , σ f ∈ S t ▶ A test is a k -tuple indexed by the factors, and the coordinate indexed by f has an entry in S . ▶ A test suit is a collection of tests. ▶ It is natural to use an N × k array A = ( a rf ) to present a test suit consisting of N tests and k factors . Masakazu Jimbo (Chubu Univ.) Locating arrays with error correcting ability Sept. 7 3 / 23
Interaction testing: Problems Assumptions: ▶ Each test gives a result 0 (pass) or 1 (fail). ▶ Failures are caused by an i -way interaction with i ≤ t . Problem: ▶ Is there an i -way interaction causing faults? ▶ Which are they? ▶ Given k and t , how many tests ( N ) are required? Combinatorial testing arrays: ▶ Covering arrays ▶ Locating arrays ▶ Detecting arrays Masakazu Jimbo (Chubu Univ.) Locating arrays with error correcting ability Sept. 7 4 / 23
Interaction testing arrays and a t -locating array ▶ Suppose A = ( a rf ) is an N × k testing array. ▶ let K be a t -subset of the column indices of A . ▶ A t -way interaction T = { } ( f , σ f ) | f ∈ K appears in the r -th row ⇔ a rf = σ f for each f ∈ K . ▶ ρ A ( T ) consists of the rows indices r of A in which the t -way interaction T appears, namely ρ A ( T ) = { r | a rf = σ f for each f ∈ K } . How can we find faults? ▶ Let T be the set of i -way interactions for i ≤ t . And assume that there is only one i -way interaction which causes failure in T . ▶ An array A can detect any single failure in T iff ρ A ( T )’s are distinct for all T in T . ▶ Such an array A is called a ¯ t -locating array. Masakazu Jimbo (Chubu Univ.) Locating arrays with error correcting ability Sept. 7 5 / 23
Interaction testing arrays and a t -locating array ▶ Suppose A = ( a rf ) is an N × k testing array. ▶ let K be a t -subset of the column indices of A . ▶ A t -way interaction T = { } ( f , σ f ) | f ∈ K appears in the r -th row ⇔ a rf = σ f for each f ∈ K . ▶ ρ A ( T ) consists of the rows indices r of A in which the t -way interaction T appears, namely ρ A ( T ) = { r | a rf = σ f for each f ∈ K } . How can we find faults? ▶ Let T be the set of i -way interactions for i ≤ t . And assume that there is only one i -way interaction which causes failure in T . ▶ An array A can detect any single failure in T iff ρ A ( T )’s are distinct for all T in T . ▶ Such an array A is called a ¯ t -locating array. Masakazu Jimbo (Chubu Univ.) Locating arrays with error correcting ability Sept. 7 5 / 23
Example: Non 1 -locating array An s -ary testing array with N = 6, k = 9 and s = 3 levels for each factor. f 1 f 2 f 3 f 4 f 5 f 6 f 7 f 8 f 9 outcome t 1 1 3 3 3 2 2 1 3 2 1 t 2 2 1 3 3 3 2 2 1 1 1 t 3 2 2 1 3 3 3 1 2 3 0 t 4 3 2 2 1 3 3 3 1 3 0 3 3 2 2 1 3 2 3 1 0 t 5 3 3 3 2 2 1 3 2 2 0 t 6 ▶ Assume there is at most one 1-way interaction causing faults. ▶ Outcome says that t 1 , t 2 have the same value σ and t 3 , t 4 , t 5 , t 6 are different from σ . ▶ Such a 1-way interaction is ( f 6 , 2). ▶ { t 1 , t 2 } = ρ (( f 6 , 2)). Masakazu Jimbo (Chubu Univ.) Locating arrays with error correcting ability Sept. 7 6 / 23
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