A Divide-and-Conquer Approach for Solving Fuzzy Max-Archimedean 𝑡-Norm Relational Equations

Algorithm 1

Algorithm for finding all irredundant coverings.

Algorithm  𝐏𝐀(𝑀)

Input: a binary matrix 𝑀=𝑚_{𝑖𝑗|𝐼|×|𝐽|} with 𝑖∈𝐼 and 𝑗∈𝐽.

Output: Φ(𝑀)

 (1)  If  𝐽 is a singleton 𝑗  then

 (2)     return  {𝑖}𝑖∈𝐼_𝑗(𝑀);

 (3)  If  𝐽=∅  then

 (4)     return  {∅};

 (5)  If  ∃𝑗∈𝐽 such that 𝐼_𝑗(𝑀) = ∅  then

 (6)     return  ∅;

 (7)  If  there exists a singleton 𝐼_𝑗(𝑀) = {𝑖} for some 𝑗∈𝐽  then

 (8)     let  𝑀^′∶=𝑀[𝐼⧵{𝑖};𝐽⧵𝐽_𝑖(𝑀)]

 (9)     return  {{𝑖}}⋈  PA(𝑀^′);                             // singleton removal

 (10)  If  Ψ(𝑀) = 𝑀¹,𝑀²,…,𝑀^𝑝 and 𝑝>1  then

 (11)     return  PA(𝑀¹)  ⋈  PA(𝑀²)  ⋈⋯⋈  PA(𝑀^𝑝);             // partitioning

 (12)  Let  row 𝑖 be the row with the most 1s in 𝑀;

 (13)  Let  𝑀^×∶= M[𝐼⧵{𝑖}; 𝐽] and 𝑀⁻∶= M[𝐼⧵{𝑖}; 𝐽⧵𝐽_𝑖(𝑀)];

 (14)  Let  Φ(𝑀⁻)∶=  PA (𝑀⁻) and Φ(𝑀^×)∶=  PA (𝑀^×);

 (15)  Let  𝑀⁺∶=𝑀[𝐼⧵{𝑖};𝐽_𝑖(𝑀)];

 (16)  Let  Δ∶=𝐶∈Φ(𝑀⁻)𝐶isnotacoveringof𝑀⁺;

 (17)  Return  ({{𝑖}}⋈Δ)⋃Φ𝑀^×;                           // forced binding