Compositional Abstraction Refinement for Component-Based Systems

Algorithm 2

Abstraction refinement.

    Input: Abstract system 𝒮^𝛼=(𝛾(𝐵^𝛼1₁,…,𝐵^𝛼𝑛_𝑛),𝐼𝑛𝑖𝑡^𝛼),

      set of counterexamples 𝐸𝑟𝑟, interaction invariant Ψ

    Output: 𝑡𝑟𝑢𝑒 or 𝑓𝑎𝑙𝑠𝑒

(1)   𝑅𝑒𝑓𝑖𝑛𝑒𝑑∶=𝑓𝑎𝑙𝑠𝑒, 𝐸𝑟𝑟_𝐼∶=0/, 𝐸𝑟𝑟_𝐼𝐼∶=0/

(2)   foreach  Φ_𝑐∈𝐸𝑟𝑟  do

(3)    𝐸𝑟𝑟∶=𝐸𝑟𝑟∖ Φ_𝑐

(4)    𝐼𝑛𝑣𝑆𝑡𝑟_Φ𝑐∶=𝐼𝑛𝑣𝑎𝑟𝑆𝑡𝑟𝑒𝑛𝑔𝑡ℎ𝑒𝑛(𝒮^𝛼,Φ_𝑐)

(5)    if  𝐼𝑛𝑣𝑆𝑡𝑟_Φ𝑐=𝑡𝑟𝑢𝑒  then

(6)     𝐸𝑟𝑟_𝐼𝐼∶=𝐸𝑟𝑟_𝐼𝐼∪Φ_𝑐

(7)    else

(8)     𝐸𝑟𝑟_𝐼∶=𝐸𝑟𝑟_𝐼∪Φ_𝑐

(9)    end

(10)  end

(11)   𝐸𝑟𝑟∶=𝐸𝑟𝑟_𝐼𝐼

(12)  if  𝐸𝑟𝑟_𝐼𝐼=0/  then

(13)    return  𝑓𝑎𝑙𝑠𝑒

(14)  else

(15)     foreach  Φ_𝑐∈𝐸𝑟𝑟_𝐼𝐼  do

(16)    if  𝑆𝑝𝑙𝑖𝑡𝐴𝑏𝑠𝑡𝑟𝑎𝑐𝑡𝑖𝑜𝑛(𝒮^𝛼,Φ_𝑐)=𝑡𝑟𝑢𝑒  then

(17)       𝑅𝑒𝑓𝑖𝑛𝑒𝑑∶=𝑡𝑟𝑢𝑒

(18)    end

(19)    end

(20)     from split refinement abstract system 𝒮^𝛼_𝑟𝑒𝑓, compute

      new compute interaction invariant Ψ

(21)    foreach  Φ_𝑐∈𝐸𝑟𝑟_𝐼  do

(22)    Ψ∶=Ψ∧𝐼𝑛𝑣𝑆𝑡𝑟_Φ𝑐

(23)    end

(24)    return  𝑅𝑒𝑓𝑖𝑛𝑒𝑑

(25)  end