Compositional Abstraction Refinement for Component-Based Systems

Algorithm 1

Iteration verification.

    Input: Component system 𝒮=(𝛾(𝐵₁,…,𝐵_𝑛),𝐼𝑛𝑖𝑡), property Φ

    Output: 𝑡𝑟𝑢𝑒 or 𝑓𝑎𝑙𝑠𝑒

(1)   Φ_𝑖=𝑡𝑟𝑢𝑒 for each 𝑖=1,…,𝑛, 𝑅𝑒𝑓𝑖𝑛𝑎𝑏𝑙𝑒∶=𝑡𝑟𝑢𝑒

(2)   for  𝑖←1  to  𝑛  do

(3)      compute the component invariant Φ^′_𝑖 based on 𝐵_𝑖

(4)      Φ_𝑖∶=Φ_𝑖∧Φ^′_𝑖

(5)      compute the corresponding abstraction 𝐵^𝛼𝑖_𝑖 based on 𝐵_𝑖 and Φ_𝑖

(6)   end

(7)   from 𝛾(𝐵^𝛼1₁,…,𝐵^𝛼𝑛_𝑛), compute 𝐿₁,𝐿₂,…,𝐿_𝑚

(8)   for  𝑘←1  to  𝑚  do

(9)      Ψ_𝑘=⋁_{𝜙∈𝐿𝑘}𝛼⁻¹(𝜙)

(10)  end

(11)   Ψ∶=⋀^𝑚_𝑘=1Ψ_𝑘

(12)  if  ¬Φ∧Ψ∧(⋀^𝑛_𝑖=1Φ_𝑖) = 𝑓𝑎𝑙𝑠𝑒  then

(13)     Φ is an invariant of 𝒮

(14)     return  𝑡𝑟𝑢𝑒

(15)  else

(16)     compute counterexample set 𝐸𝑟𝑟∶={Φ_𝑐}

         𝑅𝑒𝑓𝑖𝑛𝑎𝑏𝑙𝑒∶=𝐴𝑏𝑠𝑅𝑒𝑓𝑖𝑛𝑒𝑚𝑒𝑛𝑡(𝒮^𝛼,𝐸𝑟𝑟,Ψ)

(17)     if  𝑅𝑒𝑓𝑖𝑛𝑎𝑏𝑙𝑒=𝑓𝑎𝑙𝑠𝑒&&𝐸𝑟𝑟≠0/  then

(18)     return  𝑓𝑎𝑙𝑠𝑒

(19)     end

(20)     if  𝐸𝑟𝑟=0/  then

(21)     return  𝑡𝑟𝑢𝑒

(22)     else

(23)     goto 12

(24)     end

(25)  end