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Effect of Fairness in Model Checking of Self-stabilizing programs

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## Effect of Fairness in Model Checking of Self-stabilizing programs

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**Effect of Fairness in Model Checking of Self-stabilizing**programs Jingshu Chen, FuadAbujarad and SandeepKulkarni**Outline**• Problem Statement • Related work • Our approach • Results • Summary**Problem Statement**• To verify self-stabilizing programs by symbolic model checking • stabilization: • ability of a system to converge in finite number of steps from arbitrary states to desired state. • complex algorithm • Symbolic model checking: doesn’t require the designer to have considerable experience in logic reasoning.**Case Study-K-state Program**Two actions: • x0= xn -> x0=(x0+1) mod K • xi!=xi-1 ->xi=x(i-1) Note that: • the domain of x is [0,..,K-1] • This program is known to be self-stabilizing if K>N. In subsequent discussion, we let K=N+1. p0 p1 pn x0 p2 xn pn-1 p3 xi-1 xi pi-1 pi**Case Study-K-state Program**Legal state: • For Process 0 either x[0]=x[n] or x[0]=(x[n]+1) mod K • For i=1.. N, either x[i-1]=x[i] or x[i-1]=x[i]+1 p0 p1 pn x0 p2 xn pn-1 p3 xi-1 xi pi-1 pi**Previous work**• T’s work has demonstrate feasibility of applying symbolic model checking for verifying self-stabilizing programs. • The result shows that verification is feasible only for programs with a small number of process.**Approach(1)**• Observation: - the current approach is done under weak fairness computation; - current model checker focus on weak fairness in representation of fairness. • Our approach is to verify self-stabilization under unfair computation.**Case Study- K-state program (k=3)**Verification under weak fairness Two actions: • x0= xn -> x0=(x0+1) mod K • xi!=xi-1->xi=x(i-1)**Case Study- K-state program (k=3)**Verification under unfair computation**Results(1)**In spite of the improved hardware, the ability to verify self-stabilizing programs under weak fairness remains essentially the same. Scalability of verifying self-stabilization can be significantly improved for the case where the program is correct self-stabilizing without fairness.**Approach(2)**• For the case where weak fairness is essential for self-stabilization, • Decomposition • Utilizing the weak version of stabilization-weak stabilization**Results(3)**Verification of weak stabilization is substantially more scalable. This result also validates the suggestion in Gouda’s work that weak stabilization is easier to verify than self-stabilization.**Summary**• If self-stabilization is possible without fairness then cost of verifying self-stabilization is substantially lower. • This is the first paper that has shown feasibility of verifying the typical self-stabilizing programs, e.g., K-state program, with large number of processes. • We also identify two approaches for those cases where weak fairness is essential for self-stabilization.