On the Hardness of Adding Nonmasking Fault Tolerance

This paper investigates the complexity of adding nonmasking fault tolerance, where a nonmasking fault-tolerant program guarantees recovery from states reached due to the occurrence of faults to states from where its specifications are satisfied. We first demonstrate that adding nonmasking fault tolerance to low atomicity programs-where processes have read/write restrictions with respect to the variables of other processes--is NP-complete (in the size of the state space) on an unfair or weakly fair scheduler. Then, we establish a surprising result that even under strong fairness, addition of nonmasking fault tolerance remains NP-hard! The NP-hardness of adding nonmasking fault tolerance is based on a polynomial-time reduction from the 3-SAT problem to the problem of designing self-stabilizing programs from their non-stabilizing versions, which is a special case of adding nonmasking fault tolerance. While it is known that designing self-stabilization under the assumption of strong fairness is polynomial, we demonstrate that adding self-stabilization to non-stabilizing programs is NP-hard under weak fairness.