Complexity Analysis of Weak Multitolerance

In this paper, we classify multitolerant systems, i.e., systems that tolerate multiple classes of faults and provide potentially different levels of tolerance to them in terms of strong and weak multitolerance. Intuitively, this classification is based upon the guarantees provided by the program when one class of faults occurs while it is recovering from another class of faults. We focus on automated synthesis of weak multitolerant programs. Such weak multitolerance becomes necessary when it is impossible to provide strong multitolerance and/or when the probability of one class of faults occurring while the program is `recovering' from a fault from another class is negligible. By considering the levels of fault-tolerance provided to each class of faults, we evaluate five possible combinations for weak multitolerance. We find a counterintuitive result that if masking fault-tolerance is desired for one class of faults and masking (or failsafe) fault-tolerance is desired for another class of faults then the problem is NP-hard. This result is surprising since the corresponding problem for strong multitolerance can be solved in polynomial time. Also, we show that the problem of synthesizing weak multitolerance for other combinations is in P. More broadly, this result demonstrates the role of assumptions, e.g., independence of occurrences of faults from different classes, in the complexity of automated synthesis.