Fault detection problems for Boolean networks and Boolean control networks

In this paper we address two fault detection problems for Boolean control networks (BCNs). We assume that the BCN may exhibit only two possible configurations, a non-faulty and a faulty one. The fault is simply described as the switching from the non-faulty configuration to the faulty one, and we assume that the BCN cannot autonomously recover from the fault, unless some external intervention restores the regular working conditions. Finally, we suppose that the fault affects only the state-update, not the output measurements. In this set-up, we introduce the concepts of meaningful fault and of detectable meaningful fault. Two different situations are investigated: the case when fault detection must be performed on-line, under arbitrary working conditions, and hence corresponding to arbitrary inputs acting on the BCN, and the case when an off-line test is performed, by making use of a specific input, in order to test whether the BCN is non-faulty or faulty. Complete characterizations and an algorithm to practically perform the tests in both cases are presented. The obtained results for on-line fault detection are finally particularized to the special case of Boolean networks (BNs).