Multiple-Valued Logic Trees: Meaning and Prime Implicants

Multiple-valued logic (MVL) allows assessment of the reliability of systems where the inherent complexity leads to consider a discrete number of non-ordered classes of equivalent behaviors (CEBs). The concept of equivalent behavior is introduced to provide a comprehensive description of states and state transitions in the system and its components. MVL tree links all CEBs of interest in the engineered system by means of a combination of the CEBs of elementary components. Logic operators in the tree are defined by truth tables. The analysis of the MVL tree is aimed at eliciting the prime implicants for any one of the CEBs of the system or their groups. Prime implicants correspond to the largest set of CEBs in the smallest number of elementary components, thus extending the concept of minimal cut set which is found in binary fault-tree analysis. In reliability assessment, partitions among equivalent behaviors are often the result of subjective judgment. The methods presented in the paper for finding prime implicants are based upon criteria that recover unnecessary partitions among different CEBs. The methods indeed entail no theoretical constraint with regard to the number of CEBs that can be dealt with or their ordering. Our focus is on the tabular method, the Nelson algorithm, and generalized consensus. The tabular method has tutorial interest only. The Nelson algorithm appears easier to implement in a computer code, although the algorithm asks for the computation of the complete base of prime implicants as an unavoidable step.