Irredundant Forms and Prime Implicants of a Function with Multistate Variables

For s-coherent fault trees containing only AND/OR gates, many algorithms can be used to obtain the sum-of-products (s.o.p.), cut set, expressions for the top event. If the tree contains non-coherences such as XOR (exclusive OR) gates, these s.o.p. expressions can be reduced to irredundant prime implicant form by algorithms such as Kumamoto & Henley's or by applying simplification and consensus algorithms such as Nelson's or Quine's. If, however, the trees contain multistate variables, then the Boolean binary logic expressions on which present algorithms are based no longer apply. This paper extends the laws of binary Boolean algebra to encompass multistate variables, and develops simplification and consensus algorithms whereby prime implicants for non-coherent systems containing multistate variables can be obtained. A computer code based on this has been developed.