Modularizing, Minimizing, and Interpreting the K&H Fault-Tree

Abstract

The verification by Worrell & Stack (W&S) of results previously obtained by Kumamoto & Henley (K&H) for a fault tree of an s-noncoherent system and its inverse and their correction of the three errors in the tree makes it possible to simplify the analysis by forming modules; this facilitates Boolean algebraic operations, so that both sets are described economically in their minimal forms, a subset of the prime implicants (p.i.'s). Quine's consensus operation is used to minimize and to find the p.i.'s. Corresponding to the MOCUS output for the inverse reported by K&H, which is neither minimal nor the set of p.i.'s, instead of 32 terms, there are 15 in the modularized set. Instead of the 42 p.i.'s obtained by both K&H and W&S, we have 17; 13 of these are a unique minimal form. Instead of 352 p.i.'s for the tree per both K&H and W&S, we have a 15-term minimal form, identical to the list of p.i.'s. The results are further analyzed as a contribution to the continuing discussion of the utility of minimal forms vis-a-vis the p.i.'s and of the consensus method.