Inverting and Minimizing Boolean Functions, Minimal Paths and Minimal Cuts: Noncoherent System Analysis

An efficient technique is presented for inverting the minimal paths of a reliability logic diagram or fault tree, and then minimizing to obtain the minimal cuts, or else inverting the minimal cuts for the minimal paths. The method is appropriate for both s-coherent and s-noncoherent systems; it can also obtain the minimized dual inverse of any Boolean function. Inversion is more complex with s-noncoherence than with s-coherence because the minimal form (m.f.) is not unique. The result of inversion is the dual prime implicants (p.i.´s). The terms of a dual m.f., the dual minimal states, are obtained by a search process. First the dual p.i.´s are obtained; then a m.f. is found by an algorithmic search with a test for redundancy, reversal-absorption (r.a.). The dual p.i.´s are segregated into the ``core´´ p.i.´s [8,9] essential for every m.f. and the ``noncore´´ p.i.´s, by r.a. Then a m.f. is found by repeatedly applying r.a. to randomized rearrangements of the noncore terms. Examples are included, adapted from the fault-tree literature.