Minimal cut sets
A minimal cut set (MCS) of a fault tree is a smallest set of basic events whose joint occurrence causes the top event. "Smallest" is what makes it minimal: removing any element from the set breaks the property. Minimal cut sets are the canonical Boolean simplification of a fault tree and the input for nearly every quantitative measure that follows.
Formal definition
A cut set C is any subset of basic events whose simultaneous failure makes the top event TRUE. C is minimal iff no proper subset of C is itself a cut set. The collection of all minimal cut sets is unique for a given tree (with NOT gates absent) and is computed by algorithms like MOCUS or BDD-based methods.
Worked example
Consider TOP = OR(AND(a, b), c, AND(a, b, d)). Expanding gives the cut sets {a, b}, {c}, and {a, b, d}. The third is a strict superset of the first, so it's not minimal — drop it. Final minimal cut sets:
{a, b} ← order 2: both a and b must fail
{c} ← order 1: c alone fails the system
The order-1 cut set {c} is a single point of failure — usually a red flag for safety-critical functions. ISO 26262 ASIL-D and ARP 4761 DAL-A explicitly require that no order-1 cut set defeat a top-level safety goal.
How cut sets quantify the top event
Once you have minimal cut sets, the top-event probability is bounded above by the rare-event approximation:
P(top) ≤ Σ P(cut set i)
= Σ Π P(basic event j ∈ cut set i)
The exact form (inclusion-exclusion) is more accurate but expensive on large trees; the rare-event upper bound is conservative and is what most quantification tools default to. Importance measures like Birnbaum and Fussell-Vesely are computed directly from the cut-set list.