New mathematical derivations applicable to safety and reliability analysis

Boolean logic expressions are often derived in safety and reliability analysis. Since the values of the operands are rarely exact, accounting for uncertainty with the tightest justifiable bounds is important. Accurate determination of result bounds is difficult when the inputs have "constraints." One example of a constraint is that an uncertain variable that appears multiple times in a Boolean expression must always have the same value, although the value cannot be exactly specified. A solution for this "repeated variable" problem is demonstrated for two Boolean classes. The classes, termed functions with "unate" variables (including but not limited to unate functions), and "exclusive-or" functions, frequently appear in Boolean equations for uncertain outcomes portrayed by logic trees (event trees and fault trees).