An analytic solution for a fault tree with circular logics in which the systems are linearly interrelated

A circular logic or a logical loop is defined as the infinite circulation of supporting relations due to their mutual dependencies among the systems in the fault tree analysis. While many methods to break the circular logic have been developed and used in the fault tree quantification codes, the general solution for a circular logic is not generally known as yet. This paper presents an analytic solution for circular logics in which the systems are linearly interrelated with each other. To formulate the analytic solution, the relations among systems in the fault tree structure are described by the Boolean equations. The solution is, then, obtained from the successive substitutions of the Boolean equations, which is equivalent to the attaching processes of interrelated system's fault tree to a given fault tree. The solution for three interrelated systems and their independent fault tree structures are given as an example.