Static and Time Invariant Systems with Boolean Representation

• Published in: Systems Dependability Assessment pp. 1 - 11
• Main Authors: Brinzei, Nicolae, Aubry, Jean-François
• Format: Book Chapter
• Language: English
• Published: United States Wiley 2015; John Wiley & Sons, Inc
• Subjects: Boolean function; minimal cut‐sets; minimal tie‐sets; state diagram; stationary system; time invariant system
• ISBN: 9781848217652; 184821765X
• DOI: 10.1002/9781119053996.ch1

A system whose outputs are only dependent at any time on its variables states is generally called a time invariant system or stationary system. This chapter introduces formally two very general notions: the cut‐sets and the tie‐sets of a system. The structure function may be materialized on the Hasse diagram by affecting to the nodes different marks for the two values of Y : “1” or “0”. An interesting property of the state diagram lies in an easy identification of minimal cut‐sets or minimal tie‐sets. Two “immediate neighbors” nodes with Y = 1 correspond to adjacent minterms and represent together a reduced monomial of the Boolean function. By extension, it comes that a subgraph composed of all the paths from a minimal tie‐set to the maximum represents a prime reduced monomial. So all prime reduced monomials contain the minterm associated to the maximum of the graph.