A Systems Algebra and Its Applications

Since every system of any significant size is created by composition from smaller sub-systems or components, an attempt is made to analyze the properties of a system as a function of its composition. Using a partial ordering of system states based on reachability, system states are classified in the abstract into bad, hazardous, unsafe and safe states, and a safety function that separates these is derived. Two basic types of system composition are described, and an algebra to describe the safety and fault tolerance of composed systems is obtained. The set of systems forms monoids under the two composition operators, and a semiring when both are concerned. A partial ordering relation between systems is used to compare their fault-tolerance behaviors.