Algebraic properties of system composition in the LORAL, Ulysses and McLean trace models

G.W. Dinolt, D. McCullough, and J. McLean, provide three approaches to modeling security properties of information systems. Dinolt´s approach is to model a system as a collection of sequences of ordered pairs of inputs and outputs in which both inputs and outputs are defined as sets of primitive “information units”. McCullough´s approach is to model a system as a sequence of primitive “events” in which an event is distinguished as either an “output”, an “input”, or an “internal event”. In McLean´s model, trace sets are sequences of ordered pairs (x_i,y_i) where x_i is an input word of length j and y_i is an output word of length k. In all models we are provided with a definition of “composition”, that is, a definition which permits the combining of two systems into a more complex product system. We prove the associative laws, commutative laws, and “containment relationships” for the composition of event systems defined by Dinolt, McCullough, and McLean. These algebraic properties serve as useful characteristics in the comparisons of these three types of event systems