The foundation should be general, and it should be expressive enough to subsume the great variety of special formalisms, languages, and modes of expression prevalent in M&S practice.
The foundation should incorporate the concepts of dynamic systems theory. Dynamic systems theory has provided a uniform set of concepts that help to understand how objects change in time, that is, their dynamics, and how these behaviors are related to the objects' underlying mechanisms or structure. General systems theory represents the convergence of rich traditions, in areas such as control theory and automata theory, to a common mathematical conception of a dynamic system. 1
More specifically, models should be formulated as means to specify dynamic systems. That is, a model should be understood as a combination of equations, rules, and constraints that, when correctly interpreted, describes a unique dynamic system from the collection of all such objects.
Any theory of M&S should establish a framework identifying and defining the key elements of M&S and their relationships. As indicated, the theory can employ the powerful foundation of dynamic systems theory to express these elements and their interrelations. In choosing what to identify as key elements, the theory should draw on the actual practice of M &S so as to highlight distinctions that are indeed significant. As examples here, it is important to distinguish among the real system, a model, a simulator (e.g., a simulation program or a hardware flight simulator), and what is sometimes called the experimental frame. The model is an attempt to describe aspects of the real system in a specific context such as estimating the likely time dependence of a real-system variable for any of a specified set of initial conditions. A simulation program might generate that estimated behavior using the model's equations, rules, and constraints. The experimental frame specifies the input stimuli, outputs of interest, and context of use. Thus, it is closely related to the concept of experimental design.
Any framework for M&S should facilitate discussion of meaningful relationships among key elements. For example, it is important to be able to discuss the validity of simulated model behavior with respect to the real system in a particular experimental frame. That is, validity is a relationship measured for a context. Another example of a meaningful relationship is whether a simulator such as a simulation program has been verified as representing the model adequately, again in the context specified by the experimental frame. Numerical approximations, for example, might be entirely acceptable in one frame, but a source of unacceptable error in another.
For a review, see Pichler and Schwartzel (1992).