G
Components of a Theory of Modeling and Simulation
Bernard Zeigler, University of Arizona
The text of this report calls for further work in developing, extending, and communicating theories of modeling and simulation (M&S). This appendix sketches some key features that any theory of M&S should have. In particular, a theory should provide a basic foundation and framework, formalisms for defining and manipulating concepts, methodologies for representation and abstraction, and mechanisms for executing the models (e.g., turning them into computer programs). What follows focuses specifically on models of dynamic systems, that is, models whose variables change in value over time.
FOUNDATION
To deal with the foregoing issues, a theory of M&S needs to establish a mathematical, rigorous foundation upon which to base its formalization of the elements and relationships it has identified. Although the foundation will necessarily be more difficult to comprehend than ordinary language, its underlying concepts should be understandable to people who are not mathematical experts. The advantages of having such a rigorous foundation are readily stated. One concerns communication: many of the confusions that impede progress are due to terms, such as “model,” that have different meanings across disciplines. A universally accepted theory of M&S would provide the common conceptual framework and vocabulary for people from different backgrounds to communicate effectively. A second advantage is that rigorous principles provide the means to tackle problems beyond the reach of more informal methods. The value of this is clear from other areas such as physics.
Some of the requirements that such a foundation should satisfy are as follows:

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.
FRAMEWORK
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 realsystem 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.
^{1 } 
For a review, see Pichler and Schwartzel (1992). 
A full framework should identify just the right elements and relationships to facilitate all aspects of the practice of M&S—including aspects involving portability, reuse, and composability.
FORMALISMS
A framework should provide basic concepts, but theories must accomplish a good deal more—allowing workers to reason rigorously about issues, derive theorems, prove correctness of simulators, and so on. As a result, theories require formalisms. Formalisms are typically mathematical languages. One example is the predicate calculus.
Set theory is a common way to construct formalisms. Assuming use of set theory, a formalism for M&S should have a number of attributes:

The theory should characterize the three basic types of simulation model (differentialequation, discretetime (or timestepped), and discreteevent (or eventbased)) through use of settheoretic formalisms, which should also expose their commonalities and differences.

The theory should specify means of composing models in the basic formalisms from more elementary pieces. One means of composing models is by connecting outputs to inputs. Such coupling should work on wellspecified input and output interfaces, without reference to internal structure.

The basic formalisms should be closed under coupling. This means that coupling models expressed within a formalism should only produce composite models that can also be expressed in the formalism. This supports hierarchical construction, modular reuse, and hierarchical simulators.

The theory should support combination and extension of the basic formalisms. An example of a combination is the formalism combining differentialequation and discreteevent formalism for hybrid modeling. An example of an extension is that in which a formalism allows models to change their structure over time.

More generally, the theory should provide a methodology by which a new specialized formalism can be instituted by defining the subclass of systems that the formalism specifies.

The theory should also provide a methodology by which the coupling of a new formalism is definable and its closureundercoupling properties demonstrated.
SIMULATORS
In many cases, models will define relationships capturing key aspects of the system being treated. In themselves, however, they may not generate predictions. As an example here, Newton's laws do not themselves tell us how a falling
body's altitude will change with time. For that we need to compute the implications of the model.
Simulators are the computational devices (be they algorithms, programs, hardware, or networks) that execute models to generate their time behavior. A theory of M&S must deal with simulators:

For each modeling formalism, the theory should provide a simulator concept that can execute any model in the class (i.e., generate the system's estimated time behavior). (One way to develop such concepts is through objectoriented frameworks.)

Such frameworks should enable the development of verifiable, efficient, and interoperable implementations in an openended variety of contexts and platforms (e.g., parallel, distributed).

The theory should provide a methodology by which the simulator framework for a new formalism is definable and logical correctness demonstrated. This means that there should be a way to define the simulator for a new class of models and provide its correctness for that class.

The theory should characterize, and provide a means to estimate, the computational complexity of a model. Roughly, the computational complexity is measured by resources required by the most efficient simulator to simulate it.
REPRESENTATION AND ABSTRACTION
The theory should deal with the representation of systems as models and the abstraction of models into (usually simpler) models. There are mathematical concepts, called “morphisms” that provide the formal equivalent of the relations underlying representation and abstraction. For example, an isomorphism between two (mathematical) groups is a onetoone correspondence between their elements that preserves their group operations. Such groups are said to be “isomorphic” or “equivalent.”

The theory should permit transforming a model expressed in one formalism into an equivalent expressed in a different formalism (e.g., differential equation models can be cast into discreteevent equivalents, which are computationally more efficient).

Further, the theory should provide a methodology for characterizing the class of dynamic systems that can be represented by a formalism under a prescribed morphism relation (e.g., the class of piecewise constant inputoutput systems is known to be representable by discreteevent simulation).

The theory should provide the basis for abstraction, that is, transforming a model into an equivalent with reduced complexity, within a specified experimental frame.

The theory should provide an openended set of abstraction methods (e.g., aggregation and omission) and characterize their applicability. The theory should characterize, and provide a means to estimate, the simulational complexity of a model (typically, abstractions are intended to reduce such complexity).
ENCOMPASSING THEORIES
The theory should provide the elements of manipulation for more encompassing theories such as those of systems engineering, design, and management. ^{2}
^{2 } 
For additional reading, see Praehofer (1991), Zeigler (1976), Pichler and Schwartzel (1992) and Zeigler et al. (1993). 