Preface 2

This text is an outgrowth of different activities associated with topics of modeling, simulation, verification and validation. In the beginning these activities have been more or less 'side effects' of other main topics. But meanwhile these topics of Computer Aided Modeling, Simulation, and Verification (CAMSV) have become a central theoretical issue. This results from the fact that in science and even more in engineering it is not only necessary to elaborate explicitly formal theories (mathematical structures) as the main 'content' of these activities but also to use nearly all the time computers to enable human persons to deal with complex dynamic processes.

Very lately, between 1850 and 1950 modern logic and mathematics has arisen and more and more advanced activities in science and engineering became possible. During the 20th century we can observe a long and intensive discussion about the 'right' format of scientific theories. Under the label of theory of science or philosophy of science numerous papers and books have been published to analyze and describe different types of formal structures and associated experimental designs to define the nature of empirical and formal theories. At the beginning of the 21st century theory of science has not succeeded to establish a common formal framework for most disciplines. The 'failure' of the theory of science program had at least two reasons: (i) for a successful application of theory of science methodologies within a certain domain one needs knowledge in both areas, from meta theory as well as from the application domain. But until now there is no common methodology across all disciplines to enable such a common theory format. (ii) The practical application of theory of science only based on manual computations is mostly impossible. Therefore one needed heavily support by computers which only lately during the fourth quarter of the 20th century showed up. But a simple application of computers for theory of sciences had its own difficulties: the semantics of full blown theories in classical theory of sciences is usually far beyond the operational semantics which can be provided by a computer. This serious 'formal semantic gap' has not been solved until today.

Despite the non-success of computers associated with theory of science we can today observe a new paradigm -somehow in parallel to the theory of science program- arising and distributing to more and more disciplines: computational science. Although this 'broad' meaning is perhaps of no great help it is interesting why this paradigm has success at all. Primarily I can use computers for many applications without being forced to write a full blown theory first. I need only some simple algorithm to make a computer run. An algorithm is comparable to a function, and a function is something 'simple' compared to a complete theory. Therefore the 'entry level' for the usage of computer in the context of science and engineering is principally much more simpler than the entry level for a theory of science program. And indeed, if one looks to computational approaches they are lacking very often a clear theoretical status. Furthermore it makes some difference whether one is working in empirical sciences to find some explaining structure or in engineering to build a working solution compared to build a model of some given phenomena or concepts to make it better understandable (or make experiments which are cheaper).

This theoretical weakness of most computer applications is the reason why the broad term computational science should perhaps be replaced by the more concrete term Computer Aided Modeling, Simulation, and Verification (CAMSV). This term 'remembers' the main topics which belong to theory building and engineering. Every problem has first to be transformed into a model of the expected behavior, otherwise nobody knows exactly what has to be solved. Then, complex models can only be understood if one is able to simulate the behavior. Furthermore it must be possible to verify the simulated behavior against certain important criteria which the behavior should fulfill. In engineering one has to extend this general scheme insofar as the model of the behavior $M_{sr}$ has to be complemented by the model of a generating system $M_{sys}$ which then will become implemented into a concrete system $SYS$. The relationship $R(M_{sr}, M_{sys})$ is a kind of 'similarity' (morphism), which can be analyzed under the label of verification. The relationship $R(M_{sr},SYS)$ is that of validation. The full paradigm for an engineering context would then perhaps better be served by a term like Computer Aided Modeling3, Simulation3, Verification, and Validation (CAM3S3V2)

In the following it will be presented a lengthy example to enable a first idea of what modeling and simulation could mean. Then it follows the theory. Connected to the theoretical part there will be many examples which will be placed in a special section.

This paper will be work in progress for a long time. It started with German chapters which will be replaced step by step by new English versions.

Although there are many good papers and books related to the topic of modeling and simulation -see the references-, there is the book of Zeigler, Praehofer, and Kim (2000)[102], which we will use as main reference for modeling and simulation methodologies.