On the universal combination function and the universal difference equation for reified temporal-causal network models

The universal differential and difference equation form an important basis for reified temporal-causal networks and their implementation. In this chapter, a more in depth analysis is presented of the universal differential and difference equation. It is shown how these equations can be derived in a direct manner and they are illustrated by some examples. Due to the existence of these universal difference and differential equation, the class of temporal-causal networks is closed under reification: by them it can be guaranteed that any reification of a temporal-causal network is itself also a temporal-causal network. That means that dedicated modeling and analysis methods for temporal-causal networks can also be applied to reified temporal-causal networks. In particular, it guarantees that reification can be done iteratively in order to obtain multilevel reified network models that are very useful to model multiple orders of adaptation. Moreover, as shown in Chap. 9, the universal difference equation enables that software of a very compact form can be developed, as all reification levels are handled by one computational reified network engine in the same manner. Alternatively, it is shown how the universal difference or differential equation can be used for compilation by multiple substitution for all states, which leads to another form of implementation. The background of these issues is discussed in the current chapter.