Computational General System Theory

Dynamic Systems. Because we are not only interested in a concrete moment of a system but also in its development and change over time, we have to find a formalism to include this in our model. Here a lot of scientific research has been done in the area of automata theory: Since Alan Turing did present his model of the so called “Turing Machine”, a lot of books and papers have been written about theoretical models of computational and human machines (so automata theory is not only used in computer sciences but also for example quite important for cellular automata theory in cell biology).

Using cellular automata for dynamic system simulation.

According tour model we will introduce automata characteristics when defining a system as the sum of its possible different states Q=q₁, … , q_n while every q is mapped to a triple (Γ, Δ, Θ). It is important that we do not use time as a continuous concept of states expressed by real numbers in our model at all. So developing of a system and functions expressing partial development always must refer to states. States focus on relations (so a relation θ₁ can occur in a state q_k but disappears in another state q_l) but can also be connected to classes and properties. Having a closer look, these two cases can be reduced to the case of adding and removing relations, because a class that is not connected to the system any longer might have no effects to the related application at all. In case of instances we also have changing sets of instances and - this might be an important feature - changing values of attributes over states.

Example. Let us assume an application for a social system where a relation isFriendOf exists (isFriendOf(Person, Person)). Here it is of great interest, how the . But also on the pure TBox level (so without any instances) modelling possible states is necessary: Just assume a (very complex) system model for human beings, where all attributes and relations can be put together in different states (e.g. from embryo to adult state).    

Dynamic logic rules. Dynamic logic rules can determine upcoming states of a system using logical operations and concepts of the actual and past states. They focus on boolean attributes or discrete attributes of other types and only determine single states but no continuous development of a system.

Example. Genes are responsible for phenotypes and in sexual biological systems combinations of genetic information of two parents is interesting because there are different types of genes. For example recessive genes are only active in the phenotype if both parents have them – according to a model of a biological system this would be represented in logical rules that determine future states of the system.

Dynamic functional rules. Dynamic functional rules can determine upcoming states of a system using mathematical functions and concepts of the actual and past states. But in difference to logical rules they express continuous developments and are therefore much more complicate.

Example. Examples are the functions of Ludwig von Bertalanffy to foresee fish populations by a differential system of equations. In human body also the development of the body temperature could be expressed by such functions.