Axiomatic Systems in Concurrency Theory

There are many ways to examine the nature of time and space. Here only one possibility is pursued: the description of the numerous concepts relating to 'time' by the means of axioms and mathematical definitions. Prof. Carl Adam Petri investigated the subject when trying to present a theory about systems, signals, processes, space, and time in order to relate net theory to the principles of physics. Since net theory would then have more in common with the real world, it would probably be more suited to practical applications. In the paper Petri published his ideas for the first time, most notably he introduced the concurrency relation co for C/E-systems. This relation grew more and more important during the development of the theory and finally became the focus of interest. It also gave rise to the name 'concurrency theory'. Few results about concurrency theory were proved after Petri's initial effort, but then a connection to the theory of net systems was uncovered. To do this, five additional assumptions were made that could not be proved from the existing axioms, effectively enlarging the number of axioms. The physical background of concurrency theory is not sufficently explored up to now. The impact of each single axiom on practical applications of the theory is still unclear. Any collection of axioms that could be presented today would be tentative and would probably be incompatible with all previous axiomatic systems, further expanding the search space for axiomatic systems. Formal analysis is done which allows to observe interrelations of the old systems and new systems and new systems are developed. It is possible to reduce the number of axioms by proving equivalences and to eliminate possible inconsistencies. By proving one axiom from others the number of axioms needed for any particular system is reduced.