Algebraic nets with flexible arcs

Abstract

Algebraic Petri nets as defined by Reisig (Theoret. Comput. Sci. 80 (1991) 1–34.) lack a feature for modelling distributed network algorithms, viz. flexible arcs. In this paper, we equip algebraic Petri nets with flexible arcs and call the resulting extension algebraic system nets. We demonstrate that algebraic system nets are better suited for modelling distributed algorithms. Besides this practical motivation for introducing algebraic system nets, there is a theoretical one. The concept of place invariants introduced along with algebraic Petri nets has a slight insufficiency: There may be place invariants of the unfolded algebraic Petri net that cannot be expressed as a place invariant of the algebraic Petri net itself. By introducing algebraic system nets along with a more general concept of place invariants we eliminate this insufficiency too. Moreover, we generalize the concept of place invariants, which we call simulations. Many well-known concepts of Petri net theory such as siphons, traps, modulo-invariants, sur-invariants and sub-invariants are special cases of a simulation. Still, a simulation can be verified in the same style as classical place invariants of algebraic Petri nets.