A Disconnected 1-Safe Petri Net Whose Reachability Tree Is Homomorphic to a Complete Boolean Lattice

Boolean Petri Nets

“…Our work towards this end revealed to our surprise that there can be even such disconnected 1-safe Petri nets. We demonstrated the existence of a disconnected 1-safe Petri net which was obtained by removing the central transition from the star Petri net S n , whose reachability tree can be tactically represented as an n-dimensional complete lattice L n [7]. In this disconnected Petri net, the firing of transitions in a particular way (which we may regard as a 'tact' or 'strategy'), gives exactly 2 n marking vectors, repetitions occurring possibly only within a level, that can be arranged as a homomorphic image of the reachability tree of the Petri net, forming the n-dimensional complete Boolean lattice L n .…”

“…This problem is still open. Many specific classes of such 1-safe Petri nets have been found [5][6][7]. Also, many fundamental issues regarding Boolean Petri nets emerge from this study.…”

“…[7] Let C n =( P, T, I − , I + , µ 0 ) be the 1-safe Petri net consisting of n connected components, each isomorphic to C := −→ . Then the reachability tree of C n is homomorphic to the n-dimensional complete lattice L n .…”

Embedding an Arbitrary 1-safe Petri Net into a Boolean Petri Net

“…We It has been found and established that the reachability tree of a 1-safe Petri net can be homomorphically mappped on to the n-dimensional complete Boolean lattice, thereby yielding new techniques to represent the dynamics of these Petri nets [12]. DEFINITION 2.…”