A modification of Petri nets with anticipation on a position

We propose a modification of Petri nets with strong anticipation on a position. The extension modifies a transition rule by adding a new term that contains an integer function of the new marking in the position. The differences from classic Petri nets are found; for example, the set of markings that are reachable from a current marking by firing the enabled transition can either be empty or contain more than one marking. We consider the construction of a reachability graph and a coverability tree. We give the conditions for the existence of the coverability tree and propose the algorithm for constructing the coverability tree that generalizes the well-known classic algorithm. The main ideas and constructions are illustrated in the examples.