Computational Complexity of the Place/Transition-Net Symmetry Reduction Method

Computational complexity of the sub-tasks appearing in the symmetry reduction method for Place/Transition-nets is studied. The first task of finding the automorphisms (symmetries) of a net is shown to be polynomial time many-one equivalent to the problem of finding the automorphisms of a graph. The problem of deciding whether two markings are symmetric is shown to be equivalent to the graph isomorphism problem. Surprisingly, this remains to be the case even if the generators for the automorphism group of the net are known. The problem of constructing the lexicographically greatest marking symmetric to a given marking (a canonical representative for the marking) is classified to belong to the lower levels of the polynomial hierarchy, namely to somewhere between FPNP[logn] and FPNP. It is also discussed how the self-symmetries of a marking can be exploited. Calculation of such symmetries is classified to be as hard as computing graph automorphism groups. Furthermore, the coverability version of testing marking symmetricity is shown to be an NP-complete problem. It is shown that unfortunately canonical representative markings and the symmetric coverability problem cannot be combined in a straightforward way.