Fixed Parameter Tractability and Polynomial Time Results for the Synthesis of b-bounded Petri Nets

Abstract

Synthesis for a type of Petri nets is the problem of finding, for a given transition system (TS, for short) A, a Petri net N of this type whose state graph is isomorphic to A if such a net exists. The decision version of this search problem, called -feasibility, asks if, for a given TS A, there exists a Petri net N of type with a state graph isomorphic to A. In this case, A is called -feasible. A’s feasibility is equivalent to fulfilling two so-called separation properties. In fact, a transition system A is -feasible if and only if it satisfies the type related state separation property (SSP) and event state separation property (ESSP). Both properties, SSP and ESSP, define decision problems. In this paper, we introduce for $b\in \mathbb {N}$ the type of restricted $\mathbb {Z}_{b+1}$-extended b-bounded P/T-nets and show that synthesis and deciding ESSP and SSP for this type is doable in polynomial time. Moreover, we demonstrate that, given a TS A, deciding if A has the SSP can be done in polynomial time for the types of (pure) $\mathbb {Z}_{b+1}$-extended b-bounded P/T-nets. Finally, we exhibit that deciding if a TS A is feasible or has the ESSP for the types of (pure) $\mathbb {Z}_{b+1}$-extended b-bounded P/T-nets is fixed parameter tractable if the number of occurrences of events is considered as parameter.