Synthesis of Structurally Restricted b-bounded Petri Nets: Complexity Results

Abstract

Let $b\in \mathbb {N}^+$. A b-bounded Petri net (b-net) solves a transition system (TS) if its reachability graph and the TS are isomorphic. Synthesis (of b-nets) is the problem of finding for a TS A a b-net N that solves it. This paper investigates the computational complexity of synthesis, where the searched net is structurally restricted in advance. The restrictions relate to the cardinality of the preset and the postset of N’s transitions and places. For example, N is choice-free (CF) if the postset-cardinality of its places do not exceed one. If additionally the preset-cardinality of N’s transitions is at most one then it is fork-attribution. This paper shows that deciding if A is solvable by a pure or test-free b-net N which is choice-free, fork-attribution, free-choice, extended free-choice or asymmetric-choice, respectively, is NP-complete. Moreover, we show that deciding if A is solvable by a b-bounded weighted (m, n)-T-systems, $m,n\in \mathbb {N}$, is NP-complete if m, n belong to the input. On the contrary, synthesis for this class becomes tractable if $m,n\in \mathbb {N}$ are chosen a priori. We contrast this result with the fact that synthesis for weighted (m, n)-S-systems, being the T-systems’s dual class, is NP-complete for any fixed $m,n\ge 2$.