3 research outputs found
Towards Completely Characterizing the Complexity of Boolean Nets Synthesis
Boolean nets are Petri nets that permit at most one token per place. Research
has approached this important subject in many ways which resulted in various
different classes of boolean nets. But yet, they are only distinguished by the
allowed interactions between places and transitions, that is, the possible
effects of firing transitions. There are eight different interactions: no
operation (nop), input (inp), output (out), set, reset (res), swap, test of
occupation (used), and test of disposability (free). Considering every
combination for a possible net class yields 256 boolean classes in total. The
synthesis problem for a particular class is to take an automaton A and compute
a boolean net of that class that has a state graph isomorphic to A. To the best
of our knowledge, the computational complexity of this problem has been
analyzed for just two of the 256 classes, namely elementary nets systems (nop,
inp, out), where the problem is NP-hard, and flip-flop nets (nop, inp, out,
swap), which are synthesizable in polynomial time. However, depending on the
desired net features, like read-only places, exception handling, or hierarchy,
one has to synthesize nets with other interactions as for instance contextual
nets (nop, inp, out, used, free) or trace nets (nop, inp, out, set, res, used,
free). The contribution of this paper is a thorough investigation of the
synthesis complexity for the 128 boolean net classes that allow nop. Our main
result is a general proof scheme that identifies 77 NP-hard cases. All
remaining 51 classes are shown to be synthesizable in polynomial time where 35
of them turn out to be trivial.Comment: Corrected a minor erro
Tracking Down the Bad Guys: Reset and Set Make Feasibility for Flip-Flop Net Derivatives NP-complete
Boolean Petri nets are differentiated by types of nets based on which
of the interactions nop, inp, out, set, res, swap, used, and free they apply or
spare. The synthesis problem relative to a specific type of nets is to
find a boolean -net whose reachability graph is isomorphic to a given
transition system . The corresponding decision version of this search
problem is called feasibility. Feasibility is known to be polynomial for all
types of flip flop derivates that contain at least the interactions nop, swap
and an arbitrary selection of inp, out, used, free. In this paper, we replace
inp and out by res and set, respectively, and show that feasibility becomes
NP-complete for the types that contain nop, swap and a non empty selection of
res, set and a non empty selection of used, free. The reduction guarantees a
low degree for A's states and, thus, preserves hardness of feasibility even for
considerable input restrictions.Comment: In Proceedings ICE 2019, arXiv:1909.0524
The Complexity of Synthesizing nop-Equipped Boolean Nets from g-Bounded Inputs (Technical Report)
Boolean Petri nets equipped with nop allow places and transitions to be
independent by being related by nop. We characterize for any fixed natural
number g the computational complexity of synthesizing nop-equipped Boolean
Petri nets from labeled directed graphs whose states have at most g incoming
and at most g outgoing arcs