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Ronny Tredup

Publications and source records attributed to Ronny Tredup.

12 recordsLinked to original sources
arXiv2022

On the Complexity of Techniques That Make Transition Systems Implementable by Boolean Nets

Synthesis consists in deciding whether a given labeled transition system (TS) $A$ can be implemented by a net $N$ of type $\tau$. In case of a negative decision, it may be possible to convert $A$ into an implementable TS $B$ by applying various modification techniques, like relabeling edges that previously had the same label, suppressing edges/states/events, etc. It may however be useful to limit the number of such modifications to stay close to the original problem, or optimize the technique. In this paper, we show that most of the corresponding problems are NP-complete if $\tau$ corresponds to the type of flip-flop nets or some flip-flop net derivatives.

cs.CC↗
arXiv2021

Some Basic Techniques allowing Petri Net Synthesis: Complexity and Algorithmic Issues

In Petri net synthesis we ask whether a given transition system $A$ can be implemented by a Petri net $N$. Depending on the level of accuracy, there are three ways how $N$ can implement $A$: an embedding, the least accurate implementation, preserves only the diversity of states of $A$; a language simulation already preserves exactly the language of $A$; a realization, the most accurate implementation, realizes the behavior of $A$ exactly. However, whatever the sought implementation, a corresponding net does not always exist. In this case, it was suggested to modify the input behavior -- of course as little as possible. Since transition systems consist of states, events and edges, these components appear as a natural choice for modifications. In this paper we show that the task of converting an unimplementable transition system into an implementable one by removing as few states or events or edges as possible is NP-complete -- regardless of what type of implementation we are aiming for; we also show that the corresponding parameterized problems are $W[2]$-hard, where the number of removed components is considered as the parameter; finally, we show there is no $c$-approximation algorithm (with a polynomial running time) for neither of these problems, for every constant $c\geq 1$.

cs.FL↗
arXiv2021

Synthesis of Pure and Impure Petri nets With Restricted Place-environments: Complexity Issues

Petri net synthesis consists in deciding for a given transition system $A$ whether there exists a Petri net $N$ whose reachability graph is isomorphic to $A$. Several works examined the synthesis of Petri net subclasses that restrict, for every place $p$ of the net, the cardinality of its preset or of its postset or both in advance by small natural numbers $\varrho$ and $\kappa$, respectively, such as for example (weighted) marked graphs, (weighted) T-systems and choice-free nets. In this paper, we study the synthesis aiming at Petri nets which have such restricted place environments, from the viewpoint of classical and parameterized complexity: We first show that, for any fixed natural numbers $\varrho$ and $\kappa$, deciding whether for a given transition system $A$ there is a Petri net $N$ such that (1) its reachability graph is isomorphic to $A$ and (2) for every place $p$ of $N$ the preset of $p$ has at most $\varrho$ and the postset of $p$ has at most $\kappa$ elements is doable in polynomial time. Secondly, we introduce a modified version of the problem, namely Environment Restricted Synthesis (ERS, for short), where $\varrho$ and $\kappa$ are part of the input, and show that ERS is NP-complete, regardless whether the sought net is impure or pure. In case of the impure nets, our methods also imply that ERS parameterized by $\varrho+\kappa$ is $W[2]$-hard.

cs.FL↗
arXiv2021

The Complexity of Synthesis of $b$-Bounded Petri Nets

For a fixed type of Petri nets $\tau$, \textsc{$\tau$-Synthesis} is the task of finding for a given transition system $A$ a Petri net $N$ of type $\tau$ ($\tau$-net, for short) whose reachability graph is isomorphic to $A$ if there is one. The decision version of this search problem is called \textsc{$\tau$-Solvability}. If an input $A$ allows a positive decision, then it is called $\tau$-solvable and a sought net $N$ $\tau$-solves $A$. As a well known fact, $A$ is $\tau$-solvable if and only if it has the so-called $\tau$-\emph{event state separation property} ($\tau$-ESSP, for short) and the $\tau$-\emph{state separation property} ($\tau$-SSP, for short). The question whether $A$ has the $\tau$-ESSP or the $\tau$-SSP defines also decision problems. In this paper, for all $b\in \mathbb{N}$, we completely characterize the computational complexity of \textsc{$\tau$-Solvability}, \textsc{$\tau$-ESSP} and \textsc{$\tau$-SSP} for the types of pure $b$-bounded Place/Transition-nets, the $b$-bounded Place/Transition-nets and their corresponding $\mathbb{Z}_{b+1}$-extensions.

cs.CC↗
arXiv2020

The Complexity of Boolean State Separation (Technical Report)

For a Boolean type of nets $τ$, a transition system $A$ is synthesizeable into a $τ$-net $N$ if and only if distinct states of $A$ correspond to distinct markings of $N$, and $N$ prevents a transition firing if there is no related transition in $A$. The former property is called $τ$-state separation property ($τ$-SSP) while the latter -- $τ$-event/state separation property ($τ$-ESSP). $A$ is embeddable into the reachability graph of a $τ$-net $N$ if and only if $A$ has the $τ$-SSP. This paper presents a complete characterization of the computational complexity of \textsc{$τ$-SSP} for all Boolean Petri net types.

cs.LO↗
arXiv2020

On the Parameterized Complexity of Synthesizing Boolean Petri Nets With Restricted Dependency

Modeling of real-world systems with Petri nets allows to benefit from their generic concepts of parallelism, synchronisation and conflict, and obtain a concise yet expressive system representation. Algorithms for synthesis of a net from a sequential specification enable the well-developed theory of Petri nets to be applied for the system analysis through a net model. The problem of $τ$-synthesis consists in deciding whether a given directed labeled graph $A$ is isomorphic to the reachability graph of a Boolean Petri net $N$ of type $τ$. In case of a positive decision, $N$ should be constructed. For many Boolean types of nets, the problem is NP-complete. This paper deals with a special variant of $τ$-synthesis that imposes restrictions for the target net $N$: we investigate dependency $d$-restricted tau-synthesis (DR$τ$S) where each place of $N$ can influence and be influenced by at most d transitions. For a type $τ$, if tau-synthesis is NP-complete then DR$τ$S is also NP-complete. In this paper, we show that DR$τ$S parameterized by $d$ is in XP. Furthermore, we prove that it is W[2]-hard, for many Boolean types that allow unconditional interactions set and reset.

cs.CC↗
arXiv2020

On the Parameterized Complexity of Synthesizing Boolean Petri Nets With Restricted Dependency (Technical Report)

The problem of $τ$-synthesis consists in deciding whether a given directed labeled graph $A$ is isomorphic to the reachability graph of a Boolean Petri net $N$ of type $τ$. In case of a positive decision, $N$ should be constructed. For many Boolean types of nets, the problem is NP-complete. This paper deals with a special variant of $τ$-synthesis that imposes restrictions for the target net $N$: we investigate \emph{dependency $d$-restricted $τ$-synthesis (DR$τ$S)} where each place of $N$ can influence and be influenced by at most $d$ transitions. For a type $τ$, if $τ$-synthesis is NP-complete then DR$τ$S is also NP-complete. In this paper, we show that DR$τ$S parameterized by $d$ is in XP. Furthermore, we prove that it is $W[2]$-hard, for many Boolean types that allow unconditional interactions $set$ and $reset$.

cs.CC↗
arXiv2019

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 $τ$ 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 $τ$ is to find a boolean $τ$-net $N$ whose reachability graph is isomorphic to a given transition system $A$. 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.

cs.CC↗
arXiv2019

Hardness Results for the Synthesis of $b$-bounded Petri Nets (Technical Report)

Synthesis for a type $τ$ of Petri nets is the following search problem: For a transition system $A$, find a Petri net $N$ of type $τ$ whose state graph is isomorphic to $A$, if there is one. To determine the computational complexity of synthesis for types of bounded Petri nets we investigate their corresponding decision version, called feasibility. We show that feasibility is NP-complete for (pure) $b$-bounded P/T-nets if $b\in \mathbb{N}^+$. We extend (pure) $b$-bounded P/T-nets by the additive group $\mathbb{Z}_{b+1}$ of integers modulo $(b+1)$ and show feasibility to be NP-complete for the resulting type. To decide if $A$ has the event state separation property is shown to be NP-complete for (pure) $b$-bounded and group extended (pure) $b$-bounded P/T-nets. Deciding if $A$ has the state separation property is proven to be NP-complete for (pure) $b$-bounded P/T-nets.

cs.LO↗
arXiv2018

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.

cs.CC↗
arXiv2017

The Hardness of Synthesizing Elementary Net Systems from Highly Restricted Inputs

Elementary net systems (ENS) are the most fundamental class of Petri nets. Their synthesis problem has important applications in the design of digital hardware and commercial processes. Given a labeled transition system (TS) $A$, feasibility is the NP-complete decision problem whether $A$ can be equivalently synthesized into an ENS. It is well known that $A$ is feasible if and only if it has the event state separation property (ESSP) and the state separation property (SSP). Recently, these properties have also been studied individually for their practical implications. A fast ESSP algorithm, for instance, would allow applications to at least validate the language equivalence of $A$ and a synthesized ENS. Being able to efficiently decide SSP, on the other hand, could serve as a quick-fail preprocessing mechanism for synthesis. Although a few tractable subclasses have been found, this paper destroys much of the hope that many practically meaningful input restrictions make feasibility or at least one of ESSP and SSP efficient. We show that all three problems remain NP-complete even if the input is restricted to linear TSs where every event occurs at most three times or if the input is restricted to TSs where each event occurs at most twice and each state has at most two successor and two predecessor states.

cs.LO↗