Reachability Problem For Petri Nets Research Articles

Finding Cut-Offs in Leaderless Rendez-Vous Protocols is Easy

In rendez-vous protocols an arbitrarily large number of indistinguishable finite-state agents interact in pairs. The cut-off problem asks if there exists a number $B$ such that all initial configurations of the protocol with at least $B$ agents in a given initial state can reach a final configuration with all agents in a given final state. In a recent paper (Horn and Sangnier, CONCUR 2020), Horn and Sangnier proved that the cut-off problem is decidable (and at least as hard as the Petri net reachability problem) for protocols with a leader, and in EXPSPACE for leaderless protocols. Further, for the special class of symmetric protocols they reduce these bounds to PSPACE and NP, respectively. The problem of lowering these upper bounds or finding matching lower bounds was left open. We show that the cut-off problem is P-complete for leaderless protocols and in NC for leaderless symmetric protocols. Further, we also consider a variant of the cut-off problem suggested in (Horn and Sangnier, CONCUR 2020), which we call the bounded-loss cut-off problem and prove that this problem is P-complete for leaderless protocols and NL-complete for leaderless symmetric protocols. Finally, by reusing some of the techniques applied for the analysis of leaderless protocols, we show that the cut-off problem for symmetric protocols with a leader is NP-complete, thereby improving upon all the elementary upper bounds of (Horn and Sangnier, CONCUR 2020).

Матричный подход решения задач достижимости в стохастических сетях Петри

The purpose of the research was to develop the theory of stochastic Petri nets and consider its practical application when studying discrete systems. The paper considers the possibility of solving the reachability problem in stochastic Petri nets by means of matrix equations widely used in Petri nets; describes the stages and features of generating matrix equations for stochastic networks; formulates the rules for introducing virtual elements, i.e., positions and transitions, into the stochastic Petri net to generate and solve matrix equations. Stochastic Petri nets different in structure and composition were used to explore the possibility of applying matrix equations. Findings of the research show that the reachability of the required states of the networks is determined through the firing of transitions, which are the solution of the matrix equation. Within the study, we interpreted the obtained results and developed an algorithm that allowed us to validate the assumption made and visually determine the restrictions on the use of matrix equations for various initial states of the simulated system. The results of the proposed algorithm are presented in graphical form on the examples of stochastic Petri nets that model the process of forensic handwriting analysis. The conclusion is made about the applicability of matrix equations in stochastic Petri nets and the need for further research in this area

From strong to exact Petri net computers

Petri net paradigm of computing is concerned discussing basic types of Petri net computers and their relations. We developed a technique of parametric specification of Petri nets, built of a few connected components which are repeated depending on a parameter or a set of parameters, and composition of software to generate automatically the corresponding models in a vivid graphical form. We construct explicitly a Petri net that is a strong computer of double exponent after R.J. Lipton. Its nondeterministic computation is represented by a Petri net reachability graph to which a successful branch belongs. Then we minimise and transform the net to obtain twice smaller construct than the original one. Moreover, the obtained net possesses such a marvellous property that addingto it four priority arcs makes it an exact (deterministic) computer having a single firable sequence only. Same as the Lipton’s net, the minimized net allows the proof that the Petri net reachability problem is exponential in space. Besides, it represents an example when simple transformations allow guessing the successful branch of a nondeterministic computation. The practical significance of the approach in applications consists in a state-of-art heuristic technique of solving some NP problems in polynomial time.

Towards efficient verification of population protocols

Population protocols are a well established model of computation by anonymous, identical finite-state agents. A protocol is well-specified if from every initial configuration, all fair executions of the protocol reach a common consensus. The central verification question for population protocols is the well-specification problem: deciding if a given protocol is well-specified. Esparza et al. have recently shown that this problem is decidable, but with very high complexity: it is at least as hard as the Petri net reachability problem, which is TOWER-hard, and for which only algorithms of non-primitive recursive complexity are currently known. In this paper we introduce the class { WS}^3 of well-specified strongly-silent protocols and we prove that it is suitable for automatic verification. More precisely, we show that { WS}^3 has the same computational power as general well-specified protocols, and captures standard protocols from the literature. Moreover, we show that the membership and correctness problems for { WS}^3 reduce to solving boolean combinations of linear constraints over {mathbb {N}}. This allowed us to develop the first software able to automatically prove correctness for all of the infinitely many possible inputs.

The complexity of verifying population protocols

Population protocols (Angluin et al. in PODC, 2004) are a model of distributed computation in which indistinguishable, finite-state agents interact in pairs to decide if their initial configuration, i.e., the initial number of agents in each state, satisfies a given property. In a seminal paper Angluin et al. classified population protocols according to their communication mechanism, and conducted an exhaustive study of the expressive power of each class, that is, of the properties they can decide (Angluin et al. in Distrib Comput 20(4):279–304, 2007). In this paper we study the correctness problem for population protocols, i.e., whether a given protocol decides a given property. A previous paper (Esparza et al. in Acta Inform 54(2):191–215, 2017) has shown that the problem is decidable for the main population protocol model, but at least as hard as the reachability problem for Petri nets, which has recently been proved to have non-elementary complexity. Motivated by this result, we study the computational complexity of the correctness problem for all other classes introduced by Angluin et al., some of which are less powerful than the main model. Our main results show that for the class of observation models the complexity of the problem is much lower, ranging from varPi _2^p to PSPACE.

The Reachability Problem for Petri Nets Is Not Elementary

Petri nets, also known as vector addition systems, are a long established model of concurrency with extensive applications in modeling and analysis of hardware, software, and database systems, as well as chemical, biological, and business processes. The central algorithmic problem for Petri nets is reachability: whether from the given initial configuration there exists a sequence of valid execution steps that reaches the given final configuration. The complexity of the problem has remained unsettled since the 1960s, and it is one of the most prominent open questions in the theory of verification. Decidability was proved by Mayr in his seminal STOC 1981 work, and, currently, the best published upper bound is non-primitive recursive Ackermannian of Leroux and Schmitz from Symposium on Logic in Computer Science 2019. We establish a non-elementary lower bound, i.e., that the reachability problem needs a tower of exponentials of time and space. Until this work, the best lower bound has been exponential space, due to Lipton in 1976. The new lower bound is a major breakthrough for several reasons. Firstly, it shows that the reachability problem is much harder than the coverability (i.e., state reachability) problem, which is also ubiquitous but has been known to be complete for exponential space since the late 1970s. Secondly, it implies that a plethora of problems from formal languages, logic, concurrent systems, process calculi, and other areas, which are known to admit reductions from the Petri nets reachability problem, are also not elementary. Thirdly, it makes obsolete the current best lower bounds for the reachability problems for two key extensions of Petri nets: with branching and with a pushdown stack. We develop a construction that uses arbitrarily large pairs of values with ratio R to provide zero testable counters that are bounded by R . At the heart of our proof is then a novel gadget, the so-called factorial amplifier that, assuming availability of counters that are zero testable and bounded by k , guarantees to produce arbitrarily large pairs of values whose ratio is exactly the factorial of k . Repeatedly composing the factorial amplifier with itself by means of the former construction enables us to compute, in linear time, Petri nets that simulate Minsky machines whose counters are bounded by a tower of exponentials, which yields the non-elementary lower bound. By refining this scheme further, we, in fact, already establish hardness for h -exponential space for Petri nets with h + 13 counters.

Verification column

In this insightful and timely article, several approaches relevant for verification are discussed for combatting the prohibitively high complexity of the reachability problem for Petri nets. The author generously shares his knowledge of the state of the art, identifying promising directions for immediate further research.

The ILLTP Library for Intuitionistic Linear Logic

Benchmarking automated theorem proving (ATP) systems using standardized problem sets is a well-established method for measuring their performance. However, the availability of such libraries for non-classical logics is very limited. In this work we propose a library for benchmarking Girard's (propositional) intuitionistic linear logic. For a quick bootstrapping of the collection of problems, and for discussing the selection of relevant problems and understanding their meaning as linear logic theorems, we use translations of the collection of Kleene's intuitionistic theorems in the traditional monograph "Introduction to Metamathematics". We analyze four different translations of intuitionistic logic into linear logic and compare their proofs using a linear logic based prover with focusing. In order to enhance the set of problems in our library, we apply the three provability-preserving translations to the propositional benchmarks in the ILTP Library. Finally, we generate a comprehensive set of reachability problems for Petri nets and encode such problems as linear logic sequents, thus enlarging our collection of problems.

Verification of population protocols

Population protocols (Angluin et al. in PODC, 2004) are a formal model of sensor networks consisting of identical mobile devices. Two devices can interact and thereby change their states. Computations are infinite sequences of interactions satisfying a strong fairness constraint. A population protocol is well specified if for every initial configuration C of devices, and every computation starting at C, all devices eventually agree on a consensus value depending only on C. If a protocol is well specified, then it is said to compute the predicate that assigns to each initial configuration its consensus value. While the computational power of well-specified protocols has been extensively studied, the two basic verification problems remain open: Is a given protocol well specified? Does a given protocol compute a given predicate? We prove that both problems are decidable by reduction to the reachability problem of Petri nets. We also give a new proof of the fact that the predicates computed by well-defined protocols are those definable in Presburger arithmetic (Angluin et al. in PODC, 2006).

Structurally Cyclic Petri Nets

A Petri net is structurally cyclic if every configuration is reachable from itself in one or more steps. We show that structural cyclicity is decidable in deterministic polynomial time. For this, we adapt the Kosaraju's approach for the general reachability problem for Petri nets.

A Survey of Reachability Trees of Unbounded Petri Nets

In recent years both industry and academia have paid much attention to the theory and applications of Petri nets. Reachability is a basic property of a Petri net, and many properties can be analyzed via it. However, analyzing the reachability problem of unbounded Petri nets by finite reachability trees has been an open problem since the inception of Petri nets. Researchers began to study the problem of reachability trees over 40 years ago. However, they made only limited progress over the last 20 years due to its complexity and difficulty. We present an overview of some important contributions toward its solution. The focuses are on four novel finite reachability trees: finite reachability tree(FRT),augmented reachability tree(ART), modified reachability tree(MRT) and new modified reachailbity tree(NMRT). The paper concludes with a discussion of directions for future research of the reachability problem of unbounded Petri nets.

The Hardness of Finding Linear Ranking Functions for Lasso Programs

Finding whether a linear-constraint loop has a linear ranking function is an important key to understanding the loop behavior, proving its termination and establishing iteration bounds. If no preconditions are provided, the decision problem is known to be in coNP when variables range over the integers and in PTIME for the rational numbers, or real numbers. Here we show that deciding whether a linear-constraint loop with a precondition, specifically with partially-specified input, has a linear ranking function is EXPSPACE-hard over the integers, and PSPACE-hard over the rationals. The precise complexity of these decision problems is yet unknown. The EXPSPACE lower bound is derived from the reachability problem for Petri nets (equivalently, Vector Addition Systems), and possibly indicates an even stronger lower bound (subject to open problems in VAS theory). The lower bound for the rationals follows from a novel simulation of Boolean programs. Lower bounds are also given for the problem of deciding if a linear ranking-function supported by a particular form of inductive invariant exists. For loops over integers, the problem is PSPACE-hard for convex polyhedral invariants and EXPSPACE-hard for downward-closed sets of natural numbers as invariants.

Recent and simple algorithms for Petri nets

We show how inductive invariants can be used to solve coverability, boundedness and reachability problems for Petri nets. This approach provides algorithms that are conceptually simpler than previously published ones.

Using Constraint Programming for solving the Reachability Problem in Timed Petri Nets: Evaluation of Basic Labeling Strategies

In this paper, we propose to use a constraint programming approach to address the reachability problem in Timed Petri Nets (TPNs). TPNs can be used to model a wide class of systems, from manufacturing issues to formal verification of embedded systems. Many of the considered problems can be modeled as reachability problems in TPNs. To reduce the space state explosion brought by the exploration of the TPN behavior, we propose to follow the incremental methodology proposed by Bourdeaud'huy and Hanafi (2006), who translate the reachability graph of Timed PNs into a mathematical model. We improve this model by adding valid inequalities and search for solutions using Constraint Programming (CP). More particularly, we compare different labeling strategies and assess their respective efficiency.

ON YEN'S PATH LOGIC FOR PETRI NETS

In [19], Yen defines a class of formulas for paths in Petri nets and claims that its satisfiability problem is EXPSPACE-complete. In this paper, we show that in fact the satisfiability problem for this class of formulas is as hard as the reachability problem for Petri nets. Moreover, we salvage almost all of Yen's results by defining a fragment of this class of formulas for which the satisfiability problem is EXPSPACE-complete by adapting his proof.

A Mathematical Programming Model for safe T-Time Petri Nets

In this paper, we propose a mathematical programming model for the resolution of the reachability problem in Time Petri Nets. We are more particularly interested in unweighted Time Petri Nets with safe markings and weak time semantics.

Reachability Analysis of Variants of Communication-Free Petri Nets

Communication-free Petri nets provide a net semantics for Basic Parallel Processes, which form a subclass of Milner's Calculus of Communicating Systems (CCS) a process calculus for the description and algebraic manipulation of concurrent communicating systems. It is known that the reachability problem for communication-free Petri nets is NP-complete. Lacking the synchronization mechanism, the expressive power of communication-free Petri nets is somewhat limited. It is therefore importance to see whether the power of communication-free Petri nets can be enhanced without sacrificing their analytical capabilities. As a first step towards this line of research, in this paper our main concern is to investigate, from the decidability/complexity viewpoint, the reachability problem for a number of variants of communication-free Petri nets, including communication-free Petri nets augmented with ‘static priorities, ’ ‘dynamic priorities, ’ ‘states, ’ ‘inhibitor arcs, ’ and ‘timing constraints.’

Bouziane’s transformation of the Petri net reachability problem and incorrectness of the related algorithm

The proceedings of FOCS’98 contain a paper by Zakariae Bouziane, who sketches a new representation of the Petri net reachability problem and claims to provide a new algorithm solving the problem. In this note, the essence of Bouziane’s approach is explained, and a serious flaw of the algorithm is exposed.

An algebraic approach to revising propositional rule-based knowledge bases

One of the important topics in knowledge base revision is to introduce an efficient implementation algorithm. Algebraic approaches have good characteristics and implementation method; they may be a choice to solve the problem. An algebraic approach is presented to revise propositional rule-based knowledge bases in this paper. A way is firstly introduced to transform a propositional rule-based knowledge base into a Petri net. A knowledge base is represented by a Petri net, and facts are represented by the initial marking. Thus, the consistency check of a knowledge base is equivalent to the reachability problem of Petri nets. The reachability of Petri nets can be decided by whether the state equation has a solution; hence the consistency check can also be implemented by algebraic approach. Furthermore, algorithms are introduced to revise a propositional rule-based knowledge base, as well as extended logic programming. Compared with related works, the algorithms presented in the paper are efficient, and the time complexities of these algorithms are polynomial.

Mathematical programming approach to the Petri nets reachability problem

This paper focuses on the resolution of the reachability problem in Petri nets, using the mathematical programming paradigm. The proposed approach is based on an implicit traversal of the Petri net reachability graph. This is done by constructing a unique sequence of Steps that represents exactly the total behaviour of the net. We propose several formulations based on integer and/or binary linear programming, and the corresponding sets of adjustments to the particular class of problem considered. Our models are validated on a set of benchmarks and compared with standard approaches from IA and Petri nets community.