## Complexity Results for 1-safe Nets (1993)

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@MISC{Cheng93complexityresults,

author = {Allan Cheng and Javier Esparza and Jens Palsberg},

title = {Complexity Results for 1-safe Nets},

year = {1993}

}

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### Abstract

We study the complexity of several standard problems for 1-safe Petri nets and some of its subclasses. We prove that reachability, liveness, and deadlock are all PSPACE-complete for 1-safe nets. We also prove that deadlock is NP-complete for free-choice nets and for 1-safe free-choice nets. Finally, we prove that for arbitrary Petri nets, deadlock is equivalent to reachability and liveness. This paper is to be presented at FST&TCS 13, Foundations of Software Technology & Theoretical Computer Science, to be held 1517 December 1993, in Bombay, India. A version of the paper with most proofs omitted is to appear in the proceedings. 1 Introduction Petri nets are one of the oldest and most studied formalisms for the investigation of concurrency [33]. Shortly after the birth of complexity theory, Jones, Landweber, and Lien studied in their classical paper [24] the complexity of several fundamental problems for Place/Transition nets (called in [24] just Petri nets). Some years later, Howell,...

### Citations

3836 |
Introduction to automata theory, languages, and computation
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Citation Context ...y visits the cells of the tape containing the input. The input is bounded by a left and a right marker, say # and $, and the head can visit no cell to the left of # and no cell to the right of $ (see =-=[18]-=- for a formal definition). The problem is defined as follows: 8sGiven: a deterministic linearly bounded automaton M0 and an input x for M0, To decide: if M0 accepts x. First, we construct in polynomia... |

1437 | Petri Nets: Properties, Analysis and Applications - Murata - 1989 |

579 |
Nets: An Introduction
- Reisig, Petri
- 1985
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Citation Context ...ete polynomial time NP-complete Table 1: Summary of complexity results for Petri nets. 1 Introduction Petri nets are one of the oldest and most studied formalisms for the investigation of concurrency =-=[30]-=-. Shortly after the birth of complexity theory, Jones, Landweber, and Lien studied in their classical paper [22] the complexity of several fundamental problems for Place/Transition nets (called in [22... |

262 | Categories of models for concurrency
- Winskel
- 1984
(Show Context)
Citation Context ...8] have shown that a model of 1-safe nets, called Elementary Net Systems, has strong categorical connections with many other models of concurrency, such as event structures (another good reference is =-=[35]-=-). Finally, 1-safe nets are closer to classical language theory, and can be interpreted as a synchronisation of finite automata. These properties have motivated the design of verification methods part... |

163 |
An algorithm for the general petri net reachability problem
- Mayr
- 1981
(Show Context)
Citation Context ...bine theorems 1 and 2 with Hack’s reduction from liveness to reachability [16]. For the complexity of the deadlock problem, use the equivalence with reachability and obtain the decidability from Mayr =-=[25]-=- and the EXPSPACE-hardness from Lipton [24]. ✷ The technique of the proofs is similar to those of, for instance, Chapter 5 in Petersons book [29]. The same result holds for Place/Transition nets with ... |

162 | Petri Nets are Monoids - Meseguer, Montanari - 1990 |

159 |
Using unfoldings to avoid the state explosion problem in the verification of asynchronous circuits
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- 1992
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Citation Context ...zurkiewicz trace theory [26, 35] has been exploited to design efficient “partial-order” verification methods [34, 14]. Several other different proposals have recently been presented in the literature =-=[27, 9]-=-. In order to evaluate them, and as a guide for future research, it is necessary to know the complexity of verification problems for 1-safe nets. This paper provides the first systematic study for 1-s... |

156 | Stubborn sets for reduced state space generation, in
- Valmari
- 1989
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Citation Context ...n of verification methods particularly suited for 1-safe nets. The obvious connection to Mazurkiewicz trace theory [26, 35] has been exploited to design efficient “partial-order” verification methods =-=[34, 14]-=-. Several other different proposals have recently been presented in the literature [27, 9]. In order to evaluate them, and as a guide for future research, it is necessary to know the complexity of ver... |

146 |
Using partial orders to improve automatic verification methods
- Godefroid
- 1990
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Citation Context ...n of verification methods particularly suited for 1-safe nets. The obvious connection to Mazurkiewicz trace theory [26, 35] has been exploited to design efficient “partial-order” verification methods =-=[34, 14]-=-. Several other different proposals have recently been presented in the literature [27, 9]. In order to evaluate them, and as a guide for future research, it is necessary to know the complexity of ver... |

123 |
Trace theory
- Mazurkiewicz
- 1987
(Show Context)
Citation Context ...ed as a synchronisation of finite automata. These properties have motivated the design of verification methods particularly suited for 1-safe nets. The obvious connection to Mazurkiewicz trace theory =-=[26, 35]-=- has been exploited to design efficient “partial-order” verification methods [34, 14]. Several other different proposals have recently been presented in the literature [27, 9]. In order to evaluate th... |

114 |
Model Checking Using Net Unfoldings
- Esparza
- 1993
(Show Context)
Citation Context ...zurkiewicz trace theory [26, 35] has been exploited to design efficient “partial-order” verification methods [34, 14]. Several other different proposals have recently been presented in the literature =-=[27, 9]-=-. In order to evaluate them, and as a guide for future research, it is necessary to know the complexity of verification problems for 1-safe nets. This paper provides the first systematic study for 1-s... |

110 |
Marked directed graphs
- Commoner, Holt, et al.
- 1971
(Show Context)
Citation Context ...o interesting subclasses of Petri nets are not covered by Table 1, namely S- and T-systems [4]. For those, reachability, liveness, and deadlock are known to be polynomial in the Place/Transition case =-=[4, 6, 13]-=-, hence also in the 1-safe case. Related work concerning not the complexity of particular verification problems but the complexity of deciding different equivalence notions can be found in [21]. The p... |

110 | The Petri box calculus: a new causal algebra with multi-label communication - Best, Devillers, et al. - 1992 |

89 | Decidability issues for petri nets - a survey
- Esparza, Nielsen
- 1994
(Show Context)
Citation Context ...Lien studied in their classical paper [22] the complexity of several fundamental problems for Place/Transition nets (called in [22] just Petri nets). This paper was later followed by many others (see =-=[10]-=- for an overview). It has been observed that the nets in which a place can contain at most one token, called in the sequel 1-safe nets, have many interesting properties. Places of 1-safe nets no longe... |

82 |
The reachability problem requires exponential space
- Lipton
- 1976
(Show Context)
Citation Context ...eadlock. 3 Place/Transition Nets For Place/Transition nets, it is known that the liveness and reachability problems are recursively equivalent [16], and that they are both decidable and EXPSPACE-hard =-=[24]-=-. We complete the picture by showing that the deadlock problem is recursively equivalent to them, and thus decidable and EXPSPACE-hard. Theorem 1 Reachability is polynomial-time reducible to deadlock.... |

74 |
Analysis of production schemata by Petri nets
- Hack
- 1972
(Show Context)
Citation Context ...M(Q) > 0, and unmarked at M if M(Q) = 0. A subset Q of places of N is a siphon if • Q ⊆ Q • , and a trap if Q • ⊆ • Q. We use some well known lemmata about siphons and traps. They can all be found in =-=[15]-=- or - a more accessible reference - in [2]. Lemma 10 Let N be a net, and M a marking of N. (1) If Q is a siphon of N unmarked at M, then Q remains unmarked at all markings reachable from M. 18 N T2s(2... |

72 | Petri nets, algebras, morphisms, and compositionality - Winskel - 1987 |

42 |
Analyse von Petri-netz-modellen
- Starke
- 1990
(Show Context)
Citation Context ...laces is a siphon because Q • = T . Then, check in polynomial time that it contains no trap marked at the initial marking using Starke’s algorithm to find the maximal trap contained in a given siphon =-=[31]-=- (see [8] for a reference in English). We prove completeness by reducing the satisfiability problem of propositional formulas in conjunctive normal form (CON-SAT) to the deadlock problem. An instance ... |

42 | Fernández C, Nonsequential processes. A Petri net view - Best - 1988 |

32 | Nets, terms, and formulas - Olderog - 1991 |

27 |
Complexity of some problems in Petri nets
- Jones, Landweber, et al.
- 1977
(Show Context)
Citation Context ... are one of the oldest and most studied formalisms for the investigation of concurrency [30]. Shortly after the birth of complexity theory, Jones, Landweber, and Lien studied in their classical paper =-=[22]-=- the complexity of several fundamental problems for Place/Transition nets (called in [22] just Petri nets). This paper was later followed by many others (see [10] for an overview). It has been observe... |

25 |
Elementary Net Systems
- Thiagarajan
(Show Context)
Citation Context ...heoretical Computer Science, pages 326–337, held 15–17 December 1993, in Bombay, India [5]. 2s[3, 28], but are however difficult to extend to Place/Transition nets. Nielsen, Rozenberg and Thiagarajan =-=[33, 28]-=- have shown that a model of 1-safe nets, called Elementary Net Systems, has strong categorical connections with many other models of concurrency, such as event structures (another good reference is [3... |

20 |
The equality problem for vector addition systems is undecidable
- Hack
- 1976
(Show Context)
Citation Context ...′ , t), where 17sT1 P1 P2 P1 P2 T2 T1 N released form of Figure 3: A net and its released form. p ′ and t ′ are a new place and a new transition. The interested reader can find a formal definition in =-=[22, 17]-=-. Figure 3 shows a non-free-choice net (on the left), and its released form (on the right). Perhaps surprisingly, the liveness problem is polynomial for this class. Theorem 9 The liveness problem for ... |

18 | On Representing CCS Programs by Finite Petri Nets - Goltz - 1988 |

17 |
Problems concerning fairness and temporal logic for conflict-free Petri net
- Howell, Rosier
- 1989
(Show Context)
Citation Context ... several papers; in particular, the complexity of our three problems. A net N = (P, T, F, M0) is conflict-free if for every place p, if |p • | > 1, then p • ⊆ • p. It is shown by Howell and Rosier in =-=[19, 20]-=- that the reachability, liveness, and deadlock problems for 1-safe conflict-free nets are solvable in polynomial time. They also show that, for Place/Transition nets, the deadlock and liveness problem... |

16 |
Properties of conflict-free and persistent petri nets
- Landweber, Robertson
- 1978
(Show Context)
Citation Context ...s covered by M ′ if M ⊆ M ′ . The coverability problem for a given net N and a marking M of N is the problem of deciding whether some reachable marking of N covers M. A net N is said to be persistent =-=[23]-=- if for every reachable marking M, if two different transitions t, t ′ are enabled at M then M t −→ M ′ t ′ −→ M ′′ for some markings M ′ , M ′′ . The persistency problem for a net is the problem of d... |

13 |
A polynomial-time algorithm to decide liveness of bounded free choice nets
- Esparza, Silva
- 1992
(Show Context)
Citation Context ...s surprisingly, the liveness problem is polynomial for this class. Theorem 9 The liveness problem for free-choice 1-safe nets is solvable in polynomial time. Proof. See the paper by Esparza and Silva =-=[11]-=-, and the paper by Desel [7]. ✷ We now show that the deadlock problem for 1-safe free-choice nets is NPcomplete. Membership in NP is non-trivial, and requires to introduce some concepts and results of... |

10 |
Deciding true concurrency equivalences on finite safe nets
- Jategaonkar, Meyer
- 1993
(Show Context)
Citation Context ... [4, 6, 13], hence also in the 1-safe case. Related work concerning not the complexity of particular verification problems but the complexity of deciding different equivalence notions can be found in =-=[21]-=-. The paper is organised as follows. Section 2 contains basic definitions. In section 3 we show that the deadlock problem is recursively equivalent to the liveness and reachability problems. Section 4... |

10 | Complexity of some problems - Jones, Landweber, et al. - 1977 |

9 |
Fiorella De Cindio. A survey of basic net models and modular net classes
- Bernardinello
- 1992
(Show Context)
Citation Context ...tion/Event systems [30] and the Elementary Net Systems [33]. This multiplicity of definitions is maybe annoying but harmless: the differences among them are small, and of rather technical nature (see =-=[1]-=- for a discussion). In particular, our results are independent of the definition used. 2 Definitions We recall in this section some basic concepts about Place/Transition nets and 1-safe nets, and defi... |

9 |
A proof of the Rank theorem for extended free-choice nets
- Desel
- 1992
(Show Context)
Citation Context ...problem is polynomial for this class. Theorem 9 The liveness problem for free-choice 1-safe nets is solvable in polynomial time. Proof. See the paper by Esparza and Silva [11], and the paper by Desel =-=[7]-=-. ✷ We now show that the deadlock problem for 1-safe free-choice nets is NPcomplete. Membership in NP is non-trivial, and requires to introduce some concepts and results of net theory. Let N be a net ... |

9 |
Behavioural notions for elementary net systems
- Nielsen, Rozenberg, et al.
- 1990
(Show Context)
Citation Context ...Computer Science, 147(1-2):117–136, 1995. Also in Proc. FST&TCS 13, Foundations of Software Technology & Theoretical Computer Science, pages 326–337, held 15–17 December 1993, in Bombay, India [5]. 2s=-=[3, 28]-=-, but are however difficult to extend to Place/Transition nets. Nielsen, Rozenberg and Thiagarajan [33, 28] have shown that a model of 1-safe nets, called Elementary Net Systems, has strong categorica... |

8 |
Completeness results for conflict-free vector replacement systems
- Howell, Rosier
- 1988
(Show Context)
Citation Context ... several papers; in particular, the complexity of our three problems. A net N = (P, T, F, M0) is conflict-free if for every place p, if |p • | > 1, then p • ⊆ • p. It is shown by Howell and Rosier in =-=[19, 20]-=- that the reachability, liveness, and deadlock problems for 1-safe conflict-free nets are solvable in polynomial time. They also show that, for Place/Transition nets, the deadlock and liveness problem... |

5 | Shortest paths in reachability graphs - Desel |

4 |
The recursive equivalence of the reachability problem and the liveness problem for Petri nets and vector addition systems
- Hack
- 1974
(Show Context)
Citation Context ...oblem of deciding if any of its reachable markings is a deadlock. 3 Place/Transition Nets For Place/Transition nets, it is known that the liveness and reachability problems are recursively equivalent =-=[16]-=-, and that they are both decidable and EXPSPACE-hard [24]. We complete the picture by showing that the deadlock problem is recursively equivalent to them, and thus decidable and EXPSPACE-hard. Theorem... |

3 |
and Jorg Desel. Partial order behaviour and structure of petri nets
- Best
- 1990
(Show Context)
Citation Context ...subset Q of places of N is a siphon if • Q ⊆ Q • , and a trap if Q • ⊆ • Q. We use some well known lemmata about siphons and traps. They can all be found in [15] or - a more accessible reference - in =-=[2]-=-. Lemma 10 Let N be a net, and M a marking of N. (1) If Q is a siphon of N unmarked at M, then Q remains unmarked at all markings reachable from M. 18 N T2s(2) If Q is a trap of N marked at M, then Q ... |

3 |
On the reachability problem for some classes of Petri nets
- Stewart
- 1992
(Show Context)
Citation Context ...ms are trivial. For both acyclic Place/Transition nets and for acyclic 1-safe nets, the reachability problem is NP-complete. For acyclic 1-safe nets, NP-hardness of reachability was proved by Stewart =-=[32]-=-, and for acyclic Place/Transition nets, membership in NP follows by a straightforward reduction to INTEGER LINEAR PROGRAMMING [5], which is in NP [18]. So, there are no essential differences between ... |

3 | A distributed operational semantics for CCS based on C/E systems - Degano, Nicola, et al. - 1988 |

2 | The recursive equivalence of the reachabilty problem and the liveness problem for Petri nets and vector addition systems - Hack - 1974 |

1 |
Some classes of live and save Petri nets
- Best, Thiagarajan
- 1987
(Show Context)
Citation Context ...s have enabled us to complete Table 1. Throughout, we attribute previously known results to their authors. Two interesting subclasses of Petri nets are not covered by Table 1, namely S- and T-systems =-=[4]-=-. For those, reachability, liveness, and deadlock are known to be polynomial in the Place/Transition case [4, 6, 13], hence also in the 1-safe case. Related work concerning not the complexity of parti... |

1 |
Reachability in reversible free choice systems
- Desel, Esparza
- 1991
(Show Context)
Citation Context ... siphon because Q • = T . Then, check in polynomial time that it contains no trap marked at the initial marking using Starke’s algorithm to find the maximal trap contained in a given siphon [31] (see =-=[8]-=- for a reference in English). We prove completeness by reducing the satisfiability problem of propositional formulas in conjunctive normal form (CON-SAT) to the deadlock problem. An instance φ of CON-... |

1 |
Petri net theory and the modeling af systems
- Peterson
- 1981
(Show Context)
Citation Context ...ith reachability and obtain the decidability from Mayr [25] and the EXPSPACE-hardness from Lipton [24]. ✷ The technique of the proofs is similar to those of, for instance, Chapter 5 in Petersons book =-=[29]-=-. The same result holds for Place/Transition nets with arc weights. To see this, just observe that our constructions can still be applied and that Hack considers nets with arc weights [16]. 4 1-Safe N... |

1 | A proof of the rank theorem - Desel - 1992 |