The subject of the paper is to give an overview and latest results on the Legal Firing Sequence Problem of Petri nets (LFS for short). LFS is very fundamental in the sense that it appears as a subproblem or a simpler form of various basic problems in Petri net theory, such as the well-known marking reachability problem, the minimum initial resource allocation problem, the liveness (of level 4) problem, the scheduling problem and so on. However, solving LFS generally is not easy: it is NP -hard even for Petri nets having very simple structures. This intractability of LFS may have been preventing us from producing efficient algorithms for those problems. So research on LFS from computational complexity point of view seems to be rewarding.

The legal firing sequence problem (LFS) asks if it is possible to fire each transition some prescribed number of times in a given Petri net. It is a fundamental problem in Petri net theory as it appears as a subproblem, or as a simplified version of marking reachability, minimum initial resource allocation, liveness, and some scheduling problems. It is also known to be NP-hard, however, even under various restrictions on nets (and on firing counts), and no efficient algorithm has been previously reported for any class of nets having general edge weights. We show in this paper that LFS can be solved in polynomial time (in O(n log n) time) for a subclass of state machines, called cacti, with arbitrary edge weights allowed (if each transition is asked to be fired exactly once).

Adaptability evaluation of software systems is one of the key concerns in both software engineering and requirements engineering. In this paper, we present a formal and systematic approach to evaluate adaptability of software systems to requirements in enterprise business applications. Our approach consists of three major parts, that is, the common modeling method for both business realms and software realms, functional adaptability evaluation between the models with Σ algebra and behavioral adaptability evaluation with process algebra. By our approach, one can rigorously and uniquely determine whether a software system is adaptable to the requirements, either totally or partially. A sample application from an order processing is illustrated to show how this approach is effective in solving the adaptability evolution problem.

In this paper, a new method is proposed for solving forbidden state problems in non-ordinary controlled Petri nets (NCPNs) with uncontrollable transitions. Using a precedence subnet and a boundary subnet with decision-free properties, the behavior of markings are analyzed structurally. An efficient algorithm is presented for calculating the number of total tokens in forbidden places reachable from a marking. This paper derives necessary and sufficient conditions for identifying admissible markings and boundary markings in terms of the precedence subnet and the boundary subnet.

Unfolding originally introduced by McMillan is gaining ground as a partial-order based method for the verification of concurrent systems without state space explosion. However, it can be exposed to redundancy which may increase its size exponentially. So far, there have been trials to reduce such redundancy resulting from conflicts by improving McMillan's cut-off criterion. In this paper, we show that concurrency is also another cause of redundancy in unfolding, and present an algorithm to reduce such redundancy in live, bounded and reversible Petri nets which is independent of any cut-off algorithm.

Basic structural characteristics, which are useful in modular synthesis based on strongly connected state machines, of SMA/LBFC nets are discussed in this paper. A more convincing and direct proof of the equivalence of two structural characterization of the class of Petri nets is given. This proof will give clearer view of the structural characteristics of LBFC/SMA nets. On the other hand, however, the structural characteristics are not practically amenable in application to modular synthesis of SMA nets from a given set of SCSM's since all possible SCSM's should be examined for the verification of the given conditions. The later half of this paper is devoted into strengthening the results, i. e. , in composition of an SMA net from a given set of SCSM's the condition is also satisfied in any SCSM generated by composition.

A formal necessary and sufficient condition on the general Petri net reachability problem is presented by eliminating all spurious solutions among known nonnegative integer solutions of state equation and unifying all the causes of those spurious solutions into a maximal-strongly-connected and siphon-and-trap subnet Nw. This result is based on the decomposition of a given net (N, Mo) with Md and the concepts of "no immature siphon at the reduced initial marking Mwo" and "no immature trap at the reduced end marking Mwd" on Nw which are both extended from "no token-free siphon at the initial marking Mo" and "no token-free trap at the end marking Md" on N, respectively, which have been both effectively, explicitly or implicitly, used in the well-known fundamental and simple subclasses.

Geometric region method is one of the techniques to handle real-time systems which have potentially infinite state spaces. However, the original geometric region method gives incorrect results for the CTL model checking of time Petri nets. In this paper, we discuss the sufficient condition for the geometric region graphs to be correct with respect to the CTL model checking of time Petri nets, and then propose a technique to partition given geometric regions so that the graphs satisfy the sufficient condition. Finally, we implement the proposed algorithm, and compare it with the other methods by using small examples.

Petri nets are one of useful models for discrete event systems in which liveness problem as well as reachability problem is one of big issues. But, it has not been completely solved from the point of view of useful initial-marking-based liveness conditions in general Petri nets. In this paper, to guarantee local liveness (i. e. , liveness underMoD) for each minimal deadlock (MSDL),ND=(SD,TD,FD,MoD), with real deadlock-trap structure, it is shown that the minimum number of required live minimal structural traps (MSTRs),NT=(ST,TT,FT,MoT) s. t. SD ST, is conditionally (which means that the conditions of Lemma 4-9 are fulfilled for a bounded MSDL ND containing at least one MSTR NT s. t. SD ST and see also Remarks 4-2 (3) in Sect. 4. 3) "one. " Note that this local liveness for ND s. t. SD ST is one of useful necessary conditions for liveness condition of general Petri nets N=(S,T,F,Mo) s. t. S SD. However, because this has not been discussed in literature and is not trivial, some new concepts such as T-cornucopias and return paths are introduced into the real deadlock-trap structure s. t. SD ST in N and this is proven by dividing it into two cases: ND s. t. SD ST is live and unbounded under MoD and ND s. t. SD ST is live and bounded under MoD. Usefulness for the results obtained is also discussed.

This paper proposes a synthesis method to obtain speed-independent asynchronous circuits directly from signal transition graph (STG) specifications with single cycle signals which can be non-persistent and have free-choice operations. The resulting circuits are implemented with basic gates and asynchronous latches, and operate correctly under finite but unbounded gate delays and the zero wire delay assumptions. The proposed method introduces 5 types of lock relations to implement a non-persistent STG. A non-persistent STG can be implemented if every non-persistent signal to a signal t is super-locked with t. The resulting circuits are optimized by extracting of literals, mapping onto asymmetric C-elements, etc. Experimental results show that the proposed synthesis method outperforms the existing synthesis systems such as SYN and SIS.

Petri net is a graphical and mathematical tool for modelling, analysis, verification, and evaluation of discrete event systems. Liveness is one of the most important problems of Petri net analysis. This is concerned with a capability for firing of transitions and can be interpreted as a problem to decide whether the system under consideration is always able to reach a stationary behavior, or to decide whether the system is free from any redundant elements. An asymmetric choice (AC) net is a superclass of useful subclasses such as EFCs, FCs, SMs, and MGs, where SMs admit no synchronization, MGs admit no conflicts, FCs as well as EFCs admit no confusion, and ACs allow asymmetric confusion but disallow symmetric confusion. It is known that an AC net N is live iff it is place-live, but this is not the "initial-marking-based" condition and place-liveness is in general hard to test. For the initial-marking-based liveness for AC nets, it is only known that an AC net N is live if (but not only if) every deadlock in N contains a marked structural trap.

A formal system-level synthesis model for the concurrent object-oriented design of parallel computer systems, called Multi-token Object-oriented Bi-directional net (MOBnet), is proposed. The MOBnet model extends the standard Petri net by defining (1) multiple tokens to represent different kinds of synthesis control information, (2) object-oriented nodes (places) to denote the system parts under synthesis, and (3) bi-directional arcs to model the design completion check and synthesis rollback operations. In this paper, we first show that MOBnet can serve as a pre-fabrication design methodology analysis tool in ways such as class hierarchy construction, design specification comparison, reachability analysis, and concurrent process management and analysis. We then formally prove MOBnet to be a valid model for concurrent synthesis and give experimental application examples to verify. Finally, solution schemes for the design completion check and synthesis rollback problems are formally validated by analyzing the dynamic behavior of MOBnet, and experimentally illustrated through examples.

A minimal siphon (or alternatively a structural deadlock) of a Petri net is defined as a minimal set S of places such that existence of any edge from a transition t to a place of S implies that there is an edge from some place of S to t. The subject of the paper is to find a minimal siphon containing a given set of specified places of a general Petri net.

We develop a new class of stochastic Petri net: non-regenerative stochastic Petri net (NRSPN), which allows the firing time of its transitions with arbitrary distributions, and can automatically generate a bounded reachability graph that is equivalent to a generalization of the Markov renewal process in which some of the states may not constitute regeneration points. Thus, it can model and analyze behavior of a system whose states include some non-regeneration points. We show how to model a system by the NRSPN, and how to obtain numerical solutions for the NRSPN model. The probabilistic behavior of the modeled system can be clarified with the reliability measures such as the steady-state probability, the expected numbers of visits to each state per unit time, availability, unavailability and mean time between system failure. Finally, to demonstrate the modeling ability and analysis power of the NRSPN model, we present an example for a fault-tolerant system using the NRSPN and give numerical results for specific distributions.

A siphon (or alternatively a structutal deadlock) of a Petri net is defined as a set S of places such that existence of any edge from a transition t to a place of S implies that there is an edge from some place of S to t. A minimal siphon is a siphon such that any proper subset is not a siphon. The results of the paper are as follows. (1) The problem of deciding whether or not a given Petri net has a minimum siphon (i.e., a minimum-cardinality minimal siphon) is NP-complete. (2) A polynomial-time algorithm to find, if any, a minimal siphon or even a maximal calss of mutually disjoint minimal siphons of a general Petri net is proposed.

In this paper we present an efficient method to solve reachability problems for Petri nets based on genetic algorithms and a kind of random search which is called postpone search. Genetic algorithm is one of algorithms developed for solving several problems of optimization. We apply GAs and postpone search to approximately solving reachability problems. This approach can not determine exact solutions, however, from applicability points of view, does not directly face state space explosion problems and can extend class of Petri nets to deal with very large state space in reasonable time. First we describe how to represent reachability problems on each of GAs and postpone search. We suppose the existence of a nonnegative parickh vector which satisfies the necessary reachability condition. Possible firing sequences of transitions induced by the parickh vector is encoded on GAs. We also define fitness function to solve reachability problems. Reachability problems can be interpreted as an optimization ones on GAs. Next we introduce random reachability problems which are capable of handling state space and the number of firing sequences which enable to reach a target marking from an initial marking. State space and the number of firing sequences are considered as factors which effect on the hardness of reachability problems to solve with stochastic methods. Furthermore, by using those random reachability problems and well known dining philosophers problems as benchmark problems, we compare GAs' performance with the performance of postpone search. Finally we present empirical results that GAa is more useful method than postpone search for solving more harder reachability problems from the both points of view; reliability and efficiency.

Petri net is a graphical and mathematical modeling tool for discrete event systems. This paper treats analysis problems for Petri nets under the earliest firing rule. Under this firing rule, transitions must fire as soon as they are enabled. Marked Petri nets under the earliest firing rule are called earliest firing systems, for short. First, some relations in analysis problems between the earliest and the normal firing systems are discussed. These problems include deadlock freedom, boundedness, persistency and liveness. Then, relations among three types of reachability are considered from the viewpoint of the earliest firing rule. Since earliest firing systems can simulate register machines, they have equivalent modeling powers to Turing machines. It suggests, however, that most of the analysis problems of earliest firing systems with general net structures are undecidable. In this paper, net structures are restricted to a subclass called dissynchronous choice (DC) nets. It is shown that the reachability problem from an initial marking to dead markings (markings where no transition can fire) in earliest firing DC systems is equivalent to the usual reachability problem of the same systems under the normal firing rule. Then, the result is applied to reachability problems of controlled DC systems in which some transitions in the net have external control input places. It is shown that for systems where every transition in the net has an external control input place, one type of reachability problem is decidable. Lastly, the liveness problem of earliest firing DC systems is considered and it is shown that this problem is equivalent to that of the underlying DC system under the normal firing rule. It is also shown that this liveness problem is decidable.

Up to now, the only useful and well-known structural or initial-marking-based necessary and sufficient liveness conditions of Petri nets have only been those of an extended free-choice (EFC) net and its subclasses such as a free-choice (FC) net, a forward conflict free (FCF) net, a marked graph (MG), and a state machine (SM). All the above subclasses are activated only by deadlock-trap properties (i.e., real d-t properties in this paper), which mean that every minimal structural deadlock (MSDL ND=(SD, TD, FD, MoD)) in a net contains at least one live minimal structural trap (MSTR NT=(ST, TT, FT, MoT)) which is initially marked. However, the necessary and sufficient liveness conditions for EFCF, EBCF, EMGEFCFEBCF, AC (EFCFC), and the net with kindling traps NKT have recently been determined, in which each MSDL without real d-t properties was also activated by a new type of trap of trap, i.e., behavioral traps (BTRs), which are defined by introducing a virtual MSTR, a virtual maximal structural trap (virtual STR), a virtual MSDL, and a virtual maximal structural deadlock (virtual SDL) into a target MSDL. In this paper, a structural or initial-marking-based necessary and sufficient condition for local liveness (i.e., virtual deadlock-trap properties) of each MSDL ND s.t. SDST, SDST, SDST (but ND s.t. SDST is dead owing to real deadlock-trap properties) in a general Petri net N is presented by extending that in NKT. Specifically, live minimal behavioral traps (MBTRs) as well as live maximal behavioral traps (BTRs), i.e., virtual deadlock-trap properties, in a general Petri net N are characterized using the real d-t properties of each MSDL ND s.t. SDST for a general Petri net N, which were also obtained by extending the concept of return paths in NKT in connection with an MSDL which contains at least one MSTR and by using the concepts of T-cornucopias and absolute T-cornucopias in a subclass Ñ of N. In other words, BTRs are defined by introducing a virtual MSTR, a virtual STR, a virtual MSDL, and a virtual SDL into a target MSDL without real d-t properties. Additionally, a structural or initial-marking-based necessary and sufficient condition for liveness of a new subclass Nn of a general Petri net N (i.e., a general Petri net without time) is derived, and the usefulness of the obtained results is also discussed.

Petri nets are useful in modeling and analyzing various types of discrete-event systems such as parallel processing systems, distributed systems, and sequential control systems, because Petri nets can easily be used to represent such properties of these systems as concurrency, nondecidability, and causality. Various behavioral analytic problems on Petri nets are reduced to reachability and liveness on them. It is also known that the decidability of liveness is equivalent to that of reachability which is solvable. However, useful necessary and sufficient structural liveness conditions have been given only for extended free-choice (EFC) nets and their subclasses. Moreover recently, a necessary and sufficient structural liveness condition for a useful subclass NKT=(SKT, TKT, FKT, MoKT) (i.e., a Petri net in which each minimal structural deadlock (MSDL) contains at least one real or virtual kindling trap, each locally structural-live MSDL ND=(SD, TD, FD, MoD) is never globally dead even if all key transitions for local liveness of each MSDL are controlled by the net of SKTSD s.t. SKTSD, and there exists no singular MSDL of type (α)) has also been given. In this paper, in order to give one of the bases for a necessary and sufficient "structural" or "initial-marking-based" liveness condition for a general Petri net N, we will, first, directly present a necessary and sufficient local liveness condition for each MSDL with a real deadlock-trap structure in a subclass Ñ (N) using the net structure and initial token distribution and extending basic concepts used in NKT, where Ñ is a general Petri net without live behavioral traps, local liveness means a useful necessary condition for the above final goal, and real deadlock-trap structure means that each MSDL in Ñ contains at least one minimal structural trap. Secondly, a new subclass is shown in which, if the above locally structural liveness condition for each MSDL holds, then the whole-net liveness is also guaranteed. It is also argued that the obtained results are applicable to describing new live behavioral traps and deriving a necessary and sufficient structural liveness condition, which is the final goal in this work, for a general Petri net N.

If a general Petri net N = (S, T, F, Mo) is transition-live under Mo, it is evident that each maximal structural deadlock SDL(D) in N as well as each minimal structural deadlock MSDL (ND) in each D is also transition-live under Mo. However, since the converse of the latter of the above is not always true, it is important to obtain the conditions for this converse to be true if we want to have a useful necessary and sufficient "initial-marking-based" or "structural" liveness condition for N. Up to now, usefull and well-known structural or initial-marking-based necessary and sufficient liveness conditions of Petri nets have only been those of an asymmetric choice (AC) net and its subclasses such as an EFC net, an FC net, an FCF net, MG, and SM. However, all the above subclasses are activated only by real or virtual deadlock-trap properties which are local liveness for each minimal deadlocks; in other words, the above topics of this paper are unconditionally satisfied in those subclasses because of their special structure of nets. In this paper, a necessary and sufficient structural liveness condition for a general Petri net N with globally structural live minimal structural deadlocks is presented as follows: The next () or () is satisfied. () N has no SDL D. () If N has at least one SDL D, () or () is satisfied under the condition that each MSDL ND in N is transition-live under Mo. () N has no singular MSDL (α) (i.e., (α-) and (α-)). () If N has at least one singular MSDL (α-)((α-), resp.), every semi-MDSL ()((), resp.) NDS = (SDS, TDS, FDS, MoDS with respect to each singular MSDL (α-)((α-), resp.), is transition-live under the MoDS under the condition of "the condition (**)", where the locally structural liveness for this NDS means (1) or (2)((3), resp.) of Lemma 4-4 and "the condition (**)" is defined in Lemma 4-7 of this paper. The relationship between the above results and the liveness problem for N is also shown.