Petri Nets with Special Structures

Petri nets have traditionally been motivated by their ability to express concurrency. Subclasses of Petri nets with concurrency but without choices or conflicts have been studied extensively. One of the best known - and comparatively restricted - such class are the marked graphs [4]. However, perhaps the largest class of (intuitively) choice-free nets are the persistent nets [7], a class of nets that is significantly larger than marked graphs. Some early results about persistent nets are Keller's theorem [6], which will be recalled in a later part of this paper, and the famous semilinearity result of Landweber and Robertson [7], which states that the set of reachable markings of a persistent net is semilinear. In this paper, we show that in bounded persistent nets, the smallest cycles of its reachability graph enjoy a uniqueness property.

Join-Free Petri nets, whose transitions have at most one input place, model without synchronizations while Choice-Free Petri nets, whose places have at most one output transition, model without conflicts. These classes respectively encompass the state machines (or S-systems) and the marked graphs (or T-systems).Whereas a structurally bounded and structurally live Petri net graph is said to be well-formed, a bounded and live Petri net is said to be well-behaved. Necessary and sufficient conditions for the well-formedness of Join-Free and Choice-Free nets have been known for some time, yet the behavioral properties of these classes are still not well understood. In particular efficient sufficient conditions for liveness have not been found until now. In this paper, we extend results on weighted T-systems to the class of weighted Petri nets and present transformations which preserve the feasible sequences of transitions and reduce the initial marking. We introduce a notion of balancing that makes possible the transformation of conservative into so-called 1-conservative systems while retaining the feasible transition sequences. This transformation leads to polynomial sufficient conditions of liveness for well-formed Join-Free and Choice-Free nets.

We present two aspects of knitting technique, the structural properties (especially the P- and T-invariants), and the synchronized choice net (a new class of Petri net), that are of both theoretical importance and practical uses to the verification of structural correctness of a Petri net or to detect the structural problem of a Petri net. This work first proves that the ordinary Petri nets synthesized with knitting technique are structurally bounded, consistent, conservative and safe (when each home place holds one token) using the well-known linear algebra approach. It also provides a procedure for finding P- and T-invariants for Petri net synthesized using the knitting technique. We present examples for P-invariants and show that we can synthesize Petri nets more general than the "asymmetric-choice nets". The algorithm for finding P-invariants of ordinary Petri nets is extended to find the P-invariants for a general Petri net synthesized with knitting technique and the arc-ratio rules. We present a new class of Petri nets, called synchronized choice nets, which are the largest set of Petri nets that can be covered by both T-components and P-components. An algorithm is proposed to find its T-components and the P-components, respectively. The complexity of this algorithm is also presented. The theory of synchronized choice nets has the potential to simplify that for free choice nets.

<u>1.1 The language CAP</u>While there are a couple of programming languages which allow the representation of parallelism /1,2,5/ there is a lack for languages which allow in an as well disciplined as general manner the description of concurrency.A well defined mathematical description method for concurrent processes is given by the Petri net model /3,7,9/. It fulfils the above requirements and in addition is widely used and extremely easy to understand.This concept is integrated into the language CAP in a very elegant way: Lables are used as places, where one has in mind that a labelled statement puts a token into its label(s) <u>after</u> it has been processed. On the other hand the processability of a statement is controlld by "On-conditions" on labels. Every statement is processed if and only if its "On-condition" has become true. At this time, in accordance with the firing-rule (="On-condition"), "tokens" are withdrawn from certain labels used within the "On-condition" and placed into the labels of the statement.In addition various parameters may be associated to transitions thus offering a description power similar to the Macro-E-Nets /8/. On the other hand there are a lot of language-constructs to support structured programming.<u>1.2 The analysis of CAP-programs at compile-time</u>Since a couple of years Petri nets are object of research. For some rather restricted subclasses of Petri nets sufficiant an necessary conditions for the topology of a net to be well-formed are known /3,4,7/. Under a well-formed net we understand a net <u>without dead-locks</u> being <u>safe</u> (no loss of information) and <u>residue-free</u> (independent from a certain history). Related to results by Herzog an Yoeli /4/ for a subclass of Petri nets we prove that there are six conditions for the topology of a net of the subclass of Petrinets which are used to describe the control-structure of CAP-programs so that the net is well-formed.By this we can decide at compile-time without simulation whether a CAP-program describes a life, safe and residue-free system (i. e. a useful one) or not. After having defined, what we mean ba a "Structured CAP-program" we will have the fine result that every structured CAP-program is well-formed.<u>1.3 The simulation of systems described in CAP</u>As CAP allows the description of systems down to a level of specification which is comparable with the bit-level of digital systems inclusive the detailled description of the real-time-behaviour, for the processing of CAP one needs a system similar to a simulator for digital circuits. On the other hand also very global descriptions on a high level of abstraction are possible.This implies that the run-time system for CAP must be extremely adaptive. We tried to solve this problem by a very flexible and powerful table driven and event oriented simulator with the event-mechanism being directly adopted from the Petri net concept of CAP.

Performance evaluation is an important issue for discrete event dynamic systems, which in many cases can be described in terms of the language of Petri nets. In particular, for marked graphs, a subclass of Petri nets, a formula for performance evaluation is widely known. That is, the ratio of the total execution time to the number of tokens in a cycle gives a lower bound for the cycle time of the system, and the maximum of these ratios over all cycles determines the overall system performance. However, we need to enumerate all directed cycles in a graph to apply this formula of performance evaluation. Carrying out this enumeration for an actual system is often impractical, since the number of cycles in a graph usually grows exponentially with the size of a system. We give a linear algebraic characterization for directed cycles, and based on this result, transform the problem of performance evaluation into a simple linear programming problem. A few explanatory examples are also given.

When a Petri net is synthesised from a labelled transition system, it is frequently desirable that certain additional constraints are fulfilled. For example, in circuit design, one is often interested in constructing safe Petri nets. Targeting such subclasses of Petri nets is not necessarily computationally more efficient than targeting the whole class. For example, targeting safe nets is known to be NP-complete while targeting the full class of place/transition nets is polynomial, in the size of the transition system. In this paper, several classes of Petri nets are examined, and their suitability for being targeted through efficient synthesis from labelled transition systems is studied and assessed. The focus is on choice-free Petri nets and some of their subclasses. It is described how they can be synthesised efficiently from persistent transition systems, summarising and streamlining in tutorial style some of the authors’ and their groups’ work over the past few years.

Starts with a brief review of the history and the application areas considered in the literature. The author then proceeds with introductory modeling examples, behavioral and structural properties, three methods of analysis, subclasses of Petri nets and their analysis. In particular, one section is devoted to marked graphs, the concurrent system model most amenable to analysis. Introductory discussions on stochastic nets with their application to performance modeling, and on high-level nets with their application to logic programming, are provided. Also included are recent results on reachability criteria. Suggestions are provided for further reading on many subject areas of Petri nets.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

A class of synchronized queueing networks with deterministic routing is identified to be equivalent to a subclass of timed Petri nets called marked graphs. Some structural and behavioral properties of marked graphs are used to show interesting properties of this class of performance models. In particular, ergodicity is derived from the boundedness and liveness of the underlying Petri net representation. In the case of unbounded (i.e., nonstrongly connected) marked graphs, ergodicity is computed as a function of the average transition firing delays. For steady-state performance, linear programming problems defined on the incidence matrix of the underlying Petri nets are used to compute tight (i.e., attainable) bounds for the throughput of transitions for marked graphs with deterministic or stochastic time associated with transitions. These bounds depend on the initial marking and the mean values of the delays but not on the probability distortion functions. The benefits of interleaving qualitative and quantitative analysis of marked graph models are shown. >

The AIP manufacturing testbed discussed by Ghaffari et al. (see ibid., vol. 48, p.18-29, 2003) was previously treated using automaton-based [as distinct from Petri net (PN)-based] supervisory control theory (SCT). Problems of this general type have been transcribed from the PN literature, sometimes greatly enlarged, converted into SCT format, and solved using integer-decision-diagram (IDD) techniques with only modest expenditure of time and memory. Thus it is substantially misleading to state, that "the AIP control problem would require huge processing time and memory if addressed with the Ramadge-Wonham [SCT] approach." In contrast to the restriction in Ghaffari et al. (2003) to "marked graphs", no prior structural restrictions other than finiteness of the state set (or boundedness in the case of PNs) were placed on the models of system components.

Liveness, fairness, and recurrence in Petri nets

This study deals with deadlock problems in a system with uncontrollable and unobservable events, which is modeled with a subclass of Petri nets. We design monitors to enforce constraints for minimal siphons in a Petri net model. A key point of this paper is how to select the input and output arcs of a monitor when there exist uncontrollable and unobservable transitions. After the enforcement of constraints, an iterative algorithm is developed to obtain a controlled system with liveness. A sufficient and necessary condition is established to decide the existence of a monitor enforcing a constraint in a plant with uncontrollable and unobservable transitions.

This chapter, that can be seen as the continuation of the previous one, presents additional background material on Petri nets. In particular, the main focus is on structural analysis, i.e., algebraic tools that do not require the enumeration of the reachability set of a marked net but are based on the analysis of the state equation, on the incidence matrix, etc. Meaningful structural properties, i.e., properties that are only related to the structure of the net and not to its initial marking, are also defined and analyzed. Subclasses of Petri nets are finally defined and simplified analysis criteria that pertain to these classes are presented.

A design methodology for self-timed VLSI systems is proposed. This includes a language for describing algorithmic behaviour which is based on the data-flow concept. A subclass of Petri nets is introduced to model the designed system, and a refinement technique is presented that allows for the synthesis of correct-by-construction systems. Finally, a systematic means for implementing the design is given.

In spite of their increasing popularity for modeling and performance analysis of parallel systems, Petri nets play only a marginal role in the area of synthesis of asynchronous hardware. It is suggested that a change in the perception of their role will lead to a cost effective design method for parallel asynchronous architectures. A hardware implementable subclass of Petri nets is presented that exploits conflicts as a nondeterministic scheduling feature and that offers token evaluation as an option for path selection. The net primitives, complemented with functional primitives, constitute the lowest abstractions for a data flow oriented design paradigm.

A marked Petri net is lucent if there are no two different reachable markings enabling the same set of transitions, i.e., states are fully characterized by the transitions they enable. This paper explores the class of marked Petri nets that are lucent and proves that perpetual marked free-choice nets are lucent. Perpetual free-choice nets are free-choice Petri nets that are live and bounded and have a home cluster, i.e., there is a cluster such that from any reachable state there is a reachable state marking the places of this cluster. A home cluster in a perpetual net serves as a “regeneration point” of the process, e.g., to start a new process instance (case, job, cycle, etc.). Many “well-behaved” process models fall into this class. For example, the class of short-circuited sound workflow nets is perpetual. Also, the class of processes satisfying the conditions of the \(\alpha \) algorithm for process discovery falls into this category. This paper shows that the states in a perpetual marked free-choice net are fully characterized by the transitions they enable, i.e., these process models are lucent. Having a one-to-one correspondence between the actions that can happen and the state of the process, is valuable in a variety of application domains. The full characterization of markings in terms of enabled transitions makes perpetual free-choice nets interesting for workflow analysis and process mining. In fact, we anticipate new verification, process discovery, and conformance checking techniques for the subclasses identified.