Synthesis of Bounded Choice-Free Petri Nets

Leibniz International Proceedings in Informatics, Aug 2015

This paper describes a synthesis algorithm tailored to the construction of choice-free Petri nets from finite persistent transition systems. With this goal in mind, a minimised set of simplified systems of linear inequalities is distilled from a general region-theoretic approach, leading to algorithmic improvements as well as to a partial characterisation of the class of persistent transition systems that have a choice-free Petri net realisation.

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Synthesis of Bounded Choice-Free Petri Nets

LIPIcs.CONCUR. Synthesis of Bounded Choice-Free Petri Nets Eike Best 0 Raymond Devillers 0 0 Department of Computing Science, Carl von Ossietzky Universität Oldenburg, Germany Département d'Informatique, Université Libre de Bruxelles , Belgium This paper describes a synthesis algorithm tailored to the construction of choice-free Petri nets from finite persistent transition systems. With this goal in mind, a minimised set of simplified systems of linear inequalities is distilled from a general region-theoretic approach, leading to algorithmic improvements as well as to a partial characterisation of the class of persistent transition systems that have a choice-free Petri net realisation. and phrases Choice-Freeness; Labelled Transition Systems; Persistence; Petri Nets; System Synthesis - Introduction, some examples, and basic notation In system analysis, the main task is to examine a given system’s properties by means of a behavioural description. By contrast, in system synthesis, the task is to construct – preferably automatically – an implementing system from a given behavioural specification. The benefit of such an approach is that a successfully synthesised system is “correct by design”. There is no need to re-examine its behavioural properties, because they are known to hold by construction. If synthesis fails, this may also help to delineate the true reasons of the failure, paving the way to modifications of the given input behaviour allowing for a more successful subsequent synthesis. Synthesis is being applied in many different areas (e.g., [11, 19]). In general, however, since behavioural descriptions may be extremely (even infinitely) large, synthesis algorithms could be impossible to obtain by theoretical undecidability [14], or at least be very time-consuming. Also, synthesis suffers from nondeterminism, since for a given behavioural specification, many different implementations may exist. Moreover, if there is a desire for an implementation to enjoy further properties, detecting the existence of a suitable one (if possible) tends to increase the difficulty of a synthesis problem. We investigate a special, decidable instance of system synthesis. It is assumed that a behavioural specification is given in the form of a finite, edge-labelled transition system, or lts, for short. For example, we could be interested in the transition system T S1 shown on the left-hand side of Figure 1. We shall be asking whether or not such an lts can be implemented by an unlabelled Petri net having a specific shape. The shape we shall be aiming at is choice-freeness, meaning that every place has at most one outgoing transition. For example, both Petri nets N1 and N10 shown in Figure 1 implement T S1, in the sense that their reachability graphs are isomorphic to T S1. However, N1 is choice-free while N10 is not. Figure 1 The lts T S1 is solved by the Petri net N1. It is also solved by N10 . The net N100 is not (in this paper) accepted as a solution of T S1 because its transitions are non-injectively labelled. T S0 s0 a Figure 3 The Petri net N3 solves the lts T S3. No pure solution of T S3 exists. Being related to arbiter-freeness [16], choice-freeness is interesting in a digital design context [11]. Choice-free Petri nets are also precisely the class of nets allowing a fully distributed [1] implementation. The problem has been addressed and solved for special classes of choice-free nets in previous papers by the present authors, as follows: for connected marked graphs and T-systems in [5, 7]; for bounded, reversible choice-free nets (i.e., where it is always possible to come back to the initial state) in [6, 8]; and for connected, bounded, live choice-free nets (i.e., where no transition may become dead) in [9]. In the present paper, this framework will be generalised to bounded choice-free nets, also allowing for non-live transitions. We shall be concerned with exact synthesis, disallowing that two or more transitions carry the same label. This excludes nets such as N100 in Figure 1 as implementations. Moreover, we shall take into consideration the full class of place/transition systems [18]. For example, the lts T S2 depicted in Figure 2 can be solved by N2 with an arc having weight 2 from p3 to c, but not by any plain (meaning: having arc weights at most 1) Petri net. Similarly, the lts T S3 shown in Figure 3 can be solved by N3, but not by a pure (meaning: side-place free) Petri net. Observe that there are also specifications which cannot be implemented by any unlabelled Petri net, such as the lts T S0 shown on the right-hand side of Figure 2 ([3]). The proofs of partial or full unsolvability are not hard and are left to the reader; [10] may help. For easy reference, basic formal definitions are summarised in the remainder of this section. Important concepts with strong impact on the formal development of this paper will be introduced in-line, that is, in the text explaining their relevance. To fac (...truncated)


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Eike Best, Raymond Devillers. Synthesis of Bounded Choice-Free Petri Nets, Leibniz International Proceedings in Informatics, 2015, pp. 128-141, 42, DOI: 10.4230/LIPIcs.CONCUR.2015.128