Uli Schlachter

Analysis of Petri Nets and Transition Systems

This paper describes a stand-alone, no-frills tool supporting the analysis of (labelled) place/transition Petri nets and the synthesis of labelled transition systems into Petri nets. It is implemented as a collection of independent, dedicated algorithms which have been designed to operate modularly, portably, extensibly, and efficiently.

Characterising Petri Net Solvable Binary Words

A word is called Petri net solvable if it is isomorphic to the reachability graph of an unlabelled Petri net. In this paper, the class of finite, two-letter, Petri net solvable words is studied. A linear time, necessary condition allows for an educated guess at which words are solvable and which are not. A full decision procedure with a time complexity of \(O(n^2)\) can be built based on letter counting. The procedure is fully constructive and can either yield a Petri net solving a given word or determine why this fails. Algorithms solving the same problem based on systems of integer inequalities reflecting the potential Petri net structure are only known to be in \(O(n^3)\). Finally, the decision procedure can be adapted from finite to cyclic words.

Petri Net Synthesis for Restricted Classes of Nets

This paper first recapitulates an algorithm for Petri net synthesis. Then, this algorithm is extended to special classes of Petri nets. For this purpose, any combination of the properties plain, pure, conflict-free, homogeneous, k-bounded, generalized T-net, generalized marked graph, place-output-nonbranching and distributed can be specified. Finally, a fast heuristic and an algorithm for minimizing the number of places in the synthesized Petri net is presented and evaluated experimentally.

An adaptive penalty function with meta-modeling for constrained problems

Constraints can make a hard optimization problem even harder. We consider the blackbox scenario of unknown fitness and constraint functions. Evolution strategies with their self-adaptive step size control fail on simple problems like the sphere with one linear constraint (tangent problem). In this paper, we introduce an adaptive penalty function oriented to Rechenbergs 1/5th success rule: if less than 1/5th of the candidate population is feasible, the penalty is increased, otherwise, it is decreased. Experimental analyses on the tangent problem demonstrate that this simple strategy leads to very successful results for the high-dimensional constrained sphere function. We accelerate the approach with two regression meta-models, one for the constraint and one for the fitness function.

Bounded choice-free Petri net synthesis: algorithmic issues

This paper describes a synthesis procedure dedicated to the construction of choice-free Petri nets from finite persistent transition systems, whenever possible. Taking advantage of the properties of choice-free Petri nets, a two-step approach is proposed. A pre-synthesis step checks necessary structural properties of the transition system and constructs some data structures needed for the second step. Then, a minimised set of simplified systems of linear inequalities is distilled from a general region-theoretic approach. This leads to a substantial narrowing of the sets of states for which linear inequalities must be solved, and allows an early detection of failures, supported by constructive error messages. The performance of the resulting algorithm is measured and compared numerically with existing synthesis tools.

Factorisation of Petri Net Solvable Transition Systems

In recent papers, general conditions were developed to characterise when and how a labelled transition system may be factorised into non-trivial factors. These conditions combine a local property (strong diamonds) and a global one (separation), the latter being of course more delicate to check. Since one of the aims of such a factorisation was to speed up the synthesis of Petri nets from such labelled transition systems, the problem arises to analyse if those conditions (and in particular the global one) could be simplified, or even dropped, in the special case of Petri net solvable behaviours, i.e., when Petri net synthesis is possible. This will be the subject of the present paper.

k-Bounded Petri Net Synthesis from Modal Transition Systems.

We present a goal-oriented algorithm that can synthesise k-bounded Petri nets (k in N^+) from hyper modal transition systems (hMTS), an extension of labelled transition systems with optional and required behaviour. The algorithm builds a potential reachability graph of a Petri net from scratch, extending it stepwise with required behaviour from the given MTS and over-approximating the result to a new valid reachability graph. Termination occurs if either the MTS yields no additional requirements or the resulting net of the second step shows a conflict with the behaviour allowed by the MTS, making it non-sythesisable.

A Graph-Theoretical Characterisation of State Separation

Region theory, as initiated by Ehrenfeucht and Rozenberg, allows the characterisation of the class of Petri net synthesisable finite labelled transition systems. Regions are substructures of a transition system which come in two varieties: ones solving event/state separation problems, and ones solving state separation problems. Linear inequation systems can be used in order to check the solvability of these separation problems. In the present paper, the class of finite labelled transition systems in which all state separation problems are solvable shall be characterised graph-theoretically, rather than linear-algebraically.

Incremental Process Discovery using Petri Net Synthesis

Process discovery aims at constructing a model from a set of observations given by execution traces (a log). Petri nets are a preferred target model in that they produce a compact description of the system by exhibiting its concurrency. This article presents a process discovery algorithm using Petri net synthesis, based on the notion of region introduced by A. Ehrenfeucht and G. Rozenberg and using techniques from linear algebra. The algorithm proceeds in three successive phases which make it possible to find a compromise between the ability to infer behaviours of the system from the set of observations while ensuring a parsimonious model, in terms of fitness, precision and simplicity. All used algorithms are incremental which means that one can modify the produced model when new observations are reported without reconstructing the model from scratch.

Bounded Petri Net Synthesis from Modal Transition Systems is Undecidable

In this paper, the synthesis of bounded Petri nets from deterministic modal transition systems is shown to be undecidable. The proof is built from three components. First, it is shown that the problem of synthesising bounded Petri nets satisfying a given formula of the conjunctive nu-calculus (a suitable fragment of the mu-calculus) is undecidable. Then, an equivalence between deterministic modal transition systems and a language-based formalism called modal specifications is developed. Finally, the claim follows from a known equivalence between the conjunctive nu-calculus and modal specifications.