Bounded choice-free Petri net synthesis: algorithmic issues

Abstract

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.

Simultaneous Petri Net Synthesis

“…Thus, for instance, with meaning "less than or equal in each component, but not entirely equal", we have (0, 0, 0, 0) (0, 1, 0, 0) (0, 2, 0, 0). And with ≤ meaning "not less than or equal", (1,2,3,4) ≤ (4,4,4,3). We define and ≥ analogously.…”

“…In the remainder of this paper, we shall describe, and analyse, an algorithm solving the simultaneous choice-free synthesis problem. The following theorem, whose proof can be found in [3,4,17], specifies a number of properties a single transition system must necessarily enjoy in order to have a choice-free solution. It forms the basis of pre-synthesis.…”

“…We shall now explain these properties (more details, examples and counter-examples may be found in [3,4]). Total reachability and both versions of determinism are always satisfied by Petri net reachability graphs.…”

“…The pre-synthesis algorithms described in [4,17] check the properties listed in Theorem 1 in some clever and efficient way -more precisely, in time O(|S| · |T | 2 ) -and create, individually for every T S i , data structures containing all distances from the initial state as well as the Parikh vectors of all small cycles. Hence we can assume from now on that all transition systems satisfy these basic properties, and that such data structures are available.…”

“…Two new checks are added with respect to the ones detailed in previous papers, and the resulting algorithms are specified in full, and analysed, in Section 5. In Section 6, techniques that have been developed for individual choice-free synthesis [4,17] are extended to simultaneous choice-free synthesis. In Section 7, some concluding remarks and future plans are evoked.…”