Project : s4
Section: New Results
Keywords : Petri net , marked graphs , pathautomatic specifications , synthesis , supervisory control .
Petri nets: synthesis and control
Participants : Éric Badouel, Benoît Caillaud, Philippe Darondeau, Guillaume Feuillade.
 Marked graph

A marked graph is an ordinary Petri net where each place has exactly one input transition and one output transition.
 Pathautomatic specifications

Pathautomatic specifications are rational presentations of sets of finite or infinite graphs, given by a regular set of paths and rational relations on this set. They cover for instance trace domains, modal transition systems, and pushdown automata.
 Synthesis

The Petri net synthesis problem consists in deciding, constructively, from a given labeled transition system, whether it is isomorphic to the reachable state graph of some initialized Petri net.
 Region

The regions of a labeled transition system are the morphisms that map this graph to the Cayley graph of the group of integers, restricted on the nonnegative nodes. The regions of a graph are the places of the associated Petri net.
 Supervisory control

A supervisor is a master system that may prevent the occurrence of some controllable transitions in a slave system based on the record of observable transitions of the slave system.
The work started last year on the synthesis of Petri nets from automatic graphs has been pursued and extended. We consider now pathautomatic specifications as follows. Given an alphabet of actions, a specification comprises: a regular subset W of path labels (words on this alphabet), two rational relations on W defining which pairs of paths may not, resp. must, be confluent, and for each action, two rational relations on W defining which occurrences of this action may, resp. must, be present in a model of the specification (models are graphs). We were able to show a decision of the problem: does a given pathautomatic specification have some Petri net model (i.e., some model isomorphic to the reachable state graph of a Petri net)? This result opens a new perspective of Petri net synthesis, since it may now be applied to ambiguous specifications, halfway between transition systems and modal logic specifications. A paper coauthored by Éric Badouel and Philippe Darondeau has been submitted to a journal. Guillaume Feuillade is now trying to go further along this direction, by considering the synthesis of Petri nets from nondisjunctive modal formulas with only greatest fixpoints.
We have solved an open problem on marked graphs due to W. Reisig. The problem was to prove constructively that for any bounded marked graph (or more generally, Tsystem), there exists a labeled onesafe marked graph (resp. Tsystem) with the same language. The construction which we propose starts with a decomposition of the marked graph into sequential processes, using ideas from FIFO nets after a suitable coloring of the tokens, proceeds by a finite unfolding of the cyclic processes based on least common multiples of their periods, and ends with imposing a fixed cyclic synchronization to all the resulting processes. A paper coauthored by Philippe Darondeau and Harro Wimmel (Univ. of Oldenburg) has been submitted to a journal.
We have finally made some progress on elementary nets synthesis, by showing a universal embedding of partial 2structures (or equivalently, finite labeled transition systems) into full and forward closed set 2structures (or equivalently, elementary nets in which transitions form a partial group, where each transition has an inverse and the product of two transitions that may be fired consecutively is a transition). A paper coauthored by Andrzej Borzyszkowski (Ipi Pan, Gdansk) and Philippe Darondeau has been written and will appear soon as a research report of the Polish Academy of Sciences.