## Section: New Results

### Complete WSTS

Participant : Jean Goubault-Larrecq.

Well-structured transition systems (WSTS) are an important class of transition systems with infinitely many states, on which several verification problems remain decidable. Among WSTS one finds Petri nets and several extensions, lossy channel systems, certain abstractions of timed Petri nets, datanets, and certain process algebras.

The fundamental decidability results on WSTS, due to Finkel and
Schnoebelen in a TCS paper of 1999, is that coverability is
decidable on every WSTS (given a start state s , and a goal state
t , can we reach a state above t from s ?). This works by a
simple, set-theoretic algorithm working its way *backwards*
from the goal state.

However, some other questions, such as whether a given state is *bounded* (are there finitely many reachable states from the
state?), or *liveness* , cannot be handled this way. In the
case of Petri nets, such questions can be solved by the Karp-Miller
algorithm, which works its way *forwards* from the start state
s .

Until now, all attempts to generalize the Karp-Miller algorithm to WSTS other than Petri nets have either failed or produced ad hoc algorithms, specific to a given kind of WSTS. Moreover, e.g., for lossy channel systems, no forward algorithm can actually terminate, contrarily to the Karp-Miller algorithm. All this contributes to a lack of understanding of forward verification algorithms for WSTS in the verification community.

With Alain Finkel (LSV, ENS Cachan), we have proposed the first true
generalization of the Karp-Miller algorithm to general WSTS. This
rests, first, on a suitable definition of a *completion* of
the state space [35] , generalizing the extra
+ components used in the Karp-Miller algorithm; second, on
the design of a very short procedure that computes the so-called
*clover* of a state in a complete WSTS, which is a finite
representation of the set of states below any reachable state (see
[34] ). The clover procedure specializes to a form of
the Karp-Miller algorithm on Petri nets, but is conceptually much
simpler, and works on any complete WSTS, in particular on those
arising by completion from an ^{2} -WSTS. (All WSTS arising in
practice are ^{2} -WSTS.)

Moreover, we characterize the cases where the clover procedure
terminates exactly, as those cases where the complete WSTS is
clover-flattable, i.e., is the projection of a *flat*
transition system, that is, one whose control is ensured by a finite
automaton with no nested loop. It follows that the completion of
every Petri net, for example, is clover-flattable.

These results rest on Jean Goubault-Larrecq's discovery of the properties of Noetherian spaces (LICS 2007), which arose as a by-product from his study of semantic models mixing non-determinism and probabilities.