6 research outputs found
On the Upward/Downward Closures of Petri Nets
We study the size and the complexity of computing finite state automata (FSA) representing and approximating the downward and the upward closure of Petri net languages with coverability as the acceptance condition.
We show how to construct an FSA recognizing the upward closure of a Petri net language in doubly-exponential time, and therefore the size is at most doubly exponential.
For downward closures, we prove that the size of the minimal automata can be non-primitive recursive.
In the case of BPP nets, a well-known subclass of Petri nets, we show that an FSA accepting the downward/upward closure can be constructed in exponential time.
Furthermore, we consider the problem of checking whether a simple regular language is included in the downward/upward closure of a Petri net/BPP net language.
We show that this problem is EXPSPACE-complete (resp. NP-complete) in the case of Petri nets (resp. BPP nets).
Finally, we show that it is decidable whether a Petri net language is upward/downward closed
Regular Separability of Well-Structured Transition Systems
We investigate the languages recognized by well-structured transition systems (WSTS) with upward and downward compatibility. Our first result shows that, under very mild assumptions, every two disjoint WSTS languages are regular separable: There is a regular language containing one of them and being disjoint from the other. As a consequence, if a language as well as its complement are both recognized by WSTS, then they are necessarily regular. In particular, no subclass of WSTS languages beyond the regular languages is closed under complement. Our second result shows that for Petri nets, the complexity of the backwards coverability algorithm yields a bound on the size of the regular separator. We complement it by a lower bound construction
Separability and Non-Determinizability of WSTS
There is a recent separability result for the languages of well-structured
transition systems (WSTS) that is surprisingly general: disjoint WSTS languages
are always separated by a regular language. The result assumes that one of the
languages is accepted by a deterministic WSTS, and it is not known whether this
assumption is needed. There are two ways to get rid of the assumption, none of
which has led to conclusions so far: (i) show that WSTS can be determinized or
(ii) generalize the separability result to non-deterministic WSTS languages.
Our contribution is to show that (i) does not work but (ii) does. As for (i),
we give a non-deterministic WSTS language that we prove cannot be accepted by a
deterministic WSTS. The proof relies on a novel characterization of the
languages accepted by deterministic WSTS. As for (ii), we show how to find
finitely represented inductive invariants without having the tool of ideal
decompositions at hand. Instead, we work with closures under converging
sequences. Our results hold for upward- and downward-compatible WSTS
Comparing the Expressive Power of Well-structured Transition Systems
We compare the expressive power of a class of well-structured transition systems that includes relational automata, Petri nets, lossy channel systems, and constrained multiset rewriting systems. For each one of these models we study the class of languages generated by labelled transition systems describing their semantics. We consider here two types of accepting conditions: coverability and reachability of a given configuration. In both cases we obtain a strict hierarchy in which constrained multiset rewriting systems is the the most expressive model