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Co-Finiteness and Co-Emptiness of Reachability Sets in Vector Addition Systems with States

Jančar Petr, Leroux Jérôme, Sutre Grégoire

Fundamenta Informaticae

2019

Vol. 169, nr 1-2

123--150

The boundedness problem is a well-known exponential-space complete problem for vector addition systems with states (or Petri nets); it asks if the reachability set (for a given initial configuration) is finite. Here we consider a dual problem, the co-finiteness problem that asks if the complement of the reachability set is finite; by restricting the question we get the co-emptiness (or universality) problem that asks if all configurations are reachable. We show that both the co-finiteness problem and the co-emptiness problem are exponential-space complete. While the lower bounds are obtained by a straightforward reduction from coverability, getting the upper bounds is more involved; in particular we use the bounds derived for reversible reachability by Leroux (2013). The studied problems were motivated by a result for structural liveness of Petri nets; this problem was shown decidable by Jančar (2017), without clarifying its complexity. The structural liveness problem is tightly related to a generalization of the co-emptiness problem, where the sets of initial configurations are (possibly infinite) downward closed sets instead of just singletons. We formulate the problems even more generally, for semilinear sets of initial configurations; in this case we show that the co-emptiness problem is decidable (without giving an upper complexity bound), and we formulate a conjecture under which the co-finiteness problem is also decidable.

On the Computational Power of 1-Deterministic and Sequential P Systems

Ibarra O.H., Woodworth S., Yen H-C., Dang Z.

Fundamenta Informaticae

2006

Vol. 73, nr 1-2

133-152

The original definition of P systems calls for rules to be applied in a maximally parallel fashion. However, in some cases a sequential model may be a more reasonable assumption. Here we study the computational power of different variants of sequential P systems. Initially we look at cooperative systems operating on symbol objects and without prioritized rules, but which allow membrane dissolution and bounded creation rules. We show that they are equivalent to vector addition systems and, hence, nonuniversal. When these systems are used as language acceptors, they are equivalent to communicating P systems which, in turn, are equivalent to partially blind multicounter machines. In contrast, if such cooperative systems are allowed to create an unbounded number of new membranes (i.e., with unbounded membrane creation rules) during the course of the computation, then they become universal. We then consider systems with prioritized rules operating on symbol objects. We show two types of results: there are sequential P systems that are universal and sequential P systems that are nonuniversal. In particular, both communicating and cooperative P systems are universal, even if restricted to 1-deterministic systems with one membrane. However, the reachability problem for multi-membrane catalytic P systems with prioritized rules is NP-complete and, hence, these systems are nonuniversal.