Narrow Results

In the standard definition of a P system, a computation step consists of a parallel application of a "maximal" set of nondeterministically chosen rules. Referring to this system as a parallel P system, we consider in this paper a sequential P system, in which each step consists of an application of a single nondeterministically chosen rule. We show the following:

1. For 1-membrane purely catalytic systems (pure CS's), the sequential version is strictly weaker than the parallel version in that the former defines (i.e., generates) exactly the semilinear sets, whereas the latter is known to define nonrecursive sets.

2. For 1-membrane communicating P systems (CPS's), the sequential version can only define a proper subclass of the semilinear sets, whereas the parallel version is known to define nonrecursive sets.

3. Adding a new type of rule of the form: ab → a_xb_yc_comed_come to the CPS (a natural generalization of the rule ab → a_xb_yc_come in the original model), where x, y ∈ {here, out}, to the sequential 1-membrane CPS makes it equivalent to a vector addition system.

4. Sequential 1-membrane symport/antiport systems (SA's) are equivalent to vector addition systems, contrasting the known result that the parallel versions can define nonrecursive sets.

5. Sequential 1-membrane SA's whose rules have radius 1, (1,1), (1,2) (i.e., of the form (a, out), (a, in), (a, out; b, in), (a, out; bc, in)) generate exactly the semilinear sets. However, if the rules have radius 1, (1,1), (2,1) (i.e., of the form (ab, out; c, in)), the SA's can only generate a proper subclass of the semilinear sets.

We continue here the study of the recently introduced spiking neural P systems, which mimic the way that neurons communicate with each other by means of short electrical impulses, identical in shape (voltage), but emitted at precise moments of time. The sequence of moments when a neuron emits a spike is called the spike train (of this neuron); by designating one neuron as the output neuron of a spiking neural P system II, one obtains a spike train of II. Given a specific way of assigning sets of numbers to spike trains of II, we obtain sets of numbers computed by II. In this way, spiking neural P systems become number computing devices. We consider a number of ways to assign (code) sets of numbers to (by) spike trains, and prove then computational completeness: the computed sets of numbers are exactly Turing computable sets. When the number of spikes present in the system is bounded, a characterization of semilinear sets of numbers is obtained. A number of research problems is also formulated.

Spiking neural P systems were introduced in the end of the year 2005, in the aim of incorporating in membrane computing the idea of working with unique objects ("spikes"), encoding the information in the time elapsed between consecutive spikes sent from a cell/neuron to another cell/neuron. More than one dozen of papers where written in the meantime, clarifying many of the basic properties of these devices, especially related to their computing power.

The present paper quickly surveys the basic ideas and the basic results, presenting a complete to-date bibliography, and also giving a completing result related to the normal forms possible for spiking neural P systems: we prove that the indegree of such systems (the maximal number of incoming synapses of neurons) can be bounded by 2 without losing the computational completeness.

A series of research topics and open problems are formulated.

Semilinearity plays a key role not only in formal languages but also in the study of Petri nets. Although the reachability set of a Petri net may not be semilinear in general, there are a wide variety of subclasses of Petri nets which enjoy having semilinear reachability sets. In this paper, we develop sufficient conditions for Petri nets under which semilinearity is guaranteed. Our approach, based on the idea of path decomposition, can be used for consolidating several existing semilinearity results as well as for deriving new results all under the same framework.

We consider transition P systems as originally defined in the seminal paper of Gh. Păun, where he introduced the field of membrane computing. We show that it is decidable to determine, given a P system Π, whether it is strongly reversible (i.e., every configuration has at most one direct predecessor), resolving in the affirmative a recent open problem in the field. We also show that the set of all direct predecessors of a given configuration in a P system is an effectively computable semilinear set, which can effectively be expressed as a Presburger formula, strengthening an early result in the literature. We also prove other related results.

We consider jumping finite automata and their operational state complexity and decidability status. Roughly speaking, a jumping automaton is a finite automaton with a non-continuous input. This device has nice relations to semilinear sets and thus to Parikh images of regular sets, which will be exhaustively used in our proofs. In particular, we prove upper bounds on the intersection and complementation. The latter result on the complementation upper bound answers an open problem from [G. J. Lavado, G. Pighizzini, S. Seki: Operational State Complexity of Parikh Equivalence, 2014]. Moreover, we correct an erroneous result on the inverse homomorphism closure. Finally, we also consider the decidability status of standard problems as regularity, disjointness, universality, inclusion, etc. for jumping finite automata.