We consider here spiking neural P systems with a non-synchronized (i.e., asynchronous) use of rules: in any step, a neuron can apply or not apply its rules which are enabled by the number of spikes it contains (further spikes can come, thus changing the rules enabled in the next step). Because the time between two firings of the output neuron is now irrelevant, the result of a computation is the number of spikes sent out by the system, not the distance between certain spikes leaving the system. The additional non-determinism introduced in the functioning of the system by the nonsynchronization is proved not to decrease the computing power in the case of using extended rules (several spikes can be produced by a rule). That is, we obtain again the equivalence with Turing machines (interpreted as generators of sets of (vectors of) numbers). However, this problem remains open for the case of restricted spiking neural P systems, whose rules can only produce one spike. On the other hand we prove that asynchronous systems, with extended rules, and where each neuron is either bounded or unbounded, are not computationally complete. For these systems, the configuration reachability, membership (in terms of generated vectors), emptiness, infiniteness, and disjointness problems are shown to be decidable. However, containment and equivalence are undecidable. In short, an SN P system consists of a set of neurons placed in the nodes of a directed graph and sending signals (spikes, denoted in what follows by the symbol a) along the arcs of the graph (they are called synapses). Thus, the architecture is that of a tissue-like P system, with only one kind of object present in the cells (the reader is referred to [18] for an introduction to membrane computing and to [23] for the up-to-date information about this research area). The objects evolve by means of standard spiking rules, which are of the form E/a c → a; d, where E is a regular expression over {a} and c, d are natural numbers, c ≥ 1, d ≥ 0. The meaning is that a neuron containing k spikes such that a k ∈ L(E), k ≥ c, can consume c spikes and produce one spike, after a delay of d steps. This spike is sent to all neurons connected by an outgoing synapse from the neuron where the rule was applied. There also are forgetting rules, of the form a s → λ, with the meaning that s ≥ 1 spikes are removed, provided that the neuron contains exactly s spikes. Extended rules were considered in [4], [17]: these rules are of the form E/a c → a p ; d, with the meaning that when using the rule, c spikes are consumed and p spikes are produced. Because p can be 0 or greater than 0, we obtain a generalization of both standard spiking and forgetting rules. In this paper we consider extended spiking rules with restrictions on the type of the regular expressions used. In particular, we consider two types of rules. The first type are called bounded rules and are of the form a i /a c → a p ; d, where 1 ≤ c ≤ i, p ≥ 0, and d ≥ 0. We also consider unbounded rules of the form a i (a j) * /a c → a p ; d...
A current research topic in membrane computing is to find more realistic P systems from a biological point of view, and one target in this respect is to relax the condition of using the rules in a maximally parallel way. We contribute in this paper to this issue by considering the minimal parallelism of using the rules: if at least a rule from a set of rules associated with a membrane or a region can be used, then at least one rule from that membrane or region must be used, without any other restriction (e.g., more rules can be used, but we do not care how many). Weak as it might look, this minimal parallelism still leads to universality. We first prove this for the case of symport/antiport rules. The result is obtained both for generating and accepting P systems, in the latter case also for systems working deterministically. Then, we consider P systems with active membranes, and again the usual results are obtained: universality and the possibility to solve NP-complete problems in polynomial time (by trading space for time).
Summary. Spiking neural P systems (SN P systems, for short) are a class of distributed parallel computing devices inspired from the way neurons communicate by means of spikes. Asynchronous SN P systems are non-synchronized systems, where the use of spiking rules (even if they are enabled by the contents of neurons) is not obligatory. In this paper, with a biological inspiration (in order to achieve some specific biological functioning, neurons from the same functioning motif or community work synchronously to cooperate with each other), we introduce the notion of local synchronization into asynchronous SN P systems. The computation power of asynchronous SN P systems with local synchronization is investigated. Such systems consisting of general neurons (resp. unbounded neurons) and using standard spiking rules are proved to be universal. Asynchronous SN P systems with local synchronization consisting of bounded neurons and using standard spiking rules characterize the semilinear sets of natural numbers. These results show that the local synchronization is useful, it provides some "programming capacity" useful for achieving a desired computational power.
The spiking neural P systems are a class of computing devices recently introduced as a bridge between spiking neural nets and membrane computing. In this paper we prove a series of normal forms for spiking neural P systems, concerning the regular expressions used in the firing rules, the delay between firing and spiking, the forgetting rules used, and the outdegree of the graph of synapses. In all cases, surprising simplifications are found, without losing the computational universality -sometimes at the price of (slightly) increasing other parameters which describe the complexity of these systems. c c c c © d d d d
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