Characterizations of Some Restricted Spiking Neural P Systems

Abstract

A k-output spiking neural P system (SNP) with output neurons, O ₁, ⋯, O _k , generates a tuple (n ₁, ⋯, n _k ) of positive integers if, starting from the initial configuration, there is a sequence of steps such that during the computation, each O _i generates exactly two spikes a a (the times the pair a a are generated may be different for different output neurons) and the time interval between the first a and the second a is n _i . After the output neurons have generated their pairs of spikes, the system eventually halts. Another model, called k-train SNP, has only one output neuron. It generates a k-tuple (n ₁, ⋯, n _k ) if, starting from the initial configuration, the output neuron O generates the spike train aa ⋯a with exactly k+1 a’s such that the interval between the i ^th a and the i+1^st a is n _i , and the system eventually halts. We assume, without loss of generality, that each neuron in the SNP is either bounded or unbounded. (Bounded here means that there is a fixed constant c such that at any time during the computation, the number of spikes in the neuron is at most c. Otherwise, the neuron is unbounded.) It is known that 1-output SNPs (= 1-train SNPs) are universal, i.e., they generate exactly the recursively enumerable sets over N. Here, we show the following:

1. For k ≥1, a set Q ⊆ N ^k is semilinear if and only if it can be generated by a k-output SNP, where every unbounded neuron satisfies the property that once it starts “spiking” it will no longer receive future spikes (but can continue spiking). This result also holds for k-train SNP.

2. The set Q = {(m,2m) |  m ≥1} (which is semilinear) cannot be generated by any 2-output bounded SNP (i.e., SNP all of whose neurons are bounded). Thus, for k ≥2, there are semilinear sets over N ^k that cannot be generated by k-output bounded SNPs. This contrasts a known result that 1-output bounded SNPs generate all semilinear sets over N.

3. For k ≥2, k-output bounded SNPs are computationally more powerful than k-train bounded SNPs. (They are identical when k=1.)

4. For k ≥1, k-output bounded SNPs and k-train bounded SNPs can be characterized by certain classes of nondeterministic finite automata with strictly monotonic counters.

This research was supported in part by NSF Grants CCF-0430945 and CCF-0524136.