Characterizations of some restricted spiking neural p systems

• Authors: Oscar H. Ibarra;Sara Woodworth
• Venue: WMC'06 Proceedings of the 7th international conference on Membrane Computing
• Year: 2006

A k-output spiking neural P system (SNP) with output neurons, O1, ⋯, Ok, generates a tuple (n1, ⋯, nk) of positive integers if, starting from the initial configuration, there is a sequence of steps such that during the computation, each Oi 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 ni. 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 (n1, ⋯, nk) 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 itha and the i+1sta is ni, 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⊆Nk 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 Nk 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.