A boundary between universality and non-universality in spiking neural P systems

In this work we offer a significant improvement on the previous smallest spiking neural P systems and solve the problem of finding the smallest possible extended spiking neural P system. Paun and Paun gave a universal spiking neural P system with 84 neurons and another that has extended rules with 49 neurons. Subsequently, Zhang et al. reduced the number of neurons used to give universality to 67 for spiking neural P systems and to 41 for the extended model. Here we give a small universal spiking neural P system that has only 17 neurons and another that has extended rules with 5 neurons. All of the above mentioned spiking neural P systems suffer from an exponential slow down when simulating Turing machines. Using a more relaxed encoding technique we get a universal spiking neural P system that has extended rules with only 4 neurons. This latter spiking neural P system simulates 2-counter machines in linear time and thus suffer from a double exponential time overhead when simulating Turing machines. We show that extended spiking neural P systems with 3 neurons are simulated by log-space bounded Turing machines, and so there exists no such universal system with 3 neurons. It immediately follows that our 4-neuron system is the smallest possible extended spiking neural P system that is universal. Finally, we show that if we generalise the output technique we can give a universal spiking neural P system with extended rules that has only 3 neurons. This system is also the smallest of its kind as a universal spiking neural P system with extended rules and generalised output is not possible with 2 neurons.Comment: Version 1 (arXiv:0912.0741v1) of this paper contained some technical errors that were mainly due to the restriction of counter machines used. Definition 3 given in this version differs from the definition given in version 1. This new definition necessitated some minor adjustments in proofs of Theorems 1, 2 and 3

On the computational complexity of spiking neural P systems

It is shown that there is no standard spiking neural P system that simulates Turing machines with less than exponential time and space overheads. The spiking neural P systems considered here have a constant number of neurons that is independent of the input length. Following this we construct a universal spiking neural P system with exhaustive use of rules that simulates Turing machines in linear time and has only 10 neurons