A Boundary between Universality and Non-universality in Extended Spiking Neural P Systems

Abstract

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. Pȃun and Pȃun [15] gave a universal spiking neural P system with 84 neurons and another that has extended rules with 49 neurons. Subsequently, Zhang et al. [18] 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.Key words: spiking neural P systems, small universal spiking neural P systems, computational complexity, strong universality, weak universality Email address: tneary@cs.nuim.ie (Turlough Neary).

On the computational complexity of spiking neural P systems

“…Analogously, some small SN P systems could be described as what Korec refers to as weak universality. However, as we noted in other work [9], it could be considered that Korec's notion of strong universality is somewhat arbitrary and we also pointed out some inconsistency in his notion of weak universality. Hence, in this work we rely on time/space complexity analysis to compare the encodings used by the small SN P system in Table 1.…”

“…The brief history of small universal spiking neural P systems is given in Table 1. Note that, to simulate an arbitrary Turing machine that computes in time t, all of the small universal spiking neural P systems prior to our results require time that is exponential in t. An arbitrary Turing machine that uses space of s is simulated by the universal systems given in [4,11,18] in space that is doubly exponential in s, and by the universal systems given in [3,10,15,19] in space that is exponential in s. [15] 67 exponential standard no Zhang et al [19] 49 exponential extended † no Pȃun and Pȃun [15] 41 exponential extended † no Zhang et al [19] 12 double-exponential extended † no Neary [11] 18 exponential extended no Neary [11,12]* 17 exponential standard † no [9] 14 double-exponential standard † no [9] 5 exponential extended † no [9] 4 double-exponential extended † no [9] 3 double-exponential extended ‡ no [9] 125 exponential/ extended † yes Zhang et al [18] double-exponential 18 polynomial/exponential extended yes Neary [10] 10 linear/exponential extended yes Section 5 Table 1. Small universal SN P systems.…”

An improved universal spiking neural P system with generalized use of rules

“…The theoretical and practical usefulness of these variants were also investigated: regarding computing power the relationship between these variants and well-known models of computation, e.g. finite automata, register machines, grammars, computing numbers or strings was investigated in [23][24][25][26][27][28][29][30][31][32][33][34][35]; computing efficiency of these variants in solving hard problems was investigated in [36,37].…”

Notes on spiking neural P systems and finite automata

“…Our constructions also reduce the neurons for such SNP modules: from three neurons down to one. Our reduction relies on more involved superscripts, similar to some of the constructions in [12].…”