Small universal spiking neural P systems with exhaustive use of rules

We consider the problem of looking for small universal spiking neural P systems with exhaustive use of rules, which was formulated as an open problem by Gheorghe Paun in a survey paper. Here, spiking neural P systems are used in two versions: as devices computing functions and as devices generating sets of numbers, with two ways of encoding the result of a computation. As devices of computing functions, if we associate the result with the distance between the first two spikes emitted by the output neuron, we produce a universal computing spiking neural P system with exhaustive use of rules (without delay) having 125 neurons; if we introduce the usual way of defining the result of a computation in membrane systems to encode the result, namely, the number of spikes emitted during a computation, then a universal computing system (without delay) with 126 neurons is also obtained in the sense of the exhaustive use of rules. For spiking neural P systems used as generators of sets of numbers, we construct a universal system (without delay) by using 128 neurons under the first way of defining the computation result, and a system (without delay) by using 127 neurons under the second way of defining the computation result.