Solving Numerical NP-Complete Problems with Spiking Neural P Systems

Abstract

Starting from an extended nondeterministic spiking neural P system that solves the Subset Sum problem in a constant number of steps, recently proposed in a previous paper, we investigate how different properties of spiking neural P systems affect the capability to solve numerical NP-complete problems. In particular, we show that by using maximal parallelism we can convert any given integer number from the usual binary notation to the unary form, and thus we can initialize the above P system with the required (exponential) number of spikes in polynomial time. On the other hand, we show that this conversion cannot be performed in polynomial time if the use of maximal parallelism is forbidden. Finally, we show that by selectively using nondeterminism and maximal parallelism (that is, for each neuron in the system we can specify whether it works in deterministic or nondeterministic way, as well as in sequential or maximally parallel way) there exists a uniform spiking neural P system that solves all the instances of Subset Sum of a given size.

Solving Subset Sum Problems by Time-free Spiking Neural P Systems

“…SN P systems with cell division or budding can generate new neurons during the computation, thus provide a way to generate exponential working space in polynomial or linear time. These systems were successfully used to (theoretically) solve computationally hard problems, particular in NP-hard problems, in a feasible (polynomial or linear) time (see, e.g., [6,7,8,12]). …”

“…In the previous obtained SN P systems in [3,5,6,7,8,12,16,20], a global clock is generally assumed marking the time for the system. The systems work in a synchronized manner in the global level (all neurons apply their rules in a parallel manner), and for each neuron, only one of the enabled rules can be used on the tick of the clock.…”

Solving the ST-Connectivity Problem with Pure Membrane Computing Techniques

“…This is usually written as the problem of deciding if X belongs to PMC F or not. It has been studied for many P system models F and for many decision problems X (see, e.g., [2,3,4,5] and references therein).…”

Spiking neural P systems with neuron division and budding

“…A different idea of constructing SN P systems for solving NP-complete problems was given in [10,11], where the Subset Sum and SAT problems were considered. In these papers, the solutions are obtained in a semi-uniform or uniform way by using non-deterministic devices but without pre-computed resources.…”

“…However, several ingredients are also added to SN P systems such as extended rules, the possibility to have a choice between spiking rules and forgetting rules, etc. An alternative to the constructions of [10,11] was given in [9], where only standard SN P systems without delaying rules and having a uniform construction are used. However, it should be noted that the systems described in [9] either have an exponential size, or their computations last an exponential number of steps.…”