Uniform solutions to SAT and Subset Sum by spiking neural P systems

Abstract

We continue the investigations concerning the possibility of using spiking neural P systems as a framework for solving computationally hard problems, addressing two problems which were already recently considered in this respect: Subset Sum and SAT: For both of them we provide uniform constructions of standard spiking neural P systems (i.e., not using extended rules or parallel use of rules) which solve these problems in a constant number of steps, working in a non-deterministic way. This improves known results of this type where the construction was non-uniform, and/or was using various ingredients added to the initial definition of spiking neural P systems (the SN P systems as defined initially are called here ''standard''). However, in the Subset Sum case, a price to pay for this improvement is that the solution is obtained either in a time which depends on the value of the numbers involved in the problem, or by using a system whose size depends on the same values, or again by using complicated regular expressions. A uniform solution to 3-SAT is also provided, that works in constant time.

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.…”

Spiking neural P systems with neuron division and budding

“…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.…”

“…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. Indeed, it has been proved in [11] that a deterministic SN P system of a polynomial size cannot solve an NP-complete problem in a polynomial time unless P=NP.…”

Solving NP-Complete Problems by Spiking Neural P Systems with Budding Rules

“…The computational efficiency of SN P systems has been recently investigated in a series of works [2,6,9,11,10]. In [12] it has been proved that a deterministic SN P system of polynomial size cannot solve an NP-complete problem in a polynomial time, unless P=NP.…”