Numerical P systems are compuational models which are inspired by cellular processes and use numerical values inside the membrane structure. In this paper we will present the variants of numerical P systems that have been proposed in the literature. In particular, we will discuss the advantages introduced by these extensions in terms of Turing completeness and potential applications. Moreover, using a reference example, we will experiment a mapping between some important numerical P systems variants.
interactions and transport across the cells.
Diferent variants of the original model were then
proThe innovative and precise character of natural processes, posed, based on the interactions and positioning of the
often referred to as "the intelligence of matter" has led to cells. If cell-like P systems feature a hierarchical
conthe emergence of an interdisciplinary area at the inter- figuration of membranes as in a cell, tissue-like P
syssection of biology and computer science, called natural tems [
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License CPWrEooUrckReshdoinpgs IhStpN:/c1e6u1r3-w-0s.o7r3g ACttEribUutRion W4.0oInrtekrnsahtioonpal (PCCroBYce4.0e).dings (CEUR-WS.org) • Vari is a set of variables from membrane • Vari (0 ) is the initial values of the variables from • Pri is the set of programs from membrane
Before presenting any variant of numarical P system,
we find it useful to introduce the formal definition
of numerical P systems, the central concept of our
survey. Without reiterating any details about membrane
computing, we present a numerical P system [
the rules that transform numerical values according to the dynamics defined. In this way, numerical processes observed in economics, robotics or real-word phenomena are simulated.
Artificial Neural Networks [
of computation that mimic the functioning of neural networks. Unlike traditional ANNs, which use continuous activation functions, SNNs use discrete events known as "spikes" to transmit information. This leads to a more eficiently processing of time-dependent data.
of membrane computing model that combine concepts from Spiking Neural Networks and membrane systems.
when the neuron accumulates enough input signals to reach a threshold. SN P systems were formalized by
During each computation step, the rules within each neuron are executed in parallel. If a neuron contains multiple rules, one of them will be nondeterministically chosen and applied. At any step, the configuration of the system is described by the states of the neurons represented by the quantity of spikes present in each neuron at that time.
computation that can be done if provided with enough memory and time. Moreover, SN P systems are powerful tools in biological modeling, as one of the main motivations for developing them was to understand and model biological neural networks.
of Spiking Neural P systems as well as the computational
power of its variants can be consulted in [
organizes inter-membrane communication [
The programs are the components responsible for computing the values of the variables at each has the following form: simulation step. A program Prli ,i , 1 ≤ li ≤
mi Fli ,i (x1 ,i , . . . , xk,i ) → c1 ,i |v1 + c2 ,i |v2 + . . .
+cmi ,i |vmi tion, c1 ,i |v1 + c2 ,i |v2 + · · ·
+ cmi ,i |vmi is the
repartition protocol, and x1 ,i , . . . , xk,i are
variables from . Variables 1, 2 . . . can be
from the membrane where the programs are
located, and from its outer and inner compartments,
for a particular membrane . If a compartment
contains more than one program, a common
strategy is to execute all the programs in parallel. This
is called all-parallel mode. More details about how
the variables values are computed according to
the programs are presented in [
The membrane structure is a hierarchi- form of spikes to other neurons. These spikes occur
where Fli ,i (x1 ,i , . . . , xk,i ) is the production func- this topic. It has been shown [
one-parallel ENP systems as number generators, two
enzymatic variables are suficient. The results significantly
The use of numerical values in the compartments of a improved upon the previous data, where the numbers of
cell-like membrane structure has naturally led to possible enzymatic variables were 13 and 52 for the all-parallel
use cases in economics. Even if the economic interpre- and one-parallel systems, respectively.
tations of the variables evolution inside a numerical P Numerical P systems with thresholds (TNP systems)
system may vary, complex interactions between diferent [
conditions, other extensions of numerical P systems have also been proposed. Pavel and Dumitrache [19] introduced Hybrid Numerical P systems (HNP systems).
systems) [
proposed another extension of NP systems, called Stochastic numerical P systems (StNP systems) [20]. The main diference arises from the stochastic production function-reparticion protocol, obtaining a class of distributed parallel computing models with applications in data clustering problems.
interesting approach was proposed in [21], where a
obtained by combining the advantages ofered by numerical P systems and spiking neural P systems (SNP systems).
mains the use of numerical values as data representation.
membrane computing and spiking neural networks, the temporal aspect being one of the most attractive features.
ral P systems (NSN P systems), a new class of membrane systems that combines the advantages of NP systems and SNP systems. Similar to SNP systems, NSN P systems have a network architecture with an enhanced capability tion is encoded using numerical variables, involved in production functions that determine how the variables within the neurons evolve over time. The repartition protocol involves that after a production function in a neuron is executed, the resulted value is sent to all the variables present in adjacent neurons. Each production function can have a threshold and will be executed only when each value of the variables involved in it is not smaller than the threshold. The formal definition of NSN P systems is the following: = ( 1 , . . . , m , syn, in, out )
(2) of representing complex topologies. Also, the informa- in the production functions, values of the variables in• 1 , . . . , m represent the neurons of the form i = (Vari , Prfi , Vari (0 )), 1 ≤ i ≤
m, where: (a) Vari = {, | 1 ≤ ≤
} represents the set of variables present in i , where x1 ,i . . . xk,i can be viewed as components of a n-dimensional real space vector (b) Vari (0) = {,(0) | ,(0) ∈ R, 1 ≤ ≤ of the corresponding variables from Vari
} refers to the set of initial values (c) Prfi represents the set of production func
tions associated with each neuron • syn is the
set of synapses for each (i , j ) ∈ syn, 1 ≤ i , j ≤
m, i ̸= j • in, out are the input and output neurons
demonstrated in [23]. In terms of potential applications, NSN P systems can be suitable for applications that involve numerical information. Moreover, adding the threshold have made NSN P systems a powerful tool for applications that require deterministic mechanisms.
P systems in asynchronous manner. They also proposed an extension of the traditional threshold strategy, introducing an extended threshold strategy which involves using a threshold interval. More precisely, a production function can be executed only if all of the variables involved are within the range of the threshold interval. In this way, the obtained asynchronous numerical spiking neural P systems (ANSN P systems) will be more flexible.
of ANSN P systems has also been discussed in [24].
direction, with concrete results in recent years. In this context, Yin et al. [25] introduced in 2021 a new variant of NSN P systems, called Novel numerical spiking neural
volved having prescribed consumption rate without all being set to 0 after the execution. NSNVC P systems also use polarization of the neurons in order to control the execution of a production function.
Another variant that aims to improve the flexibility of the system has been introduced by Xu et al. [26] in 2023 by adding weights into NSN P systems. The resulted numerical spiking neural P systems with weights (NSNW P systems) are still Turing universal and the their computational power was demonstrated using fewer neurons than NSN P systems.
An interesting approach was proposed in [27]. While in the classic NSN P systems the production functions are placed inside the neurons, in numerical spiking neural P systems with production functions on synapses (NSNFS P systems), the production functions of each neuron are placed on synapses. In this way, a neuron will contain only numerical variables. Potential applications of NSNFS P systems are further investigated.
A more practical initiative was presented by Zhang et. al in [28], where enzymatic numerical spiking neural membrane systems (ENSNP systems) were introduced. In the aforementioned paper, the practicality of ENSNP systems is demonstrated by modelling ENSNP membrane controllers for robots implementing wall following.
In this section we will illustrate two numerical P systems
variants with similar evolution, using as starting point
an example of numerical P system. By analysing the Figure 1: Illustration of the numerical P system 1
evolution of the proposed NP system, we designed two
entities of diferent NP systems variants, achieving
similar functioning. We chose to use a well-known NP system 4.2. Numerical P system with Boolean
example, extracted from the article that introduced the
concept of numerical P systems [
In order to obtain another membrane system with the same functionality as 1 , we designed 3 , a NSN P system looking for the set of values taken by variable x1 ,1 , which is never consumed. Moreover, we can observe that 3 follows the same recursive formula as 1 , x1 ,1 receiving 2 (n − 1 ) + 1 at each simulation step n. The result is graphically presented in Figure 3.
The NSN P system illustrated in Figure 3 consists of four neurons, each containing one variable with an initial value of zero. Neurons 2 , 3 , 4 also contain the production functions that will be executed during the evolution of the system. It is important to mention that the first index of each variable represents the order of the variable within the neuron (as we have only one variable in each neuron, the first index will be 1 for all the variables), whilst the second index represents the label of the neuron. However, the target variable remains can be observed by analysing how x1 ,1 and x1 ,2 accumu- x1 ,1 , whilst x1 ,2 and x1 ,3 maintain similar roles as in late values at each simulation step, based on the values 1 . Table 3 illustrates the first simulation steps of the obtained at the previous steps. The first simulation steps NSN P system obtained. are presented in Table 2. We note that the values of variables x1 ,1 and x1 ,2 should be considered just for odd, respectively even values of n, as they change at diferent steps. 2 1 3 4
Initially, the value of variable x1 ,3 is zero and the production function from neuron 3 is executed. In this way, neuron 3 transmits a value of 1 to neurons 2 and 4 . Looking at the production function from the neuron 3 , it can be observed that it does not replicate the original production function from the membrane 3 of the NP system 1 , even if neuron 3 seems to be the correspondent of membrane 3. This aspect can be explained by the repartition protocol of the program from membrane 3, dividing the value between x1 ,3 and x1 ,2 . According to the distribution stage of a NSN P system, the computed production value is transmitted to each variable from all the neighboring neurons, so there is no need to divide it. After executing the production function, the value of x1 ,3 is reset to zero.
Returning to the analysis of the system, we note that neuron 3 receives 1 from neuron 4 . Neuron 4 is used as an auxiliary structure, implementing the behavior of the program inside membrane 3 of the original NP system, where the variable x1 ,3 appears both in the production function and the repartition protocol. In order to achieve this functionality in the NSN P system, we used neuron 4 to transmit the updated value of the variable after being calculated.
At the same time, neuron 2 transmits a value of 1 to variable x1 ,1 . As 1 does not contain any production function, the value of x1 ,1 will be accumulated.
One can observe that 1 and 3 use the same recursive formula, n2 = (n − 1 )2 + 2 (n − 1 ) + 1 , where (n − 1 )2 is stored in x1 ,1 of compartment 1 and 2 (n − 1 ) + 1 is computed in compartment 2 of each of them. The second model, 2 , a BNP system, computing two sets of perfect squares, uses another recurrence, n2 = (n − 2 )2 + 4 (n − 1 ), where (n − 2 )2 is stored either in compartment 1 or 2, depending on whether n is odd or even, respectively, and 4 (n − 1 ) is computed in compartment 3, the only compartment containing programs. 3 has a slightly more complex architecture, with more compartments and implicitly programs and variables.
Even for a relatively simple example, one can notice slight variations in providing models with three diferent numerical P systems. It is expected that more complex case studies may require more diverse set of features for each of the models involved, hence the need to find optimal ones, with respect to certain criteria, such as descriptional complexity measures (number of compartments, connections, rules and programs complexity).
ing numerical P systems and their variants. We briefly described each computation model, highlighting the innovations and potential applications. By exploring these extensions of the basic concept, the reader can consider using a targeted model adapted to specific requirements. We also used a numerical P system with a concise structure to illustrate the mapping to numerical P systems with Boolean condition (BNP systems) and numerical spiking neural P systems (NSN P systems). The results highlighted the main improvements introduced by each variant.
As further developments, we will include more computation models in our mappings, also considering other examples. As a testing methodology [29] for SNP systems already exists, the mapping from NP systems to NSN P systems can serve as the starting point of a promising research line to develop a testing methodology for NSN P systems. Furthermore, motivated by the applicability of numerical P systems in areas of significant importance, such as economics or robotics, we will elaborate a testing theory and some testing methods for the most relevant numerical P systems classes. Science 988 (2024) 114376. [17] L. Liu, W. Yi, Q. Yang, H. Peng, J. Wang, Small universal numerical P systems with thresholds for computing functions, Fundamenta Informaticae 176 (2020) 43–59. [18] L. Liu, W. Yi, Q. Yang, H. Peng, J. Wang, Numerical P systems with boolean condition, Theoretical Computer Science 785 (2019) 140–149. doi:10.1016/j. tcs.2019.03.021. [19] A. B. Pavel, I. Dumitrache, Hybrid numerical P systems 78 (2016) 139–146. [20] J. Yang, H. Peng, X. Luo, J. Wang, Stochastic numerical P systems with application in data clustering problems, IEEE Access 8 (2020) 31507–31518. [21] D. Reid, A. Oddie, P. Hazlewood, Parallel numerical P systems using a mimd based architecture, in: 2010 IEEE Fifth International Conference on BioInspired Computing: Theories and Applications (BIC-TA), IEEE, 2010, pp. 1646–1653. doi:10.1109/ BICTA.2010.5645257. [22] Z. Zhang, T. Wu, L. Pan, On string languages generated by sequential numerical P systems, Fundamenta Informaticae 145 (2016) 485–509. [23] T. Wu, L. Pan, Q. Yu, K. C. Tan, Numerical spiking neural P systems, IEEE Transactions on Neural Networks and Learning Systems 32 (2020) 2443– 2457. doi:10.1109/TNNLS.2020.3005538. [24] S. Jiang, Y. Liu, B. Xu, J. Sun, Y. Wang, Asynchronous numerical spiking neural P systems, Information Sciences 605 (2022) 1–14. [25] X. Yin, X. Liu, M. Sun, Q. Ren, Novel numerical spiking neural P systems with a variable consumption strategy, Processes 9 (2021) 549. [26] B. Xu, S. Jiang, Z. Shen, X. Zhu, T. Liang, Numerical spiking neural P systems with weights, Journal of Membrane Computing 5 (2023) 12–24. doi:10. 1007/s41965-022-00116-3. [27] S. Jiang, B. Xu, T. Liang, X. Zhu, T. Wu, Numerical spiking neural P systems with production functions on synapses, Theoretical Computer Science 940 (2023) 80–89. [28] L. Zhang, F. Xu, D. Xiao, J. Dong, G. Zhang, F. Neri, Enzymatic numerical spiking neural membrane systems and their application in designing membrane controllers, International Journal of Neural Systems 32 (2022) 2250055. [29] F. Ipate, M. Gheorghe, A model learning based testing approach for spiking neural P systems, Theor. Comput. Sci. 924 (2022) 1–16.