Numerical P Systems: Variants and Applications Radu Traian Bobe1,*,† , Marian Gheorghe2,† , Florentin Ipate1,† and Ionuţ Mihai Niculescu1,† 1 Department of Computer Science, Faculty of Mathematics and Computer Science University of Bucharest, Str Academiei 14, Bucharest, 010014, Romania 2 Faculty of Engineering and Informatics, University of Bradford, Bradford, West Yorkshire, BD7 1DP, United Kingdom Abstract 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. Keywords Membrane Computing, Numerical P systems, Spiking Neural P systems, Enzymatic Numerical P systems, Modelling 1. Introduction interactions and transport across the cells. Different variants of the original model were then pro- The 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 con- the emergence of an interdisciplinary area at the inter- figuration of membranes as in a cell, tissue-like P sys- section of biology and computer science, called natural tems [2] have several one-membrane cells arranged as computing. Membrane computing, evolutionary compu- nodes in an undirected graph. This arrangement gave the tation, cellular automata, as well as neural computing are model name, because the cells are organized as in a tissue. just a few of the most important areas of natural comput- Moreover, Ionescu et al. introduced in 2006 neural-like ing. P systems, that incorporates the idea of spiking neurons Membrane computing is a branch of natural comput- into the area of membrane computing. [3]. This novel ing, developed by Gh. Păun in 1998 [1]. The concept category of membrane systems has cells represented as is inspired by the biological functionality of living cells, neurons from a neural net. The communication between abstracting computational models. A cell-like membrane them is represented as electrical impulses called spikes. system, also called a P system is a formalism that ab- More definitions about membrane computing, as well as stracts and mimics the processes observed in living cells, examples and technical results can be found in [4]. specifically focusing on how membranes within cells A category of cell-like P systems that use numerical val- interact and process information. The membranes are ues in the compartments sparked interest and innova- represented as compartments that encapsulates objects tions in fields like economics or robotics. These mem- and rules. We can refer to objects as entities that in- brane systems are called numerical P sytems [5] and focus habit inside membranes and are transformed according on manipulating numerical values inside the membrane to the rules. A rule is similar to a chemical reaction, to structure. This paper aims to present the variants of nu- the extent that dictates the evolution of objects within merical P systems. In particular, we will highlight the and across membranes. Continuing the analogy with the main differences between them, analysing the computa- bio-chemical domain, this process is similar to molecular tional power. We will also demonstrate a transformation 24rd Conference ITAT: Workshop on Natural Computing between the most interesting models proposed in the lit- * Corresponding author. erature, taking the examples from the reference articles. † These authors contributed equally. This paper is structured as follows: Section 2 introduces $ radu.bobe@s.unibuc.ro (R. T. Bobe); m.gheorghe@bradford.ac.uk the formal definition of numerical P systems and spiking (M. Gheorghe); florentin.ipate@unibuc.ro (F. Ipate); numerical P systems. Section 3 presents and classifies all ionutmihainiculescu@gmail.com (I. M. Niculescu) the numerical P systems variants. Section 4 illustrates € https://www.bradford.ac.uk/staff/mgheorghe (M. Gheorghe); https://www.ifsoft.ro/~florentin.ipate/ (F. Ipate); the conversion from a numerical P system to a numerical https:\ionutmihainiculescu.ro (I. M. Niculescu) P system with Boolean condition and a numerical spiking  0009-0005-6611-3176 (R. T. Bobe); 0000-0002-2409-4959 neural P system. Section 5 states the conclusions of this (M. Gheorghe); 0000-0001-8777-3425 (F. Ipate); paper as well as future research directions. 0000-0002-6135-9135 (I. M. Niculescu) © 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings http://ceur-ws.org ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org) CEUR ceur-ws.org Workshop ISSN 1613-0073 Proceedings 2. Preliminaries A numerical P system evolves by iteratively applying the rules that transform numerical values according to 2.1. Numerical P System Definition the dynamics defined. In this way, numerical processes observed in economics, robotics or real-word phenomena Before presenting any variant of numarical P system, are simulated. we find it useful to introduce the formal definition An example of numerical P system is presented later in of numerical P systems, the central concept of our this paper, in Section 4.1. survey. Without reiterating any details about membrane computing, we present a numerical P system [6] as the following tuple: 2.2. From Artificial Neural Networks to Spiking Neural P Systems 𝛱 = (m, H , 𝜇, (Var1 , Pr1 , Var1 (0 )), . . . , (1) Artificial Neural Networks [7] are computational models (Varm , Prm , Varm (0 ))) inspired by the interaction between biological neural where: networks from the human brain. The central computing elements of an Artifical Neural Network (ANN) is • m ≥ 1 is degree of the system Π (the number of called neuron. The connections between neurons, called membranes); synapses are made through input signals. The strength • H is an alphabet of labels; of these signals is controlled by the weights associated with synapses. • 𝜇 is membrane structure; Spiking Neural Networks (SNNs) [8] are ANNs models of computation that mimic the functioning of neural net- • Vari is a set of variables from membrane works. Unlike traditional ANNs, which use continuous i, 1 ≤ i ≤ m; activation functions, SNNs use discrete events known as • Vari (0 ) is the initial values of the variables from "spikes" to transmit information. This leads to a more membrane i, 1 ≤ i ≤ m; efficiently processing of time-dependent data. Spiking Neural P systems (SN P systems) are a variant • Pri is the set of programs from membrane of membrane computing model that combine concepts i, 1 ≤ i ≤ m. from Spiking Neural Networks and membrane systems. Neurons are individual units that send out signals in The membrane structure 𝜇 is a hierarchi- form of spikes to other neurons. These spikes occur cal arrangement of membranes that can be when the neuron accumulates enough input signals to visualized as an expression of matching brackets, reach a threshold. SN P systems were formalized by each pair representing a membrane. It also Ionescu et al. in [3]. organizes inter-membrane communication [5] . The programs are the components responsible During each computation step, the rules within each for computing the values of the variables at each neuron are executed in parallel. If a neuron contains simulation step. A program Prli ,i , 1 ≤ li ≤ mi multiple rules, one of them will be nondeterministically has the following form: chosen and applied. At any step, the configuration of the system is described by the states of the neurons repre- Fli ,i (x1 ,i , . . . , xk ,i ) → c1 ,i |v1 + c2 ,i |v2 + . . . sented by the quantity of spikes present in each neuron at that time. +cmi ,i |vmi The computing power of SN P systems is significant and represents a key component in the research area of where Fli ,i (x1 ,i , . . . , xk ,i ) is the production func- this topic. It has been shown [3] that SN P systems are tion, c1 ,i |v1 + c2 ,i |v2 + · · · + cmi ,i |vmi is the Turing complete which means that they can compute any repartition protocol, and x1 ,i , . . . , xk ,i are vari- computation that can be done if provided with enough ables from 𝑉 𝑎𝑟𝑖 . Variables 𝑣1 , 𝑣2 . . . 𝑣𝑚𝑖 can be memory and time. Moreover, SN P systems are powerful from the membrane where the programs are lo- tools in biological modeling, as one of the main motiva- cated, and from its outer and inner compartments, tions for developing them was to understand and model for a particular membrane 𝑖. If a compartment biological neural networks. contains more than one program, a common strat- An in-depth survey detailing the computational process egy is to execute all the programs in parallel. This of Spiking Neural P systems as well as the computational is called all-parallel mode. More details about how power of its variants can be consulted in [9]. the variables values are computed according to the programs are presented in [5]. 3. Numerical P systems variants one-parallel ENP systems as number generators, two en- zymatic variables are sufficient. 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 different [14] employ a similar strategy to the enzymatic control in currency exchange models or economic policies can be the sense that they use evolution programs in controlled done. manner. The difference is that a program can be applied Programs are fundamental to the functionality of NP only when the values of the variables involved in the systems and define computational processes, enabling production function are not smaller (Lower Threshold powerful computing performance and numerical anal- Numerical P systems) or not greater (Upper Threshold ysis. Starting from the importance of programs within Numerical P systems) than the constant. A related a numerical P system and driven by the desire of better approach was introduced in [15] where a production control the processes, numerous variants of NP systems threshold control strategy is implemented. Instead of have been proposed. In short, researchers have employed comparing every value of variables involved in the different mtehods of using the evolution programs to find program with a constant, in numerical P systems new possible real-life applications of NP systems or to with production thresholds (PTNP systems), the entire study the universality results as well as the computa- production value is compared with a constant. The tional power of the resulted variants. production function is applied only when its value is In this section, we will briefly present the extensions in- not smaller (the lower-threshold case) or not greater troduced by each variant to the basic model, as well as (the upper-threshold case) than its associated constant. some initial results and possible use cases. Even if the usability of these two numerical P systems In the classical model of a NP system, only one production in concrete applications remains an open subject, the function can be applied from each membrane in a time language generating power of numerical P systems with unit. This case is called deterministic. In the stochastic thresholds was discussed in [16]. case, if a membrane contains more than one production Numerical P systems with thresholds were the funda- function, one of them is randomly chosen. Enzymatic Nu- mental concept for other research topics as well, with merical P systems (ENP systems) allow a better control their integration with Petri Nets being investigated in of programs applications, by incorporating enzymes into [26], where the operations of Petri Nets were associated the production functions, which allows for more accu- with the evolutions in TNP systems. Another interesting rate transformations of numerical values. As presented approach is presented in [17] where six mall universal in [10], in an ENP system a rule can only be applied (in function computing devices of TNP systems were this case, the rule is called active) if the corresponding en- constructed. zyme is present in the necessary amount. It is important to mention that an enzyme can be present in more than Taking into consideration the above mentioned one production functions. All the active rules are then findings, it is unanimously accepted that control executed in parallel. In this way, more precise and regu- conditions play a crucial role in controlling the evolution lated transformations of numerical values are obtained, of numerical P systems. Addressing potential practical reflecting the catalytic and regulatory roles of enzymes applications, the large majority of dynamic systems in biological systems. In addition to the use of numerical requires an accurate control. The above mentioned P systems in economics, enzymatic numerical P systems numerical P systems variants have introduced the can be utilized in robotics to enhance the precision and concept of control conditions. However, this conditions adaptability of control algorithms [11]. tend to have a simple logic and may not be able to Enzymatic numerical P systems were proved to be Tur- achieve the requirements in a real-world dynamic ing universal, aspects like the complexity of polynomial system scenario. In order to overcome this limitation, production functions or the number of variables being Liu et al. [18] introduced the control condition in investigated. The computational power on ENP systems Boolean form and proposed a new variant of NP was discussed in [12]. The topic was considered from systems, called Numerical P Systems with Boolean a different point of view in [13], with the focus on de- condition (BNP systems). Even if the control condition termining the smallest number of enzymatic variables in the previous variants of numerical P systems can be needed for universal ENP systems. Zhang et al. proved expressed as a Boolean condition, there were still a lot that if ENP systems are used as number acceptors in of Boolean expressions that cannot be expressed in all the all-parallel or one-parallel mode, only one enzymatic these variants. In a BNP system, the condition can be variable is needed to achieve universality. In the case of any Boolean expression using relational operators and linked by logical operators. In this way, BNP systems where: have introduced a stronger control mechanism that can be useful in real-world applications of NP systems. • 𝜎1 , . . . , 𝜎m represent the neurons of the form Moreover, it was proved that BNP systems are Turing 𝜎i = (Vari , Prfi , Vari (0 )), 1 ≤ i ≤ m, where: universal as number generating/accepting devices and (a) Vari = {𝑥𝑞,𝑖 | 1 ≤ 𝑞 ≤ 𝑘𝑖 } represents function computing devices, respectively, working in the set of variables present in 𝜎i , where all-parallel, one-parallel and sequential mode. x1 ,i . . . xk ,i can be viewed as components of a n-dimensional real space vector In addition to these variants that target control (b) Vari (0) = {𝑥𝑞,𝑖 (0) | 𝑥𝑞,𝑖 (0) ∈ R, 1 ≤ conditions, other extensions of numerical P systems 𝑞 ≤ 𝑘𝑖 } refers to the set of initial values have also been proposed. Pavel and Dumitrache [19] of the corresponding variables from Vari introduced Hybrid Numerical P systems (HNP systems). Numerical P systems with migrating variables (MNP (c) Prfi represents the set of production func- systems) [14] were inspired by the fact that in standard tions associated with each neuron P systems an object can pass through membranes, • syn is the set of synapses for each between regions of the same cell, between cells, or (i, j ) ∈ syn, 1 ≤ i, j ≤ m, i ̸= j between cells and their environment. In 2020, Yang et al. proposed another extension of NP systems, • in, out are the input and output neurons called Stochastic numerical P systems (StNP systems) [20]. The main difference arises from the stochastic Turing universality of NSN P systems has also been production function-reparticion protocol, obtaining demonstrated in [23]. In terms of potential applica- a class of distributed parallel computing models with tions, NSN P systems can be suitable for applications applications in data clustering problems. Another that involve numerical information. Moreover, adding interesting approach was proposed in [21], where a the threshold have made NSN P systems a powerful tool MIMD (Multiple Instruction Multiple Data) architecture for applications that require deterministic mechanisms. is used to parallelise the elements of a NP system. Also, In 2022, Jiang et al. [24] considered working with NSN the generative capacity of numerical P systems as P systems in asynchronous manner. They also proposed language generators was investigated in [22]. an extension of the traditional threshold strategy, intro- ducing an extended threshold strategy which involves In the following, we will focus on P systems extensions using a threshold interval. More precisely, a production obtained by combining the advantages offered by numeri- function can be executed only if all of the variables in- cal P systems and spiking neural P systems (SNP systems). volved are within the range of the threshold interval. In One of the main advantages of numerical P systems re- this way, the obtained asynchronous numerical spiking mains the use of numerical values as data representation. neural P systems (ANSN P systems) will be more flexible. At the same time, SN P systems combine the strengths of The Turing universality as well as the computing power membrane computing and spiking neural networks, the of ANSN P systems has also been discussed in [24]. temporal aspect being one of the most attractive features. The extension of NSN P systems is an active research In 2020, Wu et al. [23] introduced numerical spiking neu- direction, with concrete results in recent years. In this ral P systems (NSN P systems), a new class of membrane context, Yin et al. [25] introduced in 2021 a new variant systems that combines the advantages of NP systems and of NSN P systems, called Novel numerical spiking neural SNP systems. Similar to SNP systems, NSN P systems P systems with a variable consumption strategy (NSNVC have a network architecture with an enhanced capability P systems). This new variant has led to improvements of representing complex topologies. Also, the informa- in the production functions, values of the variables in- tion is encoded using numerical variables, involved in volved having prescribed consumption rate without all production functions that determine how the variables being set to 0 after the execution. NSNVC P systems also within the neurons evolve over time. The repartition use polarization of the neurons in order to control the protocol involves that after a production function in a execution of a production function. neuron is executed, the resulted value is sent to all the Another variant that aims to improve the flexibility of variables present in adjacent neurons. Each production the system has been introduced by Xu et al. [26] in 2023 function can have a threshold and will be executed only by adding weights into NSN P systems. The resulted nu- when each value of the variables involved in it is not merical spiking neural P systems with weights (NSNW P smaller than the threshold. The formal definition of NSN systems) are still Turing universal and the their compu- P systems is the following: tational power was demonstrated using fewer neurons than NSN P systems. 𝛱 = (𝜎1 , . . . , 𝜎m , syn, in, out) (2) An interesting approach was proposed in [27]. While in the classic NSN P systems the production functions Table 1 are placed inside the neurons, in numerical spiking neu- Evolution of NP system 𝛱1 ral P systems with production functions on synapses Simulation step n x1 ,1 x1 ,2 x1 ,3 (NSNFS P systems), the production functions of each neu- ron are placed on synapses. In this way, a neuron will 0 0 0 0 contain only numerical variables. Potential applications 1 1 1 1 of NSNFS P systems are further investigated. 2 4 2 2 A more practical initiative was presented by Zhang et. 3 9 3 3 4 16 4 4 al in [28], where enzymatic numerical spiking neural membrane systems (ENSNP systems) were introduced. In the aforementioned paper, the practicality of ENSNP 1 systems is demonstrated by modelling ENSNP membrane controllers for robots implementing wall following. 2 4. Illustrative Examples 3 4.1. Numerical P System 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 𝛱1 evolution of the proposed NP system, we designed two entities of different NP systems variants, achieving simi- lar functioning. We chose to use a well-known NP system example, extracted from the article that introduced the 4.2. Numerical P system with Boolean concept of numerical P systems [5]. Let us introduce condition the system 𝛱1 , having three membranes, each of them As presented in Section 3, by integrating Boolean condi- containing one variable. Examining the structure of 𝛱1 tions into NP systems, the control over the computation presented in Figure 1, we can observe that the rules sug- process is improved, making selective rule application gest a sequential dependency where the variable values one of the main advantages of using Numerical P sys- of each membrane influences the next. All membranes tems with Boolean condition (BNP systems). Thus, we start with an initial value of zero for the variables defined will proceed with the modelling of a BNP system that inside them. calculates the perfect squares starting with 1, similar to Analysing the system rules and starting with the third 𝛱1 . Moreover, by introducing Boolean conditions, we membrane, we can note that variable x1 ,3 increases by 1 will store the even and the odd values of perfect squares at each simulation step. The value of x1 ,3 is also trans- in different membranes, using the same number of mem- mitted to x1 ,2 , as assigned in the repartition protocol. branes and variables as the initial NP system presented In membrane 2, the value 2x1 ,2 + 1 is transmitted to above. x1 ,1 . Also, it can be observed that the value of x1 ,1 is Looking at the resulted BNP system 𝛱2 which is illus- never consumed, hence its value increases continuously. trated in Figure 2, we can observe that the number Taking this observation into account, we can see that of membranes is the same as in 𝛱1 . Compartment 3 the value of the targeted variable x1 ,1 is constructed at contains three programs. The first program is increas- each simulation step using the following algebraic iden- ing the value of the variable x1 ,3 . The next two pro- tity: n 2 = (n − 1 )2 + 2 (n − 1 ) + 1 , n > 0 . Table 1 grams distribute the value needed to obtain the next contains the values of the variables after n = 4 simulation odd/even perfect square that is calculated inside x1 ,1 , steps. respectively x1 ,2 . Both the programs have Boolean con- As stated before, we chose this example of numerical ditions based on the value of x1 ,3 and at each simulation P system and next we will present two mappings into step only one of them will be executed. It is important different numerical P systems variants, emphasizing the to mention that the obtained BNP system works in the characteristics of each one. all-parallel mode, executing all the applicable rules at each step. The distribution of the accumulated values into separate variables for odd and even perfect squares is based on a formula that is similar to the one used for 𝛱1 : n 2 = (n − 2 )2 + 4 (n − 1 ), n > 0 . The formula Table 2 Evolution of BNP system 𝛱2 2 3 Simulation step n x1 ,1 x1 ,2 x1 ,3 0 1 0 0 1 1 0 1 2 1 4 2 3 9 4 3 4 9 16 4 1 4 3 Figure 3: Illustration of the NSN P system 𝛱3 Table 3 2 Evolution of NSN P system 𝛱3 Simulation step n x1 ,1 x1 ,2 x1 ,3 x1 ,4 1 0 0 0 0 0 1 1 1 1 1 2 4 2 2 2 3 9 3 3 3 4 16 4 4 4 Figure 2: Illustration of the BNP system 𝛱2 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 𝛱1 . 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 vari- ables x1 ,1 and x1 ,2 should be considered just for odd, Initially, the value of variable x1 ,3 is zero and the respectively even values of n, as they change at different production function from neuron 𝜎3 is executed. In this steps. way, neuron 𝜎3 transmits a value of 1 to neurons 𝜎2 and 𝜎4 . Looking at the production function from the neuron 𝜎3 , it can be observed that it does not replicate 4.3. Numerical Spiking Neural P System the original production function from the membrane 3 In order to obtain another membrane system with the of the NP system 𝛱1 , even if neuron 𝜎3 seems to be same functionality as 𝛱1 , we designed 𝛱3 , a NSN P the correspondent of membrane 3. This aspect can be system looking for the set of values taken by variable x1 ,1 , explained by the repartition protocol of the program which is never consumed. Moreover, we can observe from membrane 3, dividing the value between x1 ,3 and that 𝛱3 follows the same recursive formula as 𝛱1 , x1 ,1 x1 ,2 . According to the distribution stage of a NSN P receiving 2 (n − 1 ) + 1 at each simulation step n. The system, the computed production value is transmitted to result is graphically presented in Figure 3. each variable from all the neighboring neurons, so there The NSN P system illustrated in Figure 3 consists is no need to divide it. After executing the production of four neurons, each containing one variable with an function, the value of x1 ,3 is reset to zero. initial value of zero. Neurons 𝜎2 , 𝜎3 , 𝜎4 also contain Returning to the analysis of the system, we note that the production functions that will be executed during neuron 𝜎3 receives 1 from neuron 𝜎4 . Neuron 𝜎4 is used the evolution of the system. 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