Elsevier

Theoretical Computer Science

Volume 893, 21 November 2021, Pages 60-71
Theoretical Computer Science

Control languages accepted by labeled spiking neural P systems with rules on synapses

https://doi.org/10.1016/j.tcs.2021.06.027Get rights and content

Abstract

Spiking neural P systems with rules on synapses (RSSNP systems) are a class of computation models which are inspired by the information processing and communication manner of neurons. In this work, we consider labeled RSSNP systems (lRSSNP systems), where each rule is assigned either with a label chosen from an alphabet Σ or with the empty label λ. A string over an alphabet is accepted by an lRSSNP system if the string is processed from left to right by the system in the sense that in a step only rules labeled with the processed symbol or with λ are used (not both), and the system reaches a final configuration in the moment when the whole string is processed. The set of all accepted strings is categorized as the restricted (computations are done with the application of the rules labeled with symbols from the given alphabet) control language and the unrestricted (rules labeled with λ are allowed in the computations) control language of the system. We study the language accepting power of lRSSNP systems using only standard spiking rules by comparing the family of control languages of lRSSNP systems with the families of languages in the Chomsky hierarchy. It is proved that restricted lRSSNP systems can accept no more than context-sensitive languages, and unrestricted lRSSNP systems can accept recursively enumerable languages.

Introduction

The computation models in membrane computing are called P systems, which are a class of parallel and distributed computation models inspired by cellular structure, internal states, and biological behaviors [24], [31]. P systems have been deeply and widely investigated both at the theoretical studies [2], [23], [30], [32] and application aspects [5], [7], [8], [18], [25], [43]. In the area of membrane computing, abstracting computing notions from the manner in which neurons process information and communicate with each other by sending spikes along synapses, spiking neural P systems (SNP systems) were introduced by Ionescu et al. [9]. Since then there has been an extensive research study on SNP systems and their variants inspired by neurobiology or motivated by computer science and mathematics. For instance, considering the biological phenomenon of the synapse creation and deletion, SNP systems with structural plasticity and SNP systems with neuron division and budding are introduced [6], [12], [21]; inspired by the biological characteristic of different kinds of neurons, SNP systems with astrocytes and axon P systems are respectively introduced [22], [44]; with the mathematical motivation of extending the way of neuron spiking, SNP systems with anti-spikes and SNP systems with extended rules are proposed [11], [20]; with computer science motivation of the use of rule strategies, SNP systems with I/O mode and asynchronous SNP systems with local synchronization are respectively proposed [1], [37].

A variant of SNP systems, SNP systems with rules on synapses (RSSNP systems), is introduced and investigated by Song et al. [36], where spikes are contained in neurons yet spiking and forgetting rules placed on synapses. In RSSNP systems, neurons represented by nodes could contain a number of spikes, and they are connected through synapses (shown in diagrams by directed edges) with spiking rules and forgetting rules. Each application of the enabled spiking and forgetting rules on synapses means that a piece of information (described by the multisets of the single object a) is processed and transmitted from upstream to downstream neurons. In this way, the system evolves in a parallel and nondeterministic way. Many achievements have resulted from RSSNP systems: RSSNP systems have been proved to be Turing universal working in the maximally parallel strategy [36], [38]; they have been verified to be Turing computable in other working modes, such as the maximum spikes consumption strategy [35], the asynchronous mode [39]; they also have been shown to be able to solve hard problems in feasible time [2].

The language generating/accepting power of a computation model is an essential issue in the fields of text mining, language processing, linguistic context analysis, and so forth [13]. The label controlled mechanism was introduced in P system to investigate the language generating/accepting power [16], [26], [28], [45], where each rule is associated with a label which is a symbol from a given alphabet or the empty label λ, and a computation of such a system is guided by a string over the alphabet of labels. Actually, great achievements have been made in investigating the computation power of P systems with label controlled mechanism [2], [15], [19], [27], [42].

To our best knowledge, there is no result on the language accepting power of RSSNP systems. Motivated by the study of control languages associated with classical SNP systems, we investigate the language accepting power of lRSSNP systems. For a given alphabet Σ of rule labels, a string over Σ is accepted by an lRSSNP system if the string is processed from left to right by the system in the sense that at a step only rules labeled with that processed symbol or with λ are used (not both), and the system reaches a final configuration in the moment when the whole string is processed. The set of all accepted strings is called the control language of the system. Here, we specify two cases of computations: (1) restricted mode: at each step, only rules with the same non-empty label can be used. In other words, rules with the empty label λ are not allowed to be used; (2) unrestricted mode: at each step, rules either with the same non-empty label or with the empty label λ (not both) can be used. We study the family of control languages of lRSSNP systems comparing with the families of languages in the Chomsky hierarchy [10]. Specifically, we prove the following results. In the restricted mode, lRSSNP systems of degree one can characterize the finite languages; lRSSNP systems can accept regular languages when there is no limitation on the number of neurons in the systems; control languages of spiking neural P systems are not beyond context-sensitive languages. In the unrestricted mode, recursively enumerable languages can be accepted by lRSSNP systems.

Section snippets

Labeled spiking neural P systems with rules on synapses

In this section, we introduce the labeled spiking neural P systems with rules on synapses (lRSSNP systems).

The necessary notions and notations is recalled first. An alphabet is a finite set of symbols. Let V be such an alphabet. A nonempty string w over V is a multiset of those symbols from V, and the empty string is denoted by λ. The set of all nonempty strings over V is denoted by V+, and the one including the empty string λ is denoted by V=V+{λ}. A set of strings in V denoted by a regular

Computation power of restricted lRSSNP systems

In this section, we investigate the computation power of restricted lRSSNP system by comparing the family of languages accepted by restricted lRSSNP systems (LlRSSNPm) with the families of finite (FIN), regular (REG), context-free (CF), and context-sensitive languages (CS). For the detailed notions and notations about languages, please refer to [10], [29].

Theorem 3.1

FIN=LlRSSNP1.

Proof

For a given alphabet Σ, let Lfin={w1,,wn} (wiΣ), where |wi|=ηi (1in) is the length of the string wi=bi1biηi over the

Computation power of unrestricted lRSSNP systems

In this section, we consider the computation power of unrestricted lRSSNP systems, where λ-step is allowed, that is, all the rules labeled λ are applied in a step. When a λ-step is applied, the supposed input pointer remains still processing no input symbol. The family of languages accepted by unrestricted RSSNP systems (LλlRSSNP) is proven to be the family of recursively enumerable languages.

Theorem 4.1

LλlRSSNP=RE.

Proof

Assume that LΣ is a recursive language, where Σ={b1,b2,,bp1} (p2). Define a function

Conclusion and discussion

Spiking neural P systems are a class of computation models inspired by nets of neurons. The investigation of the computation power of generating and accepting languages of spiking neural P systems is a basic topic [3], [4], [40], [41], [42]. In this work, we investigate a variant of spiking neuron P systems, spiking neural P systems with labeled rules on synapses (lRSSNP systems), where each rule on synapses is associated with a label by a labeling function; the label is either a symbol in a

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The work was supported by National Natural Science Foundation of China (62072201), China Postdoctoral Science Foundation (2020M672359), and the Fundamental Research Funds for the Central Universities (HUST: 2019kfyXMBZ056). The author K.G. Subramanian would like to thank Prof. L. Pan as well as Huazhong University of Science and Technology (HUST) for the support extended to him during a short visit to HUST in November 2019, when this work was initiated.

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