Networks of splicing processors: simulations between topologies

Journal of Membrane Computing https://doi.org/10.1007/s41965-023-00120-1 RESEARCH PAPER Networks of splicing processors: simulations between topologies José Angel Sanchez Martín1 · Victor Mitrana2 · Mihaela Păun3,4 Received: 30 November 2022 / Accepted: 3 March 2023 © The Author(s) 2023 Abstract Networks of splicing processors are one of the theoretical computational models that take inspiration from nature to efficiently solve problems that our current computational knowledge is not able to. One of the issues restricting/hindering is practical implementation is the arbitrariness of the underlying graph, since our computational systems usually conform to a predefined topology. We propose simulations of networks of splicing processors having arbitrary underlying graphs by networks whose underlying graphs are of a predefined topology: complete, star, and grid graphs. We show that all of these simulations are time efficient in the meaning that they preserve the time complexity of the original network: each computational step in that network is simulated by a fixed number of computational steps in the new topologic networks. Moreover, these simulations do not modify the order of magnitude of the network size. Keywords Splicing processor · Network of splicing processors · Underlying graph · Simulation. 1 Introduction The formal operation of splicing on strings has been introduced in [5] as an abstraction of the biological phenomenon of DNA recombination under the effect of restriction and ligases enzymes. The biological phenomenon is illustrated in Fig. 1. We give here a few informal explanations. Two DNA molecules (the blue and the red ones) are cut by a restriction enzyme (in this case the enzyme is EcoRI). This process yields fragments with Watson–Crick complementary * Victor Mitrana José Angel Sanchez Martín Mihaela Păun 1 Department of Software Engineering and Artificial Intelligence, Universidad Complutense de Madrid, Calle del Prof. José García Santesmases, 9, 28040 Madrid, Spain 2 Department of Information Systems, Universidad Politecnica de Madrid, Calle Alan Turing s/n, 28031 Madrid, Spain 3 Bioinformatics Department, National Institute for R &D for Biological Sciences, 296 Independenţei Bd., Bucharest 060031, Romania 4 Faculty of Administration and Business, University of Bucharest, Bucharest, Romania tails called “sticky ends”. These sticky ends may join again leading to the recombination of DNA. To fix the new combination, a DNA enzyme called ligase seals the gaps after the sticky ends are joint. We follow [15] with the formal definition of splicing as an operation on pairs of strings. First, we need to define what a splicing rule is: a quadruple of strings specifying the subsequences in the two strings where the strings are cut. Therefore, a splicing rule is intended to abstract the restriction enzymes and its subsequences indicate the sites where the enzymes cut. Different computational models based on the iteration of this operation may be defined. Thus, a generating splicing system initiates a computation starting from a given finite set of strings (axioms) and iteratively applying splicing rules, from a given finite set of such rules, producing eventually a language. This computational model was introduced in [5]; further on, the model and its variants have intensively been investigated. Splicing operation, as a formal operation on words and languages, has been vividly studied for more than two decades. There have been published a lot of papers as well as several books containing chapters devoted to this topic. We mention here just a few of them [6, 9, 16], containing extensive chapters about splicing, as well as [7, 8], containing chapters that intend to discuss various applications. There are two types of splicing systems: generating systems, which generates a language by iteratively applying 13 Vol.:(0123456789) J. A. S. Martín et al. Several variants of NSP have been considered so far, most of them being computationally complete, see, e.g., [2, 3, 10–12]. These networks have an ad hoc underlying graph structure. By different reasons like: possible implementations, uniformity, comparisons, etc., it would be useful to have networks with a fixed and well known topology as: complete graph, star, grid, etc. This is actually the aim of this work: to investigate the possibility of transforming a given NSP into an equivalent NSP with an underlying graph of such a predefined structure. We are interested not only in the construction of these networks but also in comparing the computational time and size of the constructed networks with those of the original ones. 2 Basic definitions Fig. 1  Splicing operation (Klug and Cummings 1997) splicing rules to the strings obtained starting from a finite set of strings, and accepting system, which starts out with just one initial string and a finite set of axioms and an iterative splicing as above is initiated. The computation halts when at least one string from a predefined set is obtained. The input string is accepted as soon as the system halts. The accepting splicing system has been introduced by Mitrana et al. in [13], while different variants have been studied in [1, 4, 14], etc. In [10] a highly parallel and distributed computational model based on the splicing operation was introduced: network of splicing processors (NSP). This model consists in an undirected graph whose nodes host a splicing processor. A splicing processor consists in a finite set of splicing rules, a finite set of strings (axioms) and four sets of symbols, such that two of them define the input filter while the other two define the output filter. A computation in a network of splicing processors (NEP, for short) is a sequence of splicing and communication steps which alternate with each other. In a splicing step, each processor applies, in parallel, the splicing rules it contains to all the strings existing at that moment in the processor. Note that we assume that each string appearing in a processor at some moment, appears actually in an unlimited number of identical copies such that different copies may be rewritten by different splicing rules. In a communication step, all the strings existing in the network nodes are simultaneously are expelled from their nodes, provided that they can pass the output filters of the nodes. In the same communication step, arbitrary large number of copies of each string expelled from one node (sender) enter all the nodes (receivers) connected to the sender, provided that the string can pass the input filters of the receivers. The computation halts as soon as a predefined node, called Halt, contains at least a string. 13 In this section we introduce the main concepts and notations that will be used in the sequel. For those notions not defined here we refer to [17]. An alphabet is a finite and nonempty set of symbols. The cardinality of a finite set A is written card(A). Any finite seque (...truncated)