AbstractInsertion and deletion are operations that occur commonly in DNA processing and RNA editing. Since biological macromolecules can be viewed as symbols, gene sequences can be represented as strings and structures can be interpreted as languages. This suggests that the bio-molecular structures that occur at different levels can be theoretically studied by formal languages. In the literature, there is no unique grammar formalism that captures various bio-molecular structures. To overcome this deficiency, in this paper, we introduce a simple grammar model called the matrix insertion–deletion system, and using it we model several bio-molecular structures that occur at the intramolecular, intermolecular and RNA secondary levels.

Lakshmanan K

Department of Analytics

School of Computer Science and Engineering

Vellore Campus

Mahendran A.

Fulltext

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International Journal of Applied Mathematics and Computer Science

A matrix ins-del system is described by a set of insertion-deletion rules presented in matrix form, which demands all rules of a matrix to be applied in the given order. These systems were introduced to model very simplistic fragments of sequential programs based on insertion and deletion as elementary operations as can be found in biocomputing. We are investigating such systems with limited resources as formalized in descriptional complexity. A traditional descriptional complexity measure of such a system is its ins-del size. Summing up the according numbers, we arrive at the sum-norm. We show that matrix ins-del systems with sum-norm 4 and (i) maximum length 3 with only one of insertion or deletion being performed under a one-sided context, or (ii) maximum length 2 with both insertion and deletion being performed under a one-sided context, can describe all recursively enumerable languages. We also show that if a matrix ins-del system of size s can describe the class of linear languages LIN, then without any additional resources, matrix ins-del systems of size s also describe the regular closure of LIN. © 2019, Springer Nature Switzerland AG.

SOFSEM 2019: Theory and Practice of Computer Science Lecture Notes in Computer Science

On Matrix Ins-Del Systems of Small Sum-Norm

Matrix insertion-deletion systems combine the idea of matrix control (a control mechanism well established in regulated rewriting) with that of insertion and deletion (as opposed to replacements). Given a matrix insertion-deletion system, the size of such a system is given by a septuple of integers (k; n, i′, i′ ′; m, j′, j′ ′). The first integer k denotes the maximum number of rules in (length of) any matrix. The next three parameters n, i′, i′ ′ denote the maximal length of the insertion string, the maximal length of the left context, and the maximal length of the right context of insertion rules, respectively. The last three parameters m, j′, j′ ′ are similarly understood for deletion rules. In this paper, we improve on and complement previous computational completeness results for such systems, showing that matrix insertion-deletion systems of size (1) (3; 1, 0, 1; 1, 0, 1), (3; 1, 0, 1; 1, 1, 0), (3; 1, 1, 1; 1, 0, 0) and (3; 1, 0, 0; 1, 1, 1) (2) (2; 1, 0, 1; 2, 0, 0), (2; 2, 0, 0; 1, 0, 1), (2; 1, 1, 1; 1, 1, 0) and (2; 1, 1, 0; 1, 1, 1), are computationally complete. Further, we also discuss linear and metalinear languages and we show how to simulate grammars characterizing them by matrix insertion-deletion systems of size (3; 1, 1, 0; 1, 0, 0), (3; 1, 0, 1; 1, 0, 0), (2; 2, 1, 0; 1, 0, 0) and (2; 2, 0, 1; 1, 0, 0). We also generate non-semilinear languages using matrices of length three with context-free insertion and deletion rules. © 2017, Springer Science+Business Media B.V., part of Springer Nature.

Natural Computing

Investigations on the power of matrix insertion-deletion systems with small sizes

A graph-controlled insertion–deletion (GCID) system is a regulated extension of an insertion–deletion system. Such a system has several components and each component has some insertion–deletion rules. The transition is performed by any applicable rule in the current component on a string and the resultant string is then moved to the target component specified in the rule. The language of the system is the set of all terminal strings collected in the final component. The parameters in the size (k;n,i′,i″;m,j′,j″) of a GCID system denote (from left to right) the maximum number of components, the maximal length of the insertion string, the maximal length of the left context for insertion, the maximal length of the right context for insertion; the last three parameters follow a similar representation with respect to deletion. In this paper, we discuss the computational completeness of the families of GCID systems of size (k;1,i′,i″;1,j′,j″) with k∈{3,5} and for (nearly) all values of i′,i″j′,j″∈{0,1}. All proofs are based on the simulation of type-0 grammars given in Special Geffert Normal Form (SGNF). The novelty in our proof presentation is that the context-free and the non-context-free rules of the given SGNF grammar are simulated by GCID systems of different sizes and finally we combine them by stitching and overlaying to characterize the recursive enumerable languages. This proof presentation greatly simplifies and unifies the proof of such characterization results. We also connect some of the obtained GCID simulations to the domain of insertion–deletion P systems. © 2017 Elsevier B.V.

Theoretical Computer Science

On the computational completeness of graph-controlled insertion–deletion systems with binary sizes

Matrix insertion-deletion systems combine the idea of matrix control (as established in regulated rewriting) with that of insertion and deletion (as opposed to replacements). We study families of multisets that can be described as Parikh images of languages generated by this type of systems, focusing on aspects of descriptional complexity. We show that the Parikh images of matrix insertion-deletion systems having length 2 matrices and context-free insertion/deletion contain only semilinear languages and when the matrices length increased to 3, they contain non-semilinear languages. We also characterize the hierarchy of family of languages that is formed with these systems having small sizes. We also introduce a new class, namely, monotone strict context-free matrix ins-del systems and analyze the results connecting with families of context-sensitive languages and Parikh images of regular and context-free matrix languages. © Springer International Publishing AG 2017.

Lecture Notes in Computer Science Theory and Applications of Models of Computation

Parikh Images of Matrix Ins-Del Systems

We study matrix insertion grammars (MIS) towards representation of recursively enumerable languages with small size. We show that pure MIS of size (3; 1, 2, 2) (i.e., having ternary matrices inserting one symbol in two symbol context) can characterize all recursively enumerable languages. This is achieved by either applying an inverse morphism and a weak coding, or a left (right) quotient with a regular language or an intersection with a regular language followed by a weak coding. The obtained results complete known results on insertiondeletion systems from DNA computing area. © Springer International Publishing AG 2017.

Unconventional Computation and Natural Computation Lecture Notes in Computer Science

Universal Matrix Insertion Grammars with Small Size

Modelling DNA and RNA secondary structures using matrix insertion–deletion systems

Insertion and deletion are considered to be the basic operations in Biology, more specifically in DNA processing and RNA editing. Based on these evolutionary transformations, a computing model has been formulated in formal language theory known as insertion-deletion systems. Since the biological macromolecules can be viewed as symbols, the gene sequences can be represented as strings. This suggests that the molecular representations can be theoretically analyzed if a biologically inspired computing model recognizes various bio-molecular structures like pseudoknot, hairpin, stem and loop, cloverleaf and dumbbell. In this paper, we introduce a simple grammar system that encompasses many bio-molecular structures including the above mentioned structures. This new grammar system is based on insertion-deletion and matrix grammar systems and is called Matrix insertion-deletion grammars. Finally, we discuss how the ambiguity levels defined for insertion-deletion grammar systems can be realized in bio-molecular structures, thus the ambiguity issues in gene sequences can be studied in terms of grammar systems. © 2011 Springer-Verlag.

Journal	Data powered by TypesetInternational Journal of Applied Mathematics and Computer Science
Publisher	Data powered by TypesetWalter de Gruyter GmbH
ISSN	1641876X
Open Access	Yes