On homomorphic images of the Szilard languages of matrix insertion–deletion systems with matrices of size 2

TLDR: In this paper, it was shown that any regular language can be represented as a homomorphic image of the Szilard language of matrix InsDel systems of size (2; 2, 2, 0, 0; 1, 1, 0), and that recursively enumerable languages can be obtained as the homomorphism of the SLL of matrix SLL with matrices of size 2.

Abstract: A Szilard language is a well-known tool in formal language theory to express the derivation process in a grammar system or grammar. Matrix InsDel (insertion–deletion) system is a well-known variant of InsDel system, where the idea of matrix control is combined with InsDel systems. The size of a matrix InsDel system is represented by a septuple of integers $$(k; p, m, m^{'};$$ $$q, n, n^{'})$$ , where k represents maximum number of rules in a matrix, i.e., size of a matrix. The parameters p, m and $$m^{'}$$ represent the maximal length of the strings inserted, the maximal length of the left context and the maximal length of the right context of the insertion rules, respectively. The parameters $$q, n, n^{'}$$ represent the same for deletion rules. In this paper, we investigate the Szilard languages of matrix InsDel systems with matrices of size 2. We give examples of regular, context-free and context-sensitive languages which cannot be the Szilard language of any matrix InsDel system. We show that any regular language can be represented as a homomorphic image of Szilard language of matrix InsDel system of size (2; 2, 0, 0; 1, 0, 0). Any linear language, meta-linear language and rational (or regular) closure of linear language can be obtained as the homomorphic image of Szilard language of matrix InsDel systems of size (2; 1, 1, 0; 1, 1, 0), and (2; 1, 0, 1; 1, 0, 1). Moreover, any recursively enumerable language can be obtained as the homomorphic image of the Szilard language of matrix InsDel systems of size (2; 1, 1, 0; 1, 1, 1), (2; 1, 0, 1; 1, 1, 1), (2; 1, 1, 1; 1, 1, 0), (2; 1, 1, 1; 1, 0, 1), (2; 1, 1, 0; 2, 0, 0), (2; 1, 0, 1; 2, 0, 0), (2; 2, 0, 0; 1, 1, 0),  and (2; 2, 0, 0; 1, 0, 1).