Parallel Contextual Array Insertion Deletion Grammar and (Context-Free : Context-Free) Matrix Grammar

Siromoney et al. introduced Siromoney matrix grammars (1973) which are of sequential-parallel type in the sense that first a horizontal string of nonterminals is derived sequentially by applying the horizontal production rules and then vertical productions are applied in parallel to get the intended two-dimensional picture. In 1999, Radhakrishnan et al. introduced and studied a variant of Siromoney matrix grammars called (X:Y)MG where \(X, Y \in \{Context-Free (CF), Regular(R)\}\). James et al. in 2018 introduced Parallel Contextual Array Insertion Deletion Grammar (PCAIDG) to generate two-dimensional array languages using insertion and deletion operations and parallel contextual mappings. In this paper, we prove that this family of languages generated by PCAIDGs properly includes the family (CF : CF) ML.