Simple matrix languages

Abstract

Simple matrix languages and right-linear simple matrix languages are defined as subfamilies of matrix languages by putting restrictions on the form and length (degree) of the rewriting rules associated with matrix grammars. For each n ⩾ 1, let L ( n ) [ ℛ ( n ) ] be the class of simple matrix languages [right-linear simple matrix languages] of degree n , and let L = ⋃ n ⩾ 1 L ( n ) [ ℛ = ⋃ n ⩾ 1 ℛ ( n ) ] . It is shown that L ( 1 ) [ ℛ ( 1 ) ] coincides with the class of context-free languages [regular sets] and that L is a proper subset of the family of languages accepted by deterministic linear bounded automata. It is proved that L ( n ) [ ℛ ( n ) ] forms a hierarchy of classes of languages in L [ ℛ ] . The closure properties and decision problems associated with L ( n ) , L , ℛ ( n ) , and ℛ are thoroughly investigated. Let L B [ ℛ B ] be the bounded languages in L [ ℛ ] . It is shown that L B = ℛ B and that most of the positive closure and decision results which are true for bounded context-free languages are carried over in L B . A characterization of L B as the smallest family of languages which contains the bounded context-free languages and which is closed under the operations of union and intersection is proved.