New operations and regular expressions for two-dimensional languages over one-letter alphabet

Abstract

We consider the problem of defining regular expressions to characterize the class of recognizable picture languages in the case of a one-letter alphabet. We define a diagonal concatenation and its star and consider two different families, L ( D ) and L ( CRD ) , of languages denoted by regular expressions involving such operations plus classical operations. L ( D ) is characterized both in terms of rational relations and in terms of two-dimensional automata moving only right and down. L ( CRD ) is included in REC and contains languages defined by three-way automata while languages in L ( CRD ) necessarily satisfy some regularity conditions. Finally, we introduce new definitions of advanced stars expressing the necessity of conceptually different definitions for iteration.