Picture array generation using pure 2d context-free grammar rules

Abstract

Pure two-dimensional context-free grammar (P2DCFG) is a simple but effective non-isometric 2D grammar model to generate picture arrays. This 2D grammar uses only one kind of symbol as in a pure string grammar and rewrites in parallel all the symbols in a column or row by a set of context-free type rules. P2DCFG and a variant called (l/u)P2DCFG, which was recently introduced motivated by the “leftmost” rewriting mode in string context-free grammars, have been investigated for different properties. In this paper we introduce another variant of P2DCFG that corresponds to “rightmost” rewriting in string contextfree grammars. The resulting grammar is called (r/d)P2DCFG and rewrites in parallel all the symbols only in the rightmost column or the lowermost row of a picture array by a set of context-free type rules. Unlike the case of string context-free grammars, the picture language families of P2DCFG and the two variants (l/u)P2DCFG and (r/d)P2DCFG are mutually incomparable, although they are not disjoint. We also examine the effect of regulating the rewriting in a (r/d)P2DCFG by suitably adapting two well-known control mechanisms in string grammars, namely, control words and matrix control.