The Ancestor Width of Grammars and Languages

The ancestor width is a new measure for the structure of derivations of arbitrary grammars. For every production used in a derivation or equivalently for every leaf we consider the strings of ancestors. The ancestors deene a complexity measure with a local avour. Obviously , context-free grammars have ancestor width one. We show that languages with ancestor width two are context-free. However, every recursively enumerable set can be generated by a grammar with ancestor width three. For-free grammars the ancestor width is closely related to nondeterministic space complexity. Then languages such as fwcwcjw 2 fa; bg g, fa n b n c n jn 1g or f(a n c) n jn 1g can be generated with ancestor width four. Moreover, any language can be generated with ancestor width three, if padding is used and the language is represented with tails.