The ancestor width of grammars and languages

AbstractThe 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 define a complexity measure with a local flavour. Obviously, context-free grammars have ancestor width one. We show that languages with ancestor width two are contextfree. However, every recursively enumerable set can be generated by a grammar with ancestor width three. For λ-free grammars the ancestor width is closely related to nondeterministic space complexity. Then languages such as {wcwc¦w ∈ {a,b}∗rcub;, {anbncn¦n⩾ 1} or {(anc)n¦n⩾ 1} 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