Pumping for ordinal-automatic structures
Martin Huschenbett, Alexander Kartzow, Philipp Schlicht, Philipp Schlicht · Computability · 2017
An α-automaton (for α some ordinal) is an automaton similar to a Muller automaton that processes words of length α. A structure is called α-automatic if it can be presented by α-automata (completely analogous to the notion of automatic structures which can be presented by the well-known finite automata). We call a structure ordinal-automatic if it is α-automatic for some ordinal α. We continue the study of ordinal-automatic structures initiated by Schlicht and Stephan as well as by Finkel and Todorčević. We develop a pumping lemma for α-automata (processing finite α-words, i.e., words of length α that have one fixed letter at all but finitely many positions). Using this pumping, we provide counterparts for the class of ordinal-automatic structures to several results on automatic structures: ∙ Every finite word α-automatic structure has an injective finite word α-automatic presentation for all [Formula: see text]. This bound is sharp. ∙ We classify completely the finite word [Formula: see text]-automatic Boolean algebras. Moreover, we show that the countable atomless Boolean algebra does not have an injective finite-word ordinal-automatic presentation. ∙ We separate the class of finite-word ordinal-automatic structures from that of tree-automatic structures by showing that free term algebras with at least 2 generators (and one binary function) are not ordinal-automatic while the free term algebra with countable infinitely many generators is known to be tree-automatic. ∙ For every ordinal [Formula: see text], we provide a sharp bound on the height of the finite word α-automatic well-founded order forests. ∙ For every ordinal [Formula: see text], we provide a structure [Formula: see text] that is complete for the class of finite-word α-automatic structures with respect to first-order interpretations. ∙ As a byproduct, we also lift Schlicht and Stephans’s characterisation of the injectively finite-word α-automatic ordinals to the noninjective setting.