Hausdorff Rank of Scattered Context-Free Linear Orders

A Context-Free Linear Ordering with an Undecidable First-Order Theory

“…In [4], it was shown that every ordinal less than ω ω ω is the order type of a deterministic context-free linear ordering and it was conjectured that a well-ordering is isomorphic to a context-free linear ordering if and only if its order type is less than ω ω ω . This conjecture was confirmed in [5] for deterministic context-free linear orderings, and in [18] for context-free linear orderings. Moreover, it was shown in [6] and [18] that the FC-rank of every scattered deterministic context-free linear ordering and in fact every scattered context-free linear ordering is less than ω ω .…”

“…This conjecture was confirmed in [5] for deterministic context-free linear orderings, and in [18] for context-free linear orderings. Moreover, it was shown in [6] and [18] that the FC-rank of every scattered deterministic context-free linear ordering and in fact every scattered context-free linear ordering is less than ω ω . Since the FC-rank of a well-ordering is less than ω ω exactly when its order type is less than ω ω ω , it follows in conjunction with results proved in [4] that a well-ordering is isomorphic to the lexicographic ordering of a context-free language or deterministic context-free language if and only if its order type is less than ω ω ω .…”

“…Moreover, an ordinal is the order type of a regular well-ordering if and only if it is strictly less than ω ω . The study of the lexicographic orderings of context-free languages (contextfree linear orderings) was initiated in [4] and further developed in [5,6,8,[16][17][18] and was extended to languages generated by deterministic higher order grammars in [7].…”

OPERATIONAL CHARACTERIZATION OF SCATTERED MCFLs

“…It is known, (see e.g. [15], proof of Proposition 1, paragraph 4) that in this case maxNodes(t) is the union of finitely many maximal paths. Clearly, the set {π 1 , .…”

On Context-Free Languages of Scattered Words

“…Regular words were later studied in [4,6,7,32,40] and more recently in [35]. Context-free words were introduced in [8] and their underlying linear orderings were investigated in [9,10,14,15,[22][23][24].…”