The Order Type of Scattered Context-Free Orderings of Rank One Is Computable

Abstract

A linear ordering is called context-free if it is the lexicographic ordering of some context-free language and is called scattered if it has no dense subordering. Each scattered ordering has an associated ordinal, called its rank. It is known that the isomorphism problem of scattered context-free orderings is undecidable, if one of them has a rank at least two. In this paper we show that it is decidable whether a context-free ordering has rank at most one, and if so, its order type is effectively computable.

Scattered one-counter languges have rank less than $ω^2$

“…We assume the reader has some background with formal language theory and linear orderings (e.g. with the textbook [10,14]), but we try to list the notations we use in the paper to settle the notation (which is the same as we used in [6] and [7]). We assume each alphabet (finite, nonempty set) comes with a fixed total ordering.…”

“…At the other hand, it is undecidable for a context-free grammar whether it generates a dense language, hence the isomorphism problem of context-free orderings in general is undecidable [4]. It is unknown whether the isomorphism problem of scattered context-free orderings is decidable -a partial result in this direction is that if the rank of such an ordering is at most one (that is, the order type is a finite sum of the terms ω, −ω and 1), then the order type is effectively computable from a context-free grammar generating the language [6,7]. Also, it is also decidable whether a context-free grammar generates a scattered language of rank at most one.…”

“…In this paper we prove the second conjecture of [11]: ω 2 is a strict upper bound for the rank of scattered one-counter languages. The contents of the paper contain new results only: instead of reproving the results of [6] and the subsequent, more general [7] (these papers already contain full proofs and examples as well to their respective results), we push the boundaries of the knowledge of scattered context-free orderings by applying some of the tools we developed in the earlier papers to the class of one-counter languages. It turns out that it is enough to study restricted one-counter languages to prove the conjecture, and for this, a crucial step is to reason about the cycles in a generalized sequential machine -so at the end, we can again use some graph-theoretic methods.…”