On the Order Type of Scattered Context-Free Orderings

Abstract

We show that if a context-free grammar generates a language whose lexicographic ordering is wellordered of type less than ω 2 , then its order type is effectively computable. NotationA linear ordering is a pair (Q, <), where Q is some set and the < is a transitive, antisymmetric and connex (that is, for each x, y ∈ Q exactly one of x < y, y < x or x = y holds) binary relation on Q. The pair *

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

“…Since w is the infimum of these words w i , we should have w ′′ ≤ ℓ w but this contradicts to w ′ < p w and w ′ < s w ′′ as these two imply w < s w ′′ . Now we recall from [7] that for any context-free language, we can compute a supremum or infimum of the language.…”

“…it is decidable whether (L, < ℓ ) is well-ordered or scattered [3] and the two algorithms are quite similar. In an earlier paper [7] we showed that it is undecidable for a scattered context-free ordering of rank 2 whether its order type is ω + (ω + ζ) × ω, even if it is given by a prefix grammar -so the complexity of the isomorphism problem is quite different when one makes the step from well-ordered languages to scattered ones.…”

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.…”