Decidability, complexity, and expressiveness of first-order logic over the subword ordering

Abstract

We consider first-order logic over the subword ordering on finite words, where each word is available as a constant. Our first result is that the Σ 1 theory is undecidable (already over two letters).We investigate the decidability border by considering fragments where all but a certain number of variables are alternation bounded, meaning that the variable must always be quantified over languages with a bounded number of letter alternations. We prove that when at most two variables are not alternation bounded, the Σ 1 fragment is decidable, and that it becomes undecidable when three variables are not alternation bounded. Regarding higher quantifier alternation depths, we prove that the Σ 2 fragment is undecidable already for one variable without alternation bound and that when all variables are alternation bounded, the entire first-order theory is decidable.

Matching Patterns with Variables

“…Such problems become even more involved when we consider equations with various types of constraints (e.g., length or regular). For instance, the decidability of general word equations with length constraints is a long standing open problem, but it is already an interesting open question for simpler cases (once again: regular or quadratic equations); see, e.g., [44,17,59], and the references therein. It seems interesting to us whether some of the ideas used in matching patterns can be transferred to solving (simplified) word equations, with or without constraints.…”

Languages Ordered by the Subword Order

“…In this paper, we concentrate on these positive results, i.e., we did not try to find new undecidable theories. It would, in particular, be nice to understand what properties of the regular language L determine the decidability of the Σ 1 -theory of the structure (L, ⊑ , (w) w∈L ) (it is undecidable for L = Σ * [5] and decidable for, e.g., L = {ab, baa} * ∪ bb{abb} * by our third result). Another open question concerns the complexity of our decidability results.…”

“…For this, it suffices to construct the matrix M effectively, i.e., to compute the infinite sum in Eq. (5). Using a pumping argument, one first shows the equivalence of the following two statements for all i, j ∈ {1, 2, .…”

“…The Σ 1 -theory of the structure (L, ⊑, (w) w∈L ) is decidable provided L is regular and almost every word from L contains a non-negligible number of occurrences of every letter (see below for a precise definition, Theorem 25). Note that, by [5,Theorem 3.3], this theory is undecidable if L = Σ * .…”

“…Note that the regular language L = {a, b} * is not covered by this result since it is unbounded. And, indeed, the Σ 2 -theory of ({a, b} * , ⊑) is undecidable [5]. On the positive side, we know that the Σ 1 -theory of ({a, b} * , ⊑) is decidable [15].…”