Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages

The purpose of this paper is to provide efficient algorithms that decide membership for classes of several Boolean hierarchies for which efficiency (or even decidability) were previously not known. We develop new forbidden-chain characterizations for the single levels of these hierarchies and obtain the following results: - The classes of the Boolean hierarchy over level $\Sigma_1$ of the dot-depth hierarchy are decidable in $NL$ (previously only the decidability was known). The same remains true if predicates mod $d$ for fixed $d$ are allowed. - If modular predicates for arbitrary $d$ are allowed, then the classes of the Boolean hierarchy over level $\Sigma_1$ are decidable. - For the restricted case of a two-letter alphabet, the classes of the Boolean hierarchy over level $\Sigma_2$ of the Straubing-Th'erien hierarchy are decidable in $NL$. This is the first decidability result for this hierarchy. - The membership problems for all mentioned Boolean-hierarchy classes are logspace many-one hard for $NL$. - The membership problems for quasi-aperiodic languages and for $d$-quasi-aperiodic languages are logspace many-one complete for $PSPACE$.

💡 Research Summary

The paper “Efficient Algorithms for Membership in Boolean Hierarchies of Regular Languages” addresses a long‑standing gap in the theory of regular languages: the lack of efficient (log‑space) decision procedures for membership in several Boolean hierarchies (BH) built over well‑studied logical classes. While decidability was known for the Boolean hierarchy over Σ₁ (the first level of the dot‑depth hierarchy) and for some related classes, no efficient algorithms were available, and for the Boolean hierarchy over Σ₂ (the third level of the Straubing‑Thérien hierarchy) nothing was known at all.

The authors introduce a novel structural tool called a “forbidden chain.” Unlike traditional forbidden‑pattern characterizations that forbid a particular subgraph in a DFA’s transition graph, a forbidden chain is an alternating sequence of words ordered by a specially defined partial order ≤⁽ᵈ⁾ₖ. For a language L to belong to the n‑th level D(n) of a Boolean hierarchy over a base class D, L must not contain a 1‑alternating chain of length n with respect to ≤⁽ᵈ⁾ₖ. The order ≤⁽ᵈ⁾ₖ refines the classic k‑embedding order by additionally requiring that positions respect a modulo‑d congruence. This (k,d)-embedding captures the effect of modular predicates (P⁽ᵈ⁾_j) that appear in the signatures τ_d, which extend the basic signature σ (used for dot‑depth) with modular tests.

Using these characterizations, the authors design nondeterministic logarithmic‑space (NL) algorithms for the following classes:

• Σ_σ₁(n) – the n‑th Boolean level over Σ₁ of the dot‑depth hierarchy.
• Σ_τ_d₁(n) – the n‑th Boolean level over Σ₁ of the dot‑depth hierarchy enriched with modular predicates of a fixed modulus d.
• Σ₂(n) for a two‑letter alphabet – the n‑th Boolean level over Σ₂ of the Straubing‑Thérien hierarchy, a result that had never been obtained before.

The algorithm works as follows. Given a DFA M, it first computes in log space for each state the minimal and maximal positions of occurrences of each letter, as well as the length‑k prefixes and suffixes needed for the embedding. Then, nondeterministically guesses a starting word (represented implicitly by a state and a position) and iteratively extends a candidate chain by checking whether a (k,d)-embedding exists from the current word to a longer word while preserving the alternating membership condition (i.e., the next word must lie outside L if the current one lies inside, and vice‑versa). If a chain of length n is found, the algorithm rejects (the language is not in D(n)); otherwise, after exhausting all possibilities, it accepts, confirming membership. All checks involve only constant‑size counters and local transitions, guaranteeing O(log |M|) space usage.

Complexity results are tight. The authors prove NL‑hardness by reducing the classic NL‑complete graph reachability problem to the existence of a forbidden chain, establishing NL‑completeness for all the Boolean‑hierarchy membership problems considered (except Σ_τ₁(n), whose exact complexity remains open).

In addition, the paper studies quasi‑aperiodic and d‑quasi‑aperiodic languages, which correspond to FO