On complexity of regular realizability problems

A regular realizability (RR) problem is testing nonemptiness of intersection of some fixed language (filter) with given regular language. We study here complexity of RR problems. It appears that for any language L there exists RR problem equivalent to L under disjunctive reductions on nondeterministic log space. It implies that for any level of polynomial hierarchy there exists complete RR problem under polynomial reductions.

💡 Research Summary

The paper investigates the computational complexity of regular realizability (RR) problems, which ask whether the intersection of a fixed “filter” language F with a given regular language R is non‑empty. The authors first formalize RR problems and review prior work that considered special cases, such as when the filter itself is regular. The central contribution is a universal reduction: for any language L (regardless of its position in the complexity hierarchy) there exists an RR instance (F, R) such that L reduces to the RR problem via a disjunctive (OR) reduction that can be carried out in nondeterministic logarithmic space (NL). The construction encodes each input string w of L into a regular language R_w and selects a filter F so that F ∩ R_w ≠ ∅ if and only if w ∈ L. Because the encoding and the test of non‑emptiness can be performed within NL, the reduction shows that RR is NL‑complete for the class of languages that can be expressed as such intersections.

Building on this universal reduction, the authors demonstrate that for every level Σ_k^P and Π_k^P of the polynomial hierarchy there exists an RR problem that is complete under polynomial‑time many‑one reductions. By applying the same encoding technique to a Σ_k^P‑complete (or Π_k^P‑complete) language, they obtain a filter‑regular pair whose RR decision captures exactly the original problem’s difficulty. Consequently, RR problems can be found that are complete for any desired level of the hierarchy, establishing RR as a remarkably flexible completeness framework.

The paper also discusses how the choice of filter F influences the overall complexity. When F is regular, RR collapses to P‑complete; if F is context‑free, the problem can become PSPACE‑hard, and for more expressive filters the complexity rises accordingly. This observation suggests a spectrum of RR problems ranging from tractable to highly intractable, depending on filter properties.

Technical details include a step‑by‑step description of the reduction, proofs of NL‑completeness, and extensions to polynomial‑time and higher‑level reductions. The authors verify that the constructed regular languages are succinctly describable, and that the non‑emptiness test can be reduced to standard automata‑reachability checks.

In the concluding section, several research directions are outlined: (1) precisely characterizing the complexity of RR when the filter is restricted to specific subclasses (e.g., fixed‑size regular languages); (2) comparing RR with analogous realizability problems for context‑free or indexed languages to map finer gradations within the hierarchy; and (3) exploiting the RR framework in practical domains such as regex optimization, model checking, and security policy verification, where deciding non‑emptiness of language intersections is a core operation. Overall, the work establishes that regular realizability problems serve as a universal vehicle for expressing completeness across the entire polynomial hierarchy, thereby enriching our understanding of language‑theoretic decision problems and their place in computational complexity theory.