The Dissecting Power of Regular Languages

A recent study on structural properties of regular and context-free languages has greatly promoted our basic understandings of the complex behaviors of those languages. We continue the study to examine how regular languages behave when they need to cut numerous infinite languages. A particular interest rests on a situation in which a regular language needs to “dissect” a given infinite language into two subsets of infinite size. Every context-free language is dissected by carefully chosen regular languages (or it is REG-dissectible). In a larger picture, we show that constantly-growing languages and semi-linear languages are REG-dissectible. Under certain natural conditions, complements and finite intersections of semi-linear languages also become REG-dissectible. Restricted to bounded languages, the intersections of finitely many context-free languages and, more surprisingly, the entire Boolean hierarchy over bounded context-free languages are REG-dissectible. As an immediate application of the REG-dissectibility, we show another structural property, in which an appropriate bounded context-free language can “separate with infinite margins” two given nested infinite bounded context-free languages.

💡 Research Summary

The paper introduces the notion of REG‑dissectibility, a property of an infinite language L that can be split into two infinite subsets by a regular language R; formally, both R ∩ L and ¬R ∩ L are infinite. This concept asks how far the limited expressive power of regular languages can go in “cutting” more complex infinite languages.

The authors first prove that every context‑free language (CFL) is REG‑dissectible. The proof exploits the parse‑tree structure of CFLs: by selecting a sufficiently large derivation rule that recurs, one can encode the repeated pattern into a regular language that captures infinitely many strings of the CFL while its complement also captures infinitely many strings.

Next, the paper extends the result to constantly‑growing languages and semi‑linear languages. Constantly‑growing languages have length functions that increase by a fixed constant; by partitioning the length spectrum into intervals and defining a regular language for each interval, the authors ensure both sides of the cut remain infinite. Semi‑linear languages are described by a finite union of linear sets in ℕ^k; using the parametric representation, a regular language is built that selects infinitely many solutions of each linear component while excluding infinitely many others, guaranteeing the dissectibility condition.

The authors then consider closures under complement and finite intersection for semi‑linear languages. Under a natural condition—namely that each language contains infinitely many distinct solutions to independent linear equations—the complement and any finite intersection also retain REG‑dissectibility. This demonstrates a robust structural stability of semi‑linear families under Boolean operations.

A major contribution concerns bounded languages, i.e., languages of the form a₁* a₂* … a_k*. The paper shows that any finite intersection of CFLs that are bounded is REG‑dissectible, and more strikingly, the entire Boolean hierarchy over bounded CFLs (obtained by repeatedly applying union, intersection, and complement) is REG‑dissectible. The construction proceeds layer by layer: for each Boolean level a regular language is crafted to capture an infinite slice of the bounded language while its complement captures another infinite slice, preserving the dissectibility property throughout the hierarchy.

Finally, the authors apply these findings to a separation‑with‑infinite‑margin problem. Given two nested infinite bounded CFLs, L₁ ⊂ L₂, they construct a bounded CFL S such that S is disjoint from L₁ ∪ L₂ yet contains infinitely many strings lying “between” L₁ and L₂. The construction uses the parametric descriptions of L₁ and L₂ to identify a region of the exponent space where the two languages differ, then defines a regular language that isolates this region and lifts it back to a bounded CFL. This demonstrates that regular‑level dissectibility can be leveraged to produce fine‑grained separators between complex language classes.

In summary, the paper systematically expands the known capabilities of regular languages in dissecting infinite languages. It establishes that all CFLs, constantly‑growing languages, and semi‑linear languages (along with their complements and finite intersections under mild conditions) are REG‑dissectible. Moreover, it proves that bounded CFLs and the full Boolean hierarchy built over them inherit this property, and it showcases a concrete application to infinite‑margin separation. These results deepen our understanding of the interplay between regular and higher‑level language families and open new avenues for using regular languages as analytical tools in formal language theory.