Geometric Characterization of Context-Free Intersections via the Inner Segment Dichotomy

TL;DR

This paper introduces the inner segment measure as a geometric criterion to characterize when the intersection of two context-free languages remains context-free, providing a dichotomy based on boundedness of inner segments.

Contribution

It establishes a geometric characterization of CFL intersections using the inner segment measure, resolving the crossing gap's limitations and providing a complete dichotomy for block-counting CFLs.

Findings

Bounded inner segments imply CFL via finite buffer construction.

Unbounded inner segments with pump-sensitive linkages imply non-CFL.

Complete characterization for block-counting CFLs based on joint well-nestedness.

Abstract

The intersection of two context-free languages is not generally context-free, but no geometric criterion has characterized when it remains so. The crossing gap (max(i'-i, j'-j) for two crossing push-pop arcs) is the natural candidate. We refute this: we exhibit CFLs whose intersection is CFL despite unbounded-gap crossings. The governing quantity is the inner segment measure: for crossing arcs inducing a decomposition w = P1 P2 P3 P4, it is max(|P2|,|P3|), the length of the longer inner segment between interleaved crossing endpoints. We prove a dichotomy for this measure: bounded inner segments imply context-freeness via a finite buffer construction; growing inner segments with pump-sensitive linkages imply non-context-freeness. The inner segment concept applies to all CFL intersections; the strictness of the resulting characterization depends on the language class. For block-counting CFLs (languages requiring equality among designated pairs of block lengths), the dichotomy is complete: the intersection is CFL if and only if the combined arcs are jointly well-nested. For general CFLs, the CFL direction is unconditional; the non-CFL direction requires pump-sensitive linkages whose necessity is the main open problem, reducing the general CFL intersection problem to a specific property of pump-sensitive decompositions.