通过内段二元性几何地刻画上下文无关语言的交集
Geometric Characterization of Context-Free Intersections via the Inner Segment Dichotomy
AI总结：
通过内段二元性几何方法，研究上下文无关语言交集的上下文无关性判定问题。
摘要：
两个上下文无关语言的交集通常不是上下文无关语言，但尚未有几何标准能确定其何时保持上下文无关。交叉间隙（两个交叉推弹弧的max(i'-i, j'-j)）是自然候选。我们反驳这一点：我们展示了某些上下文无关语言的交集仍为上下文无关语言，尽管存在无界间隙交叉。主导量是内段度量：对于交叉弧诱导的分解w = P1 P2 P3 P4，它是max(|P2|,|P3|)，即交错交叉端点之间较长的内段长度。我们证明了该度量的二元性：有界内段意味着通过有限缓冲构造实现上下文无关性；增长的内段与泵敏感链接意味着非上下文无关性。内段概念适用于所有上下文无关语言的交集；所得到的严格性取决于语言类。对于块计数上下文无关语言（要求指定的块长度对之间相等），二元性是完整的：交集是上下文无关语言当且仅当联合弧是联合良好嵌套的。对于一般上下文无关语言，上下文无关方向是无条件的；非上下文无关方向需要泵敏感链接，其必要性是主要开放问题，将一般上下文无关语言交集问题减少到特定泵敏感分解的性质。
英文摘要：
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.
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