Intersection and Union Hierarchies of Deterministic Context-Free Languages and Pumping Lemmas

TL;DR

The paper investigates finite intersections and finite unions of deterministic context-free languages ($\mathrm{DCFL}$) and demonstrates that the corresponding hierarchies $\mathrm{DCFL}(d)$ and $\mathrm{DCFL}[d]$ are infinite for all $d\ge2$. It introduces two practical pumping lemmas for $\mathrm{DCFL}[d]$, extending Yu's pumping lemma and enabling non-membership proofs beyond bounded languages. Using these tools, the authors prove that languages like $L_d^{(\le)}$ and $NPal^{\#}_d$ are not in $\mathrm{DCFL}(d-1)$, and deduce that $Pal$ is outside $\mathrm{DCFL}[\omega]$, while also relating $\mathrm{DCFL}$ hierarchies to deterministic limited automata ($d$-LDA). The results provide a versatile, machine- and grammar-based framework to separate hierarchical levels in deterministic CFLs, offering new nonmembership proofs and shedding light on the structure of $\mathrm{DCFL}$ families with potential implications for parsing and automata theory.

Abstract

We study the computational complexity of finite intersections and finite unions of deterministic context-free (dcf) languages. Earlier, Wotschke [J. Comput. System Sci. 16 (1978) 456--461] demonstrated that intersections of $(d+1)$ dcf languages are in general more powerful than intersections of $d$ dcf languages for any positive integer $d$ based on the separation result of the intersection hierarchy of Liu and Weiner [Math. Systems Theory 7 (1973) 185--192]. The argument of Liu and Weiner, however, works only on bounded languages of particular forms, and therefore Wotschke's result is not directly extendable to other non-bounded languages. To deal with a wide range of languages for the non-membership to the intersection hierarchy, we circumvent the specialization of their proof technics and devise a new and practical technical tool: two pumping lemmas for finite unions of dcf languages. Since the family of dcf languages is closed under complementation and also under intersection with regular languages, these pumping lemmas help us establish the non-membership relation of languages formed by finite intersections of target languages. We also concern ourselves with a relationship to deterministic limited automata of Hibbard [Inf. Control 11 (1967) 196--238] in this regard.

Figures (5)

• Figure 1: Containment of language families. All families in this illustration are known to be distinct.
• Figure 2: A history of the changes of stack heights.
• Figure 3: A turn partitioning of a stack history. Each $x^{(i)}$ is the left area of a turn and $y^{(i)}$ is the right area of the same turn.
• Figure 4: An example of two right canonical partial grammatical trees given in Lemma \ref{['left-part-theorem']}.
• Figure 5: Graphical descriptions of the cases (i) and (ii)

Theorems & Definitions (19)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Corollary 1.4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3