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Anti-Context-Free languages

Carles Cardó

TL;DR

The paper develops a dual framework for context-free languages using simple dependency trees and projective linearisations. It formalizes locality ($\mathsf{LC}$) and projectivity ($\mathsf{PR}$), proving that $\mathsf{CF} = \mathsf{LC}/\mathsf{PR}$, and introduces anti-context-free languages ($-\mathsf{CF}$) via anti-locality and anti-projectivity, with explicit dual pairs such as $L_{\mathrm{squa}}$ and $L_{\mathrm{copy}}$. It further shows that dual languages are semi-linear under Parikh mappings, preserving a key algebraic property of CF languages, and discusses connections to cross-serial dependencies in natural language. The work highlights linguistic relevance of duality, explores implications for parsing, and raises questions about extending the framework to broader dependency structures and potential formalisms capable of generating $-\mathsf{CF}$.

Abstract

Context-free languages can be characterized in several ways. This article studies projective linearisations of languages of simple dependency trees, i.e., dependency trees in which a node can govern at most one node with a given syntactic function. We prove that the projective linearisations of local languages of simple dependency trees coincide with the context-free languages. Simple dependency trees suggest alternative dual notions of locality and projectivity, which permits defining a dual language for each context-free language. We call this new class of languages anti-context-free. These languages are related to some linguistic constructions exhibiting the so-called cross-serial dependencies that were historically important for the development of computational linguistics. We propose that this duality could be a relevant linguistic phenomenon.

Anti-Context-Free languages

TL;DR

The paper develops a dual framework for context-free languages using simple dependency trees and projective linearisations. It formalizes locality () and projectivity (), proving that , and introduces anti-context-free languages () via anti-locality and anti-projectivity, with explicit dual pairs such as and . It further shows that dual languages are semi-linear under Parikh mappings, preserving a key algebraic property of CF languages, and discusses connections to cross-serial dependencies in natural language. The work highlights linguistic relevance of duality, explores implications for parsing, and raises questions about extending the framework to broader dependency structures and potential formalisms capable of generating .

Abstract

Context-free languages can be characterized in several ways. This article studies projective linearisations of languages of simple dependency trees, i.e., dependency trees in which a node can govern at most one node with a given syntactic function. We prove that the projective linearisations of local languages of simple dependency trees coincide with the context-free languages. Simple dependency trees suggest alternative dual notions of locality and projectivity, which permits defining a dual language for each context-free language. We call this new class of languages anti-context-free. These languages are related to some linguistic constructions exhibiting the so-called cross-serial dependencies that were historically important for the development of computational linguistics. We propose that this duality could be a relevant linguistic phenomenon.
Paper Structure (11 sections, 6 theorems, 34 equations, 7 figures)

This paper contains 11 sections, 6 theorems, 34 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Historical and linguistic motivation
  3. The dependency approach
  4. Projective word-orderings
  5. Structure of the article
  6. Notation and Definitions
  7. Locality
  8. Projectivity
  9. Anti-classes
  10. Anti-Context-Free Languages
  11. Discussion

Key Result

Proposition 2.5

Let $S$ be a tree, $p\geq 0$ an integer and $\varphi \in \zeta^*$:

Figures (7)

  • Figure 1: (a) Dependency tree and a projective word-ordering. (b) An example of subtree transformed into a substring.
  • Figure 2: Example of operators from Example \ref{['ExampleOperators']}.
  • Figure 4: The tree language $W_{\mathrm{mult}}$.
  • Figure 5: The dual tree language $\overline{W}_{\mathrm{squa}}$ from the Example \ref{['ExampleAntiloc']}.
  • Figure 6: Trees from Example \ref{['multiduality']}.
  • ...and 2 more figures

Theorems & Definitions (38)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Example 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 3.1
  • ...and 28 more