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Context-Free Trees

Jan Philipp Wächter

TL;DR

This paper investigates the subclass of bona fide context-free trees and shows that they have a finite-state description using multi-edge NFAs and that this specializes to certain partial DFAs in the case of deterministic graphs.

Abstract

Muller and Schupp introduced the concept of context-free graphs (originating from Cayley graphs of context-free groups). These graphs are always tree-like (i.e. quasi-isometric to a tree) and in this paper we investigate the subclass of bona fide context-free trees. We show that they have a finite-state description using multi-edge NFAs and that this specializes to certain partial DFAs in the case of deterministic graphs. We investigate this form of encoding algorithmically and show that the isomorphism problem for deterministic context-free trees is NL-complete in the rooted and the non-rooted case.

Context-Free Trees

TL;DR

This paper investigates the subclass of bona fide context-free trees and shows that they have a finite-state description using multi-edge NFAs and that this specializes to certain partial DFAs in the case of deterministic graphs.

Abstract

Muller and Schupp introduced the concept of context-free graphs (originating from Cayley graphs of context-free groups). These graphs are always tree-like (i.e. quasi-isometric to a tree) and in this paper we investigate the subclass of bona fide context-free trees. We show that they have a finite-state description using multi-edge NFAs and that this specializes to certain partial DFAs in the case of deterministic graphs. We investigate this form of encoding algorithmically and show that the isomorphism problem for deterministic context-free trees is NL-complete in the rooted and the non-rooted case.
Paper Structure (8 sections, 4 theorems, 3 equations, 5 figures, 1 table, 1 algorithm)

This paper contains 8 sections, 4 theorems, 3 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Context-Free Trees
  4. The General Case
  5. The Deterministic Case
  6. The Isomorphism Problem
  7. The Rooted Case
  8. The Non-Rooted Case

Key Result

Proposition 3.2

Let $\Gamma$ be an involutive $A$-tree and $u$ and $v$ be two of its nodes. Then, $\Gamma$ is regular with respect to $u$ as the root if and only if it is regular with $v$ as the root.

Figures (5)

  • Figure 1: Example of an mNFA with an associated tree.
  • Figure 2: An mNFA for \ref{['sfig:generatedTree']} with the gray orientation.
  • Figure 3: Another example of an mNFA with the associated tree.
  • Figure 4: Schematic drawing for the semantics of $U(p, q)$ and $U(p, q, b_0)$: The left tree must be isomorphic to the right tree by mapping $\varepsilon$ to $v$. For $U(p, q, b_0)$ we ignore the gray subtree for this isomorphism.
  • Figure 5: Drawing of $\Gamma(p)$ and $\Gamma(\check{q}, v)$. We have the edges and subtrees for all $b \in \mathop{\mathrm{out}}\nolimits q$

Theorems & Definitions (16)

  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • proof
  • proof
  • Remark
  • ...and 6 more