Integer-Based CFG Tree Counting: What You Need to Know

Written by treemap | Published 2026/03/22
Tech Story Tags: algorithms | integerizedstack-algorithm | tree-integer-bijection | grammar-based-compression | godel-numbering-trees | memoryless-tree-generation | context-free-grammar-logic | memoryless-enumeration

TLDRWrap up the guide to CFG tree enumeration. Learn how the IntegerizedStack creates simple tree-to-number bijections for logic and linguistics.via the TL;DR App

Abstract and 1. Introduction

2. Pairing functions

  1. Enumerating trees
  2. LZ-trees
  3. Conclusion and References

5 Conclusion

This work describes a simple algorithm that enumerates the trees generated by a CFG by forming a bijection between these trees and integers. The key abstraction, an IntegerizedStack, allowed us to encode arbitrary information into a single integer through the use of pairing functions.

References

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[12] G. Cantor, “Ein beitrag zur mannigfaltigkeitslehre,” Journal f¨ur die reine und angewandte Mathematik (Crelles Journal), vol. 1878, no. 84, pp. 242–258, 1878.

[13] M. P. Szudzik, “The rosenberg-strong pairing function,” arXiv preprint arXiv:1706.04129, 2017.

[14] M. A. Vsemirnov, “Two elementary proofs of the fueter–polya theorem on pairing polynomials,” Algebra i Analiz, vol. 13, no. 5, pp. 1–15, 2001.

[15] P. W. Adriaans, “A simple information theoretical proof of the fueter-p\’olya conjecture,” arXiv preprint arXiv:1809.09871, 2018.

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[18] M. Johnson, T. L. Griffiths, and S. Goldwater, “Adaptor grammars: A framework for specifying compositional nonparametric bayesian models,” in Advances in neural information processing systems, 2007, pp. 641–648.

[19] T. J. O’Donnell, Productivity and reuse in language: A theory of linguistic computation and storage. MIT Press, 2015.

[20] J. Ziv and A. Lempel, “A universal algorithm for sequential data compression,” IEEE Transactions on information theory, vol. 23, no. 3, pp. 337–343, 1977.

Author:

(1) Steven T. Piantadosi.


This paper is available on arxiv under CC BY 4.0 license.


Written by treemap | Visualizing data's branching connections, TreeMap cultivates insight and understanding, nurturing a greener future.
Published by HackerNoon on 2026/03/22