A new algorithm provides a simple, memoryless way to enumerate trees generated by arbitrary context-free grammars using integer-based bijections.

CFG Tree Enumeration: A Simple Integer-Based Bijection Algorithm

TLDR: A new algorithm provides a simple, memoryless way to enumerate trees generated by arbitrary context-free grammars using integer-based bijections.

Abstract

I present a simple algorithm for enumerating the trees generated by a Context Free Grammar (CFG). The algorithm uses a pairing function to form a bijection between CFG derivations and natural numbers, so that trees can be uniquely decoded from counting. This provides a general way to number expressions in natural logical languages, and potentially can be extended to other combinatorial problems. I also show how this algorithm may be generalized to more general forms of derivation, including analogs of Lempel-Ziv coding on trees.

1. Introduction

While context-free grammars (CFGs) are important in computational linguistics and theoretical computer science, there is no simple, memoryless algorithm for enumerating the trees generated by an arbitrary CFG. One approach is to maintain a priority queue of partially expanded trees according to probability, and expand them through (e.g.) the leftmost unexpanded nonterminal in the tree. This, however, requires storing multiple trees in memory, which can become slow when enumerating many trees. Incremental polynomial time algorithms are also known [1] and related questions have been studied for lexicographic enumeration [2–4]. These algorithms are not particularly well-known, and the tools required to state and analyze them are complex.

In contrast, simple techniques exist for enumerating binary trees with a fixed grammar (e.g. S → SS | x). A variety of techniques and history is reviewed in Section 7.2.1.6 of [5], including permutation-based methods and gray codes [6–9]. These algorithms, however, do not obviously generalize to arbitrary CFGs.

The goal of the present paper is to present an variant of integer-based enumeration schemes that works for arbitrary CFGs. The algorithm is itself very basic—just a few lines—but relies on a abstraction here called an IntegerizedStack that may be useful in other combinatorial problems.

The proposed algorithm does not naturally enumerate in lexicographic order (though variants may exist) but it is efficient: its time complexity is linear in the number of nodes present in the next enumerated tree, and it does not require additional data structures or pre-computation of anything from the grammar.

Because the algorithm constructs a simple bijection between a the natural numbers N and trees, it also provides a convenient scheme for Gödel-numbering [10, 11], when the CFG is used to describe formulas.

We then extend this algorithm to a tree-based algorithms analogous to LZ compression.

2. Pairing Functions

A pairing function is a bijection between N×N and N. The simplest example is the Cantor pairing function:

π(a,b) = (a+b)(a+b+1)/2 + b

This allows us to encode pairs of natural numbers as single natural numbers, and decode them back. The inverse functions are:

w = ⌊((sqrt(8n+1)-1)/2)⌋

t = n - w(w+1)/2

u = w - t

Where ⌊•⌋ denotes the floor function.

3. Enumerating Trees

The core insight is to use pairing functions recursively to encode tree structures as integers. Each non-terminal in the CFG can be assigned a unique natural number. The algorithm maintains an "integerized stack" that represents the current state of tree expansion.

Starting from the start symbol, we push its encoding onto the stack. At each step, we pop the top element:

• If it's a terminal, we output it
• If it's a non-terminal, we pop it and push the encodings of its production's right-hand side in reverse order

By using pairing functions to encode the order of expansion, we can map each possible tree to a unique integer, and vice versa.

The algorithm is memoryless in the sense that it only needs to maintain the current stack state, not a priority queue of partial trees. The time complexity is linear in the size of the output tree.

4. LZ-Trees

We can extend this approach to create tree-based analogs of LZ compression. Instead of expanding non-terminals according to a fixed grammar, we can use previously seen subtrees as "productions" that can be referenced by integer codes.

This creates a variable-length code where common subtrees are assigned shorter codes, similar to how LZ algorithms assign shorter codes to frequently occurring substrings in text.

The bijection between trees and integers still holds, but now the "grammar" is adaptive and based on the data being compressed.

5. Conclusion

The presented algorithm provides a simple, efficient way to enumerate trees from arbitrary CFGs using integer-based bijections. Its simplicity and memoryless nature make it attractive for applications in logic, combinatorics, and compression.

The IntegerizedStack abstraction may be useful for other combinatorial enumeration problems beyond CFGs.

Future work could explore lexicographic variants, optimization for specific grammar classes, and further applications in logic and compression.

Author: (1) Steven T. Piantadosi

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