The CFG Complexity of Singleton Sets

TL;DR

This paper investigates the minimal size of context-free grammars that generate a single string, showing upper bounds for all strings and establishing the existence of strings requiring large grammars, with both nonconstructive and constructive proofs.

Contribution

It provides tight bounds on the size of CFGs for singleton languages, revealing fundamental limits and constructions for minimal grammar complexity.

Findings

Upper bound of O(n/log n) rules for all singleton strings

Existence of strings requiring Omega(n/log n) rules in any CFG

Two proofs: one nonconstructive, one constructive

Abstract

Let G be a context-free grammar (CFG) in Chomsky normal form. We take the number of rules in G to be the size of G. We also assume all CFGs are in Chomsky normal form. We consider the question of, given a string w of length n, what is the smallest CFG such that L(G)={w}? We show the following: 1) For all w, |w|=n, there is a CFG of size with O(n/log n) rules, such that L(G)={w}. 2) There exists a string w, |w|=n, such that every CFG G with L(G)={w} is of size Omega(n/log n). We give two proofs of: one nonconstructive, the other constructive.