A Lower Bound on Unambiguous Context Free Grammars via Communication Complexity

TL;DR

This paper establishes exponential lower bounds on the size of unambiguous context-free grammars for finite languages, demonstrating they can be exponentially larger than general CFGs, with implications for automata representations.

Contribution

It proves the first exponential lower bounds for unambiguous CFGs representing finite languages, confirming a conjecture and linking grammar complexity to communication complexity.

Findings

Unambiguous CFGs can be exponentially larger than general CFGs for finite languages.

Finite languages may have exponentially smaller nondeterministic finite automaton representations.

The proof introduces a novel communication complexity lower bound approach.

Abstract

Motivated by recent connections to factorised databases, we analyse the efficiency of representations by context free grammars (CFGs). Concretely, we prove a recent conjecture by Kimelfeld, Martens, and Niewerth (ICDT 2025), that for finite languages representations by general CFGs can be doubly-exponentially smaller than those by unambiguous CFGs. To do so, we show the first exponential lower bounds for representation by unambiguous CFGs of a finite language that can efficiently be represented by CFGs. Our proof first reduces the problem to proving a lower bound in a non-standard model of communication complexity. Then, we argue similarly in spirit to a recent discrepancy argument to show the required communication complexity lower bound. Our result also implies that a finite language may admit an exponentially smaller representation as a nondeterministic finite automaton than as an unambiguous CFG.