Breaking the $4^k$ Barrier for the $k$-Distinct Language

2026-07-28

AI Summary

This work addresses the minimization of nondeterministic finite automata (NFAs) recognizing the language $L_{k,n}$ of words of length at most $k$ over an $n$-symbol alphabet with no repeated symbols. Breaking through the long-standing $4^k$ state-size barrier, the authors introduce a novel construction that integrates Witt-design-based local NFA gadgets with hashing techniques within a product automaton framework. They further develop a gadget-amplification paradigm and a compose-and-compress path compression method, which collectively eliminate intermediate layers and incorporate single-symbol shortcut transitions. This approach yields an NFA of size $3.918^k n^{O(1)}$, significantly improving upon the previous best bound of $4^{k+o(k)} n^{O(1)}$.

Abstract

For integers $k\le n$, let $L_{k,n}$ be the set of words over $[n]$ of length at most $k$ in which no symbol is repeated. We present a nondeterministic finite automaton (NFA) of size $3.918^k n^{O(1)}$, improving on the $4^{k+o(k)}n^{O(1)}$ construction of Ben-Basat, Gabizon, and Zehavi. Our proof organizes several classical ingredients---product automata, hashing, and coefficient estimates---into a gadget-amplification framework: We take the product of many copies of a small local NFA gadget, whose language is a subset of $L_{r,c}$, and hash the $k$ input symbols to copies and local colors. The hash family guarantees that, for every repetition-free input, some hash sends at most $r$ symbols to each copy such that the resulting projection in every copy is accepted by the local gadget. Taking the nondeterministic union of the corresponding product NFAs yields a global NFA. Amplifying a $200$-state gadget for $L_{6,11}$ obtained from the small Witt design $S(4,5,11)$, this framework gives a $3.967^k n^{O(1)}$-size NFA. We then introduce the compose-and-compress technique, which deletes the expensive middle layers of these products and replaces paths across the deleted bands with sound one-symbol shortcut transitions. We apply it twice, once for enhancing the amplification framework and again for the local gadget, obtaining the stated result.

Problem

• k-distinct language
• nondeterministic finite automaton
• state complexity
• automata construction
• combinatorial design

Innovation

• nondeterministic finite automaton
• gadget-amplification
• compose-and-compress
• hashing
• product automata