Exact versus unique nondeterministic automatic complexity

2026-08-23

AI Summary

研究解决了两种非确定性自动复杂度能否不同的问题，通过构造特定的六状态自动机验证了这一差异，并利用AI代理找到了分离词。

Abstract

The exact nondeterministic automatic complexity $A_{Ne}(x)$ of a word $x$ is the minimum number of states of a nondeterministic finite automaton that accepts $x$ and no other word of length $|x|$; the unique nondeterministic automatic complexity $A_N(x)$ additionally requires the accepting computation to be unique. Chen, Kjos-Hanssen, Koswara, Richter, and Stephan (FSTTCS 2025) asked whether the two measures can differ. We answer this affirmatively: the binary word $w=1101000000100$ satisfies $A_{Ne}(w)=6<7=A_N(w)$. The separation is witnessed by a six-state automaton whose sole accepted word of length $13$ is $w$, accepted along exactly three computation paths. The matching lower bounds were established by exhaustive search and by SAT solvers, with proofs of unsatisfiability certified in the DRAT format. An exhaustive census shows that $13$ is the least length of a separating word over any alphabet, binary or not, and that up to renaming of letters the separating words of length $13$ are precisely $365$ letter-mergings of the single seven-letter word $0123444445126$, all with $A_{Ne}=6<7=A_N$; exactly eight of them are binary. The separating word and the core of the verification were found by an artificial intelligence agent.

Problem

• nondeterministic automatic complexity
• exact versus unique
• finite automaton

Innovation

• nondeterministic automatic complexity
• AI agent
• SAT solvers
• DRAT format