On the complexity of automatic complexity

TL;DR

This paper explores the computational complexity of automatic complexity for equivalence relations and strings, establishing NP-completeness results and analyzing the structure of sets of strings with maximal automatic complexity.

Contribution

It generalizes automatic complexity to equivalence relations, proves complexity classifications for related decision problems, and characterizes the complexity of sets of strings with maximal automatic complexity.

Findings

Determining if $A(E)=|E|$ is NP-complete.

Deciding if $A(E)=|E|+k$ is $ ext{BH}_2$-complete.

Sets of strings with maximal automatic complexity are co-context-free but not context-free.

Abstract

Generalizing the notion of automatic complexity of individual strings due to Shallit and Wang, we define the automatic complexity $A(E)$ of an equivalence relation $E$ on a finite set $S$ of strings. We prove that the problem of determining whether $A(E)$ equals the number $|E|$ of equivalence classes of $E$ is $\mathsf{NP}$-complete. The problem of determining whether $A(E) = |E| + k$ for a fixed $k\ge 1$ is complete for the second level of the Boolean hierarchy for $\mathsf{NP}$, i.e., $\mathsf{BH}_2$-complete. Let $L$ be the language consisting of all strings of maximal nondeterministic automatic complexity. We characterize the complexity of infinite subsets of $L$ by showing that they can be co-context-free but not context-free, i.e., $L$ is $\mathsf{CFL}$-immune, but not $\mathsf{coCFL}$-immune. We show that for each $\epsilon>0$, $L_\epsilon\not\in\mathsf{coCFL}$, where $L_\epsilon$ is the set of all strings whose deterministic automatic complexity $A(x)$ satisfies $A(x)\ge |x|^{1/2-\epsilon}$.