An incompressibility theorem for automatic complexity

TL;DR

This paper improves the lower bound on the automatic complexity of binary strings, showing it can be made arbitrarily close to 2 times the string length, refining previous bounds and applying to nondeterministic complexity.

Contribution

It establishes a nearly optimal incompressibility lower bound for automatic complexity, reducing the constant from 7 to 2+epsilon, and extends the result to nondeterministic automatic complexity.

Findings

Lower bound for automatic complexity is close to 2 times the string length.

The bound applies to both deterministic and nondeterministic automatic complexity.

The result is tight for nondeterministic automatic complexity, which is at most n/2+1.

Abstract

Shallit and Wang showed that the automatic complexity $A(x)$ satisfies $A(x)\ge n/13$ for almost all $x\in{\{\mathtt{0},\mathtt{1}\}}^n$. They also stated that Holger Petersen had informed them that the constant 13 can be reduced to 7. Here we show that it can be reduced to $2+\epsilon$ for any $\epsilon>0$. The result also applies to nondeterministic automatic complexity $A_N(x)$. In that setting the result is tight inasmuch as $A_N(x)\le n/2+1$ for all $x$.