Short lists with short programs in short time

TL;DR

This paper demonstrates methods to efficiently compute small lists containing near-optimal programs for given strings, establishing bounds on list sizes and short program lengths for standard Turing machines.

Contribution

It introduces polynomial-time algorithms for generating small lists with short programs and proves optimality bounds on list sizes for such programs.

Findings

Polynomial-time algorithms produce lists with O(log|x|)-short programs

Existence of lists of size |x|^2 containing O(1)-short programs

Lower bounds showing list sizes must grow exponentially for some machines

Abstract

Given a machine $U$, a $c$-short program for $x$ is a string $p$ such that $U(p)=x$ and the length of $p$ is bounded by $c$ + (the length of a shortest program for $x$). We show that for any standard Turing machine, it is possible to compute in polynomial time on input $x$ a list of polynomial size guaranteed to contain a O$(\log |x|)$-short program for $x$. We also show that there exists a computable function that maps every $x$ to a list of size $|x|^2$ containing a O$(1)$-short program for $x$. This is essentially optimal because we prove that for each such function there is a $c$ and infinitely many $x$ for which the list has size at least $c|x|^2$. Finally we show that for some standard machines, computable functions generating lists with $0$-short programs, must have infinitely often list sizes proportional to $2^{|x|}$.