Computing rank of finite algebraic structures with limited nondeterminism

TL;DR

This paper improves the complexity bounds for computing the rank of finite algebraic structures like groups and quasigroups, showing they can be verified with highly restricted computational models using very short certificates.

Contribution

It establishes new upper bounds on the complexity of the rank problem for groups and quasigroups using limited nondeterminism and shallow circuits, advancing understanding of their computational hardness.

Findings

Rank for quasigroups decidable by circuits with $O(\log \log n)$ depth and $O(\log^2 n)$ nondeterministic bits.

For groups, the rank problem decidable by a space-efficient Turing machine with $O(\log n)$ space and $O(\log^2 n)$ nondeterminism.

Results extend to related algebraic structures and other rank variants.

Abstract

The rank of a finite algebraic structure with a single binary operation is the minimum number of elements needed to express every other element under the closure of the operation. In the case of groups, the previous best algorithm for computing rank used polylogarithmic space. We reduce the best upper bounds on the complexity of computing rank for groups and for quasigroups. This paper proves that the rank problem for these algebraic structures can be verified by highly restricted models of computation given only very short certificates of correctness. Specifically, we prove that the problem of deciding whether the rank of a finite quasigroup, given as a Cayley table, is smaller than a specified number is decidable by a circuit of depth $O(\log \log n)$ augmented with $O(\log^2 n)$ nondeterministic bits. Furthermore, if the quasigroup is a group, then the problem is also decidable by a Turing machine using $O(\log n)$ space and $O(\log^2 n)$ bits of nondeterminism with the ability to read the nondeterministic bits multiple times. Finally, we provide similar results for related problems on other algebraic structures and other kinds of rank. These new upper bounds are significant improvements, especially for groups. In general, the lens of limited nondeterminism provides an easy way to improve many simple algorithms, like the ones presented here, and we suspect it will be especially useful for other algebraic algorithms.