Improved bounds on the size of the smallest representation of relation algebra $32_{65}$

TL;DR

This paper provides new polynomial upper bounds and improved lower bounds on the size of the smallest representations of a specific relation algebra, using computational tools like SAT solvers to advance the understanding of its spectrum.

Contribution

It introduces the first polynomial upper bound for the spectrum of the algebra $A_{n}$ and improves the lower bounds, employing SAT solvers for representation analysis.

Findings

Minimum spectrum value is at most 2n^{6 + o(1)}

Lower bound on spectrum is improved to 2n^{2} + 4n + 1

1024 is in the spectrum of $A_2$, with no smaller number than 26 in the spectrum

Abstract

In this paper, we shed new light on the spectrum of the relation algebra we call $A_{n}$, which is obtained by splitting the non-flexible diversity atom of $6_{7}$ into $n$ symmetric atoms. Precisely, we show that the minimum value in $\text{Spec}(A_{n})$ is at most $2n^{6 + o(1)}$, which is the first polynomial bound and improves upon the previous bound due to Dodd \& Hirsch (\textit{J. Relational Methods in Computer Science} 2013). We also improve the lower bound to $2n^{2} + 4n + 1$, which is asymptotically double the trivial bound of $n^{2} + 2n + 3$. In the process, we obtain stronger results regarding $\text{Spec}(A_{2}) =\text{Spec}(32_{65})$. Namely, we show that $1024$ is in the spectrum, and no number smaller than 26 is in the spectrum. Our improved lower bounds were obtained by employing a SAT solver, which suggests that such tools may be more generally useful in obtaining representation results.