Notes about decidability of exponential equations

TL;DR

This paper explores the decidability of exponential equations within certain algebraic structures, demonstrating a constructed group where one-variable equations are decidable but two-variable equations are not.

Contribution

It introduces a finitely presented group that differentiates the decidability of exponential equations based on the number of variables, highlighting a nuanced boundary in algebraic decision problems.

Findings

Decidability for one-variable exponential equations in the constructed group.

Undecidability for two-variable exponential equations in the same group.

Shows a clear boundary in decidability related to the number of variables.

Abstract

We study relationship among versions of the Knapsack Problem where variables take values in Z and the number of them is fixed. In particular, we construct a finitely presented group where the problem of solvability of exponential equations with one variable is decidable but the corresponding problem for two variables is undecidable.