On the Diophantine problem related to power circuits

Abstract

Myasnikov, Ushakov, and Won introduced power circuits in 2012 to construct a polynomial-time algorithm for the word problem in the Baumslag group, which has a non-elementary Dehn function. Power circuits are computational structures that support addition and the operation $(x,y) \mapsto x \cdot 2^y$ on integers. They also posed the question of decidability of the Diophantine problem over the structure $\langle \mathbb{N}_{>0}; +, x \cdot 2^y, \leq, 1 \rangle$, which is closely related to power circuits. In this paper, we prove that the Diophantine problem over this structure is undecidable.