On the Diophantine problem related to power circuits

TL;DR

This paper proves that the Diophantine problem over a structure involving addition, multiplication by powers of two, and order is undecidable, building on the concept of power circuits.

Contribution

It establishes the undecidability of the Diophantine problem over a specific structure related to power circuits, advancing understanding in computational number theory.

Findings

Diophantine problem over the structure is undecidable

Power circuits support complex integer operations

Decidability status of related problems is clarified

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.