On the Diophantine problem related to power circuits
Alexander Rybalov
TL;DR
This paper studies the Diophantine problem in the power-circuit–related structure $\tilde{N} = \langle \mathbb{N}_{>0}; +, x \cdot 2^y, \leq, 1 \rangle$, motivated by MUW's power circuits and their use in the Baumslag group. It proves undecidability by reducing $\mathcal{DP}(\mathbb{N}_{>1})$ to $\mathcal{DP}(\tilde{N})$ and by establishing Diophantine definability of core operations including divisibility (via $2^m-1$), order, integer logarithm, squaring, and multiplication. A key step is showing that multiplication on $\mathbb{N}_{>1}$ is Diophantine definable in $\tilde{N}$, for example through $z = xy \Leftrightarrow 2z = (x+y)^2 - x^2 - y^2$. This reduction yields undecidability of $\mathcal{DP}(\tilde{N})$, and implies that $\tilde{N}$ is not automatic since automatic structures would have decidable Diophantine problems.
Abstract
Myasnikov, Ushakov and Won introduced power circuits in 2012 to construct a polynomial algorithm for the word problem in the Baumslag group, which has a non-elementary Dehn function. Power circuits are circuits supporting addition and operation $(x,y) = x \cdot 2^y$ for integer numbers. Also they posed a question about 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 undecidability of the Diophantine problem over this structure.