On the Diophantine problem related to power circuits

Alexander Rybalov

arxiv: 2601.00835 · v4 · submitted 2025-12-26 · 🧮 math.LO · math.GR· math.NT· math.RA

On the Diophantine problem related to power circuits

Alexander Rybalov This is my paper

Pith reviewed 2026-05-16 19:49 UTC · model grok-4.3

classification 🧮 math.LO math.GRmath.NTmath.RA

keywords Diophantine problemundecidabilitypower circuitsnatural numbersadditionexponentiationBaumslag group

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The pith

The Diophantine problem over positive integers with addition and multiplication by powers of two is undecidable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that the Diophantine problem is undecidable over the structure of positive integers equipped with addition, the operation of multiplying by a power of two, the order relation, and the constant one. This structure models the power circuits introduced in 2012 to obtain a polynomial-time algorithm for the word problem in the Baumslag group. The result settles the decidability question posed at the time of that introduction. A sympathetic reader would care because it shows that equations built from these operations have no general algorithmic solution, even though the operations themselves are computationally simple in isolation.

Core claim

The Diophantine problem over the structure ⟨ℕ>0; +, x · 2^y, ≤, 1⟩ is undecidable.

What carries the argument

A reduction from a known undecidable Diophantine problem to the solvability of equations in this structure.

If this is right

No algorithm exists that can decide the solvability of arbitrary equations in this structure.
Power circuits cannot be used to obtain a decision procedure for the associated Diophantine problem.
The word-problem algorithm for the Baumslag group does not extend to a general solver for equations in the underlying arithmetic.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar undecidability may hold for structures that replace the base 2 with other fixed integers.
The result separates the polynomial-time word-problem tractability from Diophantine decidability in groups defined by power circuits.

Load-bearing premise

The reduction from the known undecidable problem correctly preserves the existence or non-existence of solutions.

What would settle it

An explicit algorithm that takes any equation built from addition, multiplication by a power of two, order, and the constant one and correctly outputs whether it has a solution in positive integers would falsify the undecidability claim.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims to prove undecidability for the Diophantine problem over the power circuit structure, directly addressing the 2012 open question.

read the letter

The paper settles an open question from Myasnikov, Ushakov, and Won in 2012 by proving that the Diophantine problem is undecidable over the structure with addition, the operation x times 2 to the y, the order, and the constant 1. That's the core result. It does a good job connecting this back to power circuits and their use in solving the word problem for the Baumslag group. The structure is precisely defined, and the claim is stated cleanly without extra claims. The main limitation is that we only have the abstract, so the actual proof steps for the reduction are not visible. If the construction shows how to represent configurations or computations using the given signature without assuming extra operations, it should be fine. But any gap in showing that the target set is existentially definable would undermine the undecidability transfer. The stress test note correctly flags that the encoding step needs careful checking to ensure no implicit use of full multiplication. This work is aimed at people working on algorithmic problems in group theory and computability in algebraic structures. A reader interested in Diophantine problems over exponential structures would get value from seeing how this particular signature leads to undecidability. I think it deserves a serious referee to check the encoding details and make sure the reduction is tight. The result, if correct, would be a useful negative answer in this area.

Referee Report

1 major / 0 minor

Summary. The paper proves that the Diophantine problem over the structure ⟨ℕ_{>0}; +, x·2^y, ≤, 1⟩ is undecidable. The proof proceeds by reduction from a known undecidable problem (such as the halting problem) to the solvability of systems of equations and inequalities in this signature.

Significance. If the reduction is valid, the result resolves the open question posed by Myasnikov, Ushakov, and Won (2012) on decidability for this structure tied to power circuits. It shows that the combination of addition and the exponential operation x·2^y is already sufficient to encode undecidable problems, with direct implications for algorithmic questions in group theory.

major comments (1)

The reduction step must contain an explicit lemma establishing that full multiplication (or the necessary auxiliary predicates for encoding configurations) is existentially definable from +, x·2^y, ≤, and 1. Without a self-contained verification of this definability, the many-one reduction from an undecidable set does not go through.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the need for greater explicitness in the reduction. We address the major comment below and will revise the paper accordingly.

read point-by-point responses

Referee: The reduction step must contain an explicit lemma establishing that full multiplication (or the necessary auxiliary predicates for encoding configurations) is existentially definable from +, x·2^y, ≤, and 1. Without a self-contained verification of this definability, the many-one reduction from an undecidable set does not go through.

Authors: We agree that an explicit lemma would strengthen the presentation and make the many-one reduction fully self-contained. In the revised version we will insert a dedicated lemma (immediately preceding the main undecidability theorem) that proves the existential definability of full multiplication (and the auxiliary predicates needed to encode Turing-machine configurations) from the signature {+, x·2^y, ≤, 1}. The lemma will contain a complete, self-contained verification that the required predicates are existentially definable, thereby confirming that the reduction from the halting problem proceeds via existential formulas over the given structure. revision: yes

Circularity Check

0 steps flagged

No circularity; standard reduction from known undecidable problem

full rationale

The paper establishes undecidability of the Diophantine problem over ⟨ℕ>0; +, x·2^y, ≤, 1⟩ by reduction from a known undecidable problem. This is a conventional many-one reduction in computability theory and does not rely on self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations whose content reduces to the present claim. No equations or steps in the provided abstract or description collapse the result to its own inputs by construction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, invented entities, or non-standard axioms are stated. The result presumably rests on standard computability reductions from known undecidable problems such as Hilbert's tenth problem.

axioms (1)

standard math Standard axioms of first-order arithmetic and computability theory
Any undecidability proof via reduction relies on the background theory of recursive functions and Diophantine sets.

pith-pipeline@v0.9.0 · 5412 in / 1083 out tokens · 22293 ms · 2026-05-16T19:49:58.442179+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat ≃ Nat (recovery theorem, embed_add) unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We reduce DP(ℕ>1) to the Diophantine problem over eN ... By Lemma 2.8, we can replace every equation of type (1) by an equivalent system of equations over eN

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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