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On the Diophantine problem related to power circuits

This paper proves that the Diophantine problem over the structure $\langle \mathbb{N}_{>0}; +, x \cdot 2^y, \leq, 1 \rangle$ , which is closely related to power circuits introduced by Myasnikov, Ushakov, and Won, is undecidable.

Original authors: Alexander Rybalov

Published 2026-03-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alexander Rybalov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Puzzle That Can't Be Solved

Imagine you have a special set of Lego bricks. These bricks represent numbers. You are allowed to do two specific things with them:

Stack them: You can add two numbers together (like stacking two towers).
The "Power-Up" Move: You can take a number and multiply it by a power of two (like $x \cdot 2^y$ ). This is a very efficient way to make numbers grow huge, very quickly.

Mathematicians Myasnikov, Ushakov, and Won invented a system called Power Circuits using these rules. They used it to solve a very difficult puzzle (the "word problem" in a specific group of numbers) much faster than anyone thought possible.

However, they left a question on the table: "Is there a universal rulebook that can tell us, for any equation we write using these bricks, whether a solution exists?"

This is called the Diophantine Problem. It's like asking: "If I give you a recipe made of these specific ingredients, can you prove that a cake can be baked, or that it's impossible?"

The Answer: Alexander Rybalov, the author of this paper, says NO. There is no such rulebook. The problem is undecidable. It is impossible to create a computer program that can always tell you if a solution exists.

The Analogy: The "Magic Kitchen"

To understand why this is impossible, let's imagine a Magic Kitchen.

1. The Ingredients (The Structure)

In our kitchen, we have a special stove.

We can mix ingredients (Addition).
We have a "Super-Booster" button that takes a number and multiplies it by $2^y$ (The Power Circuit operation).
We have a ruler to compare sizes ( $\le$ ).
We have a standard unit block ($1$).

The question is: If I give you a complex recipe (an equation) using only these tools, can you always figure out if the dish is cookable?

2. The Trap: The Missing Ingredient (Multiplication)

The problem is that our kitchen doesn't have a "Multiply" button. We can add, and we can do the Super-Booster thing, but we can't just say "Multiply $x$ by $y$ ."

If you can't multiply, you can't build complex structures. It seems like a limitation that might make the problem easier to solve.

3. The Trick: Faking Multiplication

Rybalov's paper is a masterclass in cooking up a fake ingredient. He proves that even though the kitchen doesn't have a "Multiply" button, you can simulate multiplication using the tools you do have.

Here is how he does it, step-by-step:

Step A: The "Divisibility" Detective.
He shows that you can figure out if one number divides another (e.g., is 4 a factor of 12?) using the Super-Booster. It's like having a special metal detector that beeps if a number is a "multiple" of another.
- The Magic: He uses the fact that if $m$ divides $n$ , then $2^m - 1$ divides $2^n - 1$ . It's a hidden code in the numbers themselves.
Step B: The "Square" Trick.
Once you can detect divisibility, you can start building shapes. Rybalov shows you can define a "Square" ( $x^2$ ).
- The Metaphor: Imagine you have a pile of bricks. You want to know if they can form a perfect square. Rybalov proves that using the Super-Booster and divisibility, you can write a recipe that says, "This pile of bricks is a perfect square if..."
Step C: The "Multiplication" Illusion.
Here is the grand finale. In math, there is a famous trick: $(x+y)^2 = x^2 + 2xy + y^2$ .
If you can make squares, and you can add, you can rearrange this formula to isolate the multiplication ($xy$).
- The Result: Even though the kitchen has no "Multiply" button, Rybalov proves you can build a "Multiply" machine out of the other tools.

The Final Verdict: The Impossible Puzzle

Now that Rybalov has proven you can fake multiplication inside this Power Circuit kitchen, the game changes completely.

The Known Fact: Mathematicians have known for decades (since Hilbert's Tenth Problem) that if you have a kitchen with Addition and Multiplication, you can write recipes that are impossible to solve. There is no algorithm that can check every single recipe to see if it works.
The Connection: Since Rybalov proved that the Power Circuit kitchen can simulate Multiplication, it is effectively the same as the "Impossible Kitchen."
The Conclusion: Therefore, the Diophantine problem for Power Circuits is undecidable. There is no computer program that can look at a Power Circuit equation and say "Yes, it has a solution" or "No, it doesn't" for every case.

Why Does This Matter?

The paper also answers a side question: "Is this system 'Automatic'?"

In computer science, an "Automatic" structure is like a machine with a very simple, predictable brain. If a system is automatic, you can always solve its puzzles.

Rybalov's result is a "No."
Because the system is so powerful (it can fake multiplication), its brain is too complex to be "Automatic." It's too chaotic to be fully predictable.

Summary in One Sentence

Alexander Rybalov proved that even though the "Power Circuit" system looks simple and limited, it is actually powerful enough to simulate standard multiplication, which makes its mathematical puzzles impossible to solve with a universal rulebook.

1. Problem Statement

The paper addresses the Diophantine problem over the structure $\tilde{\mathbb{N}} = \langle \mathbb{N}_{>0}; +, x \cdot 2^y, \leq, 1 \rangle$ .

Context: Power circuits, introduced by Myasnikov, Ushakov, and Won (2012), are computational structures supporting addition and the operation $(x, y) \mapsto x \cdot 2^y$ . These were used to solve the word problem for the Baumslag group $G(1,2)$ in polynomial time.
The Open Question: Myasnikov, Ushakov, and Won posed the question of whether the Diophantine problem (determining if a system of equations has a solution) over $\tilde{\mathbb{N}}$ is decidable.
Related Structures:
- The classical Diophantine problem over $\langle \mathbb{N}; +, \cdot, 1 \rangle$ (Hilbert's Tenth Problem) is undecidable (Matiyasevich).
- The first-order theory of $\langle \mathbb{N}; +, x \mapsto 2^x, 1 \rangle$ is decidable (Semenov), implying its Diophantine problem is decidable.
- The structure $\tilde{\mathbb{N}}$ sits between these, featuring addition and the "power circuit" operation $x \cdot 2^y$ , but lacking general multiplication.

2. Methodology

The author proves undecidability by constructing a reduction from a known undecidable problem to the Diophantine problem over $\tilde{\mathbb{N}}$ . The proof proceeds through the following logical steps:

A. Reduction to Positive Integers ( $\mathbb{N}_{>1}$ )

The author first establishes that the Diophantine problem over $\mathbb{N}_{>k}$ (natural numbers greater than $k$ ) is undecidable for any $k$ .

Lemma 2.1: A Cook/Turing reduction is constructed from $DP(\mathbb{N})$ to $DP(\mathbb{N}_{>k})$ . This is achieved by enumerating all possible assignments of variables to values $\{0, \dots, k\}$ and checking if the remaining system has a solution in $\mathbb{N}_{>k}$ .
Corollary 2.2: Consequently, $DP(\mathbb{N}_{>1})$ is undecidable.

B. Diophantine Definability in $\tilde{\mathbb{N}}$

To reduce $DP(\mathbb{N}_{>1})$ to $DP(\tilde{\mathbb{N}})$ , the author demonstrates that standard arithmetic operations can be defined using only the operations available in $\tilde{\mathbb{N}}$ ( $+$ , $x \cdot 2^y$ , $\leq$ , $1$) via systems of equations.

Divisibility ( $x | y$ ):
- Utilizes the number-theoretic property: $m | n \iff 2^m - 1 | 2^n - 1$ .
- Since $2^x - 1$ can be expressed using the available operations, divisibility is shown to be Diophantine definable.
- Note: The author notes that while divisibility is definable, the Diophantine problem for $\langle \mathbb{N}; +, | \rangle$ is actually decidable (Beltjukov, Lipshitz), so this alone is insufficient for the main proof.
Strict Order ( $x < y$ ):
- Defined trivially as $\exists z (x + z = y)$ .
Integer Binary Logarithm ( $\lfloor \log_2 x \rfloor$ ):
- Defined via inequalities: $y = \lfloor \log_2 x \rfloor \iff (2^y \leq x) \land (x < 2^{y+1})$ .
Squaring ( $x^2$ ):
- This is the most complex step. The author defines a set $S(x) = \{kx(x+1) : k \in \mathbb{N}\}$ using divisibility conditions.
- A subset $S'(x)$ is constructed using the logarithm to bound the search space to $\{x^2, 2x^2+x, 3x^2+2x\}$ .
- A specific Diophantine condition $P(x, y)$ involving divisibility by $x+2$ and $x+3$ is used to isolate $x^2$ from the other candidates.
Multiplication ($xy$):
- Once squaring is definable, multiplication is derived using the algebraic identity:
  $z = xy \iff z + z + x^2 + y^2 = (x+y)^2$
- Since addition and squaring are definable, multiplication is definable.

3. Key Contributions and Results

Main Theorem (Theorem 2.9): The Diophantine problem over the structure $\tilde{\mathbb{N}} = \langle \mathbb{N}_{>0}; +, x \cdot 2^y, \leq, 1 \rangle$ is undecidable.
- Proof Strategy: The author reduces the undecidable problem $DP(\mathbb{N}_{>1})$ to $DP(\tilde{\mathbb{N}})$ by showing that multiplication (restricted to numbers $>1$ ) can be simulated within $\tilde{\mathbb{N}}$ . Any system of Diophantine equations over $\mathbb{N}_{>1}$ can be transformed into an equivalent system over $\tilde{\mathbb{N}}$ using the definable operations.
Resolution of Automaticity Question:
- The paper answers a secondary question from Myasnikov, Ushakov, and Won: "Is $\tilde{\mathbb{N}}$ automatic?"
- Corollary 2.10: $\tilde{\mathbb{N}}$ is not automatic.
- Reasoning: Every automatic structure has a decidable first-order theory (and thus a decidable Diophantine problem). Since the Diophantine problem for $\tilde{\mathbb{N}}$ is undecidable, the structure cannot be automatic.

4. Significance

Complexity of Power Circuits: The result highlights the computational power of power circuits. While they allow for efficient (polynomial-time) solutions to the word problem in specific groups (like the Baumslag group), the underlying arithmetic structure is rich enough to encode general Diophantine equations, making the general decision problem undecidable.
Boundary of Decidability: The paper clarifies the boundary between decidable and undecidable theories involving exponentiation and addition. It shows that adding the "power circuit" operation $x \cdot 2^y$ to addition creates a structure significantly more complex than $\langle \mathbb{N}; +, 2^x \rangle$ (which is decidable) but structurally similar to full arithmetic regarding Diophantine solvability.
Methodological Innovation: The construction of the squaring function $x^2$ using only addition, order, and the $x \cdot 2^y$ operation is a non-trivial technical achievement that bridges the gap between the limited operations of power circuits and full multiplication.

In summary, Rybalov proves that despite the utility of power circuits in group theory, the arithmetic structure they induce is sufficiently complex to render the Diophantine problem undecidable, thereby ruling out the possibility of the structure being automatic.