Collatz Conjecture Simulator

A program to efficiently determine the total stopping time of the Collatz sequence of any 128-bit integer starting value, and output all starting values $n$ whose total stopping time is the greatest of all integers in the interval $[1, n]$.

Mathematical Notation

For the purposes of this document, the following mathematical symbols will represent the corresponding concepts.

• $\mathbb{Z}$ represents the set of all integers.
• $\mathbb{Z}^+$ represents the set of all positive integers.
• $\mathbb{Z}^{0+}$ represents the set of all non-negative integers.
• $f(n)$ represents the Collatz function applied to $n$.
• $f^k(n)$ represents the Collatz function applied recursively $k$ times to $n$.
• $s(n)$ represents the total stopping time of the starting value $n$.

The Collatz Conjecture

The Collatz Conjecture is a famous unsolved mathematical problem. It regards the Collatz function, a mathematical function which takes a positive integer input $n$ and gives a positive integer output $f(n)$. If the input is even, the function returns half the input. If the input is odd, the function returns triple the input, plus one.

$$f(n) = \begin{cases} \frac{1}{2} n & \text{if } n \equiv 0 \pmod 2 \\\ 3n + 1 & \text{if } n \equiv 1 \pmod 2 \end{cases}$$

The Collatz function can be applied recursively, meaning given an initial input $n$ and resultant output $f(n)$, this first output can be used as an input, resulting in a second output $f(f(n))$. This second output can again be used as an input, resulting in a third output $f^3(n)$. And so on.

By applying the Collatz function recursively, the sequence of successive inputs and outputs will form a Collatz sequence. If a Collatz sequence includes the value $1$, then the number of elements in the sequence from the starting value to the first instance of the value $1$ is the total stopping time. That is, given a starting value $n$ and total stopping time $k$, $f^k(n) = 1$.

The Collatz Conjecture states that for all positive integer starting values $n$, finite recursive application of the Collatz function will eventually result in the value $1$. Or, expressed with mathematical logic:

$$\forall n \in \mathbb{Z}^+, \exists k \in \mathbb{Z}^{0+} : f^k(n) = 1$$

The Simulation

Collatz Conjecture Simulator aims to find the Collatz sequences with the greatest total stopping times. That is, positive integer values $n$ such that of the set of integers in the interval $[1, n]$, the starting value with the greatest total stopping time is $n$.

$$n \in \mathbb{Z}^+ : \forall x \in \{ p \in \mathbb{Z}^+ : p < n \}, s(n) > s(x)$$

Total stopping times are calculated by iterating through Collatz sequences and counting each step (application of the Collatz function) until a value of $1$ is found. Due to every iteration through a Collatz sequence being computationally independent of any other, it is possible to calculate the total stopping times of multiple starting values simultaneously. As such, the program uses the GPU to iterate through multiple Collatz sequences in parallel. The GPU is accessed via the Vulkan API, and uses compute shaders to perform the iterations. Collatz Conjecture Simulator is primarily written in C, however the shaders are written in GLSL and compiled to SPIR-V.

Program Requirements

The general environment and system requirements that must be met for Collatz Conjecture Simulator to build and run correctly. The full requirements of the GPU are given in device_requirements.md.

• CMake 3.23
• pthreads
• glslang
• SPIR-V Tools
  • spirv-link
  • spirv-opt
  • spirv-dis
• C17
  • _Atomic (Optional C11 feature)
  • __int128 (GNU C extension)
• Vulkan 1.1
  • synchronization2
  • timelineSemaphore

Building and Running

Collatz Conjecture Simulator is built via CMake. Comprehensive documentation regarding usage of CMake can be found here. To generate the build system, navigate the terminal to the project directory and execute the following command.

cmake -S . -B build

Several options can be optionally specified to customise the build system by appending -D OPTION=CONFIG to the above command.

• CMAKE_BUILD_TYPE specifies the build variant and can be set to Debug, Release, MinSizeRel, or RelWithDebInfo. If not set, it defaults to Debug.
• EXCESS_WARNINGS specifies whether to compile the program with a potentially excessive amount of warnings, and defaults to OFF.
• STATIC_ANALYSIS specifies whether to statically analyse the program during compilation if compiling with GCC, and defaults to OFF.
• DEBUG_SHADERS specifies whether to include debug information in generated SPIR-V, and defaults to OFF.
• OPTIMISE_SHADERS specifies whether to optimise generated SPIR-V using spirv-opt, and defaults to ON.
• DISASSEMBLE_SHADERS specifies whether to disassemble generated SPIR-V using spirv-dis, and defaults to OFF.

Once the above command has finished, a build directory will have been created containing the build system. To now build Collatz Conjecture Simulator, execute the following command.

cmake --build build

By default, only the executable will be built. To instead build the SPIR-V, add --target Spirv. To build both, also add --target CollatzSim. To specify the build configuration, add --config CONFIG (only applies for multi-config generators).

The above command will create a bin directory containing the SPIR-V and executable. If built in debug, the executable will be named CollatzSimDebug. Otherwise, it will be named CollatzSim. The executable must be run from within the bin directory, else it will be unable to locate the generated SPIR-V.

Inout-buffers

To facilitate this use of the GPU, inout-buffers are used. Inout-buffers are ranges of GPU memory within VkBuffer objects and consist of an in-buffer and out-buffer. In-buffers are shader storage buffer objects (SSBOs) and contain an array of 128-bit unsigned integer starting values. Out-buffers are also SSBOs and contain an array of 16-bit unsigned integer total stopping times (step counts).

The main loop consists of the CPU writing starting values to in-buffers; the GPU reading starting values from in-buffers, iterating through Collatz sequences, and writing step counts to out-buffers; and the CPU reading step counts from out-buffers. The number of inout-buffers is dependent on the system's specifications. There are one or more inout-buffers per VkBuffer object, one VkBuffer object per VkDeviceMemory object, and two or more VkDeviceMemory objects.

Collatz Conjecture Simulator attempts to minimise the time spent idle by the CPU and GPU due to one waiting for the other to complete execution. Such as the GPU waiting for starting values, or the CPU waiting for step counts. This is done by having an even number of VkDeviceMemory objects, where half contain memory close to the GPU (device local memory), and half contain memory visible to both the CPU and GPU (host visible memory). There are therefore four types of memory ranges: host visible in-buffers (HV-in), host visible out-buffers (HV-out), device local in-buffers (DL-in), and device local out-buffers (DL-out).

Rather than the CPU and GPU taking turns executing, both processors spend time running in parallel. The CPU reads and writes host visible inout-buffers, and the GPU reads and writes device local inout-buffers, simultaneously. Starting values are written to HV-in, copied from HV-in to DL-in, and read from DL-in. Step counts are written to DL-out, copied from DL-out to HV-out, and read from HV-out.

flowchart LR
  CPU-->In-buffer
  In-buffer-->GPU
  GPU-->Out-buffer
  Out-buffer-->CPU
  subgraph In-buffer
    direction LR
    HV-in-->DL-in
  end
  subgraph Out-buffer
    direction BT
    DL-out-->HV-out
  end

Loading

Starting Value Selection

It can be mathematically demonstrated that particular sets of starting values will always generate Collatz sequences that contain a smaller value. The step counts of these starting values can thus be expressed as the sum of the step count of the smaller value, and the distance between the starting value and smaller value in the Collatz sequence.

$$\exists n, k \in \mathbb{Z}^+ : n > f^k(n) \implies s(n) = s(f^k(n)) + k$$

For example, the set of even starting values.

$$\begin{align*} \forall n \in \mathbb{Z}^+, f(2n) = n \\\ \implies s(2n) = s(n) + 1 \end{align*}$$

The Collatz sequences of all even starting values $2n$ must have exactly one step between $2n$ and the first value less than it, namely $n$. So by knowing $s(n)$, $s(2n)$ can be calculated as $s(n) + 1$.

Another set of interest is the set of starting values congruent to 1 modulo 4.

$$\begin{align*} \forall n \in \mathbb{Z}^+, f(4n + 1) = 12n + 4 \\\ f(12n + 4) = 6n + 2 \\\ f(6n + 2) = 3n + 1 \\\ \implies s(4n + 1) = s(3n + 1) + 3 \end{align*}$$

These two patterns remove the requirement to iterate through the Collatz sequence of every 3 in 4 starting values, leaving only those congruent to 3 modulo 4. The step counts of all other starting values can be determined with the previously calculated step counts of lesser starting values.

Hence, Collatz Conjecture Simulator only iterates through the Collatz sequences of starting values $n \in \mathbb{Z}^+ : n \equiv 3 \pmod 4$. This improves performance by allowing larger intervals of starting values to be tested per dispatch command, and by more evenly distributing the workload between the CPU and GPU.

Artificial Intelligence

The author of Collatz Conjecture Simulator is not a lawyer, but strongly believes the usage of GPLv3-licensed works in the training and development of proprietary AI is necessarily violating of said licence. However, in the event the GPLv3 does not in itself encompass the above restriction, the following shall apply.

Collatz Conjecture Simulator includes in its terms and conditions regarding copying, distribution, and modification, in addition to those provided by version 3 of the GNU General Public Licence, the strict prohibition of its usage by artificial intelligence software not licensed in their entirety, as a whole, under version 3 or later of the GNU General Public Licence, including but not limited to the training, prompting, or generation of such artificial intelligence models or algorithms.