# What happens at the Boundary of Computation?

TLDRThe video explores the enigmatic Busy Beaver function, a measure of the computational complexity of Turing machines. It delves into the concept of non-computable functions, where the function grows faster than any computable function. The video also introduces a connection between the Busy Beaver function and the Collatz conjecture, highlighting the unexpected link between two complex mathematical problems. It discusses the implications of the halting problem and Gödel's incompleteness theorems, suggesting that the Busy Beaver function encodes answers to famous unsolved problems in mathematics, making it a fascinating subject at the boundary of computation and mathematical understanding.

### Takeaways

- 🧠 The Busy Beaver function represents a mysterious and complex area of mathematics where the deeper you look, the more mysterious it becomes.
- 🤖 A Turing machine is a mathematical model that can simulate any computation, using an infinitely long tape of ones and zeros and a state table to determine its behavior.
- 📈 The Busy Beaver function, denoted as Σ_n, is defined by considering all n-state Turing machines and finding the maximum count of ones written before they halt.
- 🚀 Σ_n grows faster than any computable function, as it encompasses the behavior of all Turing machines with n states, beyond a certain point.
- 🎲 The Busy Beaver function is likened to a game where all tools of science and mathematics are at our disposal, but the ultimate tool seems to come from number theory.
- 🔗 There's a surprising connection between the Busy Beaver function and the Collatz conjecture, linking two seemingly unrelated but fundamental mathematical concepts.
- 🔍 For n=5, the Busy Beaver machine that halts with the most ones likely runs for tens of millions of steps, suggesting a deep complexity in its operation.
- 🔢 The operations of the five-state Busy Beaver machine are similar to the Collatz function, both applying simple functions based on remainders, but with different integers.
- 🌐 The Busy Beaver function and its variants, like the shift function, encode answers to famous open problems in mathematics, turning questions of infinity into finite computations.
- 🚫 The halting problem is central to the Busy Beaver function, as it prevents a general finite algorithm from computing the function, making it a fundamental challenge in mathematics.
- 💥 Gödel's incompleteness theorems imply that there exists a point beyond which our mathematical systems cannot prove the Busy Beaver function, indicating a limit to what mathematics can deduce.

### Q & A

### What is the Busy Beaver function and why is it significant in the context of computation?

-The Busy Beaver function, denoted as Sigma n, is a function that considers all end-state Turing machines with a given number of states 'n'. It measures the maximum count of ones that can be written on an initially all-zeros tape when each machine is run. The function is significant because it represents a non-computable function, meaning that there is no algorithm that can determine the value of Sigma n for any given n, which highlights the limits of computation and the inherent complexity of certain problems in computer science.

### What is a Turing machine and how does it relate to the Busy Beaver function?

-A Turing machine is a theoretical computational model that can simulate the logic of any computer algorithm. It consists of an infinitely long tape divided into cells, a tape head that can read and write on the tape, and a finite set of states that determine the machine's behavior. The Busy Beaver function is related to Turing machines because it evaluates the behavior of these machines when they are run with a specific number of states, measuring the maximum number of ones they can write on a tape before halting.

### Why is the Busy Beaver function considered to grow faster than any computable function?

-The Busy Beaver function grows faster than any computable function because it takes into account the behavior of all Turing machines with a given number of states. As 'n' increases, the complexity and potential behavior of Turing machines also increases, leading to outputs that can grow at an incredibly fast rate. No single computable function can match this growth because each computable function is defined by a fixed set of rules and states, whereas the Busy Beaver function encompasses the combined potential of all machines with 'n' states.

### What is the connection between the Busy Beaver function and the Collatz conjecture mentioned in the script?

-The connection between the Busy Beaver function and the Collatz conjecture is that the behavior of the Turing machine that is believed to be the five-state Busy Beaver resembles the operations of the Collatz function. Both functions apply simple operations based on the remainder when dividing by a certain number (three for the Busy Beaver machine and two for the Collatz function). This similarity suggests a link between the chaotic behavior of the Busy Beaver function and the well-known but unsolved problem of the Collatz conjecture.

### What does it mean for a Turing machine to 'halt' and why is this concept important in the context of the Busy Beaver function?

-For a Turing machine to 'halt' means that it reaches a terminal state where it stops performing any further operations. This is important in the context of the Busy Beaver function because the function measures the maximum number of ones a Turing machine can write on a tape before it halts. The halting behavior is crucial for determining the output of the function and understanding the limits of computation.

### Can you explain the concept of the 'shift function' s of n as mentioned in the script?

-The 'shift function' s of n is a variation of the Busy Beaver function that measures the maximum number of times a Turing machine can shift its tape head (move left or right) before it halts. Unlike the original Busy Beaver function which counts the number of ones written, the shift function focuses on the number of steps taken in terms of tape head movements. This function is also non-computable and highlights different aspects of the computational process.

### How does the script relate the Busy Beaver function to the halting problem in computer science?

-The script relates the Busy Beaver function to the halting problem by illustrating that the function cannot be computed for arbitrary 'n' due to the undecidability of the halting problem. The halting problem asks whether, given a description of a Turing machine and an input, the machine will eventually halt or continue running indefinitely. The Busy Beaver function requires knowledge of when a Turing machine halts, which is inherently unknowable for certain machines, thus making the function non-computable.

### What is the significance of the theorem mentioned in the script that states no consistent, arithmetically sound axiomatic system can prove the Busy Beaver function beyond a certain point?

-The significance of this theorem is that it demonstrates the fundamental limits of mathematics and formal systems in proving certain properties of computations. It shows that there exists a point beyond which even a consistent and arithmetically sound axiomatic system cannot prove whether a Turing machine will halt or not, indicating that the Busy Beaver function, and by extension many problems in computation, are inherently unknowable within the confines of such systems.

### How does the script connect the Busy Beaver function to open problems in mathematics?

-The script connects the Busy Beaver function to open problems in mathematics by suggesting that the function encodes answers to famous unsolved problems like the Goldbach conjecture. It explains that if the shift function s of 27 were known, it could be used to determine the truth of the Goldbach conjecture, as a Turing machine exists that halts if and only if the conjecture is false. This connection illustrates how the Busy Beaver function can turn questions of infinity into finite computations, highlighting the central role of the halting problem in mathematics.

### What is the role of Gödel's incompleteness theorems in the discussion of the Busy Beaver function as presented in the script?

-Gödel's incompleteness theorems play a crucial role in the discussion of the Busy Beaver function by establishing the inherent limitations of formal systems in proving certain truths. The second incompleteness theorem, in particular, states that no consistent system strong enough for arithmetic can prove its own consistency. This theorem is used to prove that there exists a Turing machine for which we cannot prove whether it halts or not, contributing to the unknowability of the Busy Beaver function beyond a certain point.

### Outlines

### 🧩 The Enigma of the Busy Beaver Function

The video script introduces the concept of the Busy Beaver function, a non-intuitive mathematical function that grows faster than any computable function. It discusses the Turing machine, a mathematical model of computation, and explains how the Busy Beaver function, denoted as Sigma n, is defined by running all n-state Turing machines on a blank tape and recording the maximum number of ones that can be written before the machine halts. The function is highlighted as being inexplicably complex, with its growth surpassing any computable function, and it is likened to a game where all tools of science and mathematics can be used, but the ultimate tool appears to come from number theory, hinting at a connection with the Collatz conjecture.

### 🔍 Unraveling the Five-State Busy Beaver Mystery

This paragraph delves into the specifics of the five-state Busy Beaver problem, a machine that halts with the maximum number of ones written, likely being the machine that halts with 4098 ones. The script explains the complex behavior of this machine, which involves operations that are reminiscent of the Collatz function, a simple yet chaotic mathematical sequence. The comparison between the operations of the Busy Beaver machine and the Collatz function is made, highlighting their similarities and the implications this has for understanding the behavior of the Busy Beaver. The paragraph also discusses the implications of this for larger n-state machines, where the behavior becomes increasingly complex and potentially encodes solutions to famous open problems in mathematics.

### 🛰️ The Busy Beaver's Connection to Unsolved Mathematical Problems

The final paragraph of the script explores the profound implications of the Busy Beaver function for mathematics as a whole. It introduces the shift function, another aspect of the Busy Beaver problem that counts the maximum number of shifts a Turing machine can make before halting. The script then connects the Busy Beaver function to the halting problem, a central issue in computability theory, and suggests that the Busy Beaver function encodes answers to a vast array of mathematical questions, including the Goldbach conjecture. The video concludes with a discussion of Gödel's incompleteness theorems, which imply that there are statements about the Busy Beaver function that cannot be proven within any consistent, arithmetically sound axiomatic system, thus highlighting the fundamental limits of mathematics and the enigmatic nature of the Busy Beaver function.

### Mindmap

### Keywords

### 💡Busy Beaver function

### 💡Turing machine

### 💡Computation

### 💡Halting problem

### 💡Goldbach conjecture

### 💡Collatz conjecture

### 💡Non-computable function

### 💡Axiomatic system

### 💡Kurt Gödel

### 💡Recursive function

### 💡Incomprehensibly explosive algorithms

### Highlights

The Busy Beaver function is a wild function that computation itself cannot touch, growing faster than any computable function beyond a certain point.

A Turing machine is a mathematical object that can represent any computation with a state table determining its behavior.

Sigma n, or the Ones function, is defined as the maximum count of ones written by Turing machines with n states when run on a tape of all zeros.

The concept of the Busy Beaver function is presented as a game where all tools of science and mathematics are at our disposal.

The Collatz conjecture is linked to the Busy Beaver function, showing a fundamental connection between two remarkable, seemingly unrelated mathematical problems.

The five-state Busy Beaver machine is believed to halt with 4098 ones after running for tens of millions of steps.

The behavior of the five-state Busy Beaver machine is analogous to a tuned version of the Collatz operation, involving simple multiplication, division, and addition.

The Busy Beaver game involves finding a short program that does a significant amount of work and then halts, exploiting recursion to generate exponential growth.

The six-state Busy Beaver machine applies an exponential version of a Collatz-like function, indicating a source of explosive growth.

The Busy Beaver function encodes answers to famous open problems in mathematics, suggesting a massive set of questions may be framed in this way.

The halting problem is central to the Busy Beaver function, as it prevents a general finite algorithm from computing the function.

The shift function s of n counts the maximum number of times a machine shifts before halting, providing insights into the halting problem.

If the shift function s of 27 were known, it could provide a finite algorithm to solve the Goldbach conjecture, turning a question of infinity into a finite calculation.

The halting problem is elevated as the core of many mathematical problems, highlighting its foundational role in mathematics.

Gödel's incompleteness theorems imply that there are statements about the Busy Beaver function that cannot be proved within a consistent, arithmetically sound axiomatic system.

The Busy Beaver function demonstrates that there are aspects of computation that are fundamentally unknowable within our current mathematical frameworks.

The video discusses the potential of the Busy Beaver function to provide expert intuitions and a deeper understanding of its bizarre nature.