# The Boundary of Computation

TLDRThe video explores the concept of Busy Beaver numbers, which represent an unbounded growth function surpassing any computable function. It explains the function using binary Turing machines and the halting problem, demonstrating that Busy Beaver numbers are uncomputable and can't be determined for certain values. The video also discusses the implications of Busy Beaver numbers for mathematics and algorithms, suggesting they contain information about unsolved problems like the Goldbach conjecture and the limitations of mathematical proofs.

### Takeaways

- 🐻 The Busy Beaver function is a mind-blowing concept in mathematics that represents an upper bound for the number of ones a Turing machine can write before halting.
- 🔍 The Busy Beaver numbers are calculated by considering all possible Turing machines with a given number of states and determining the maximum number of ones they can print on an all-zero tape before halting.
- 🤖 Turing machines are abstract computational models that operate on an infinitely long tape of zeros and ones, with a finite set of states and rules for reading, writing, moving, and transitioning between states.
- 🚫 The halting problem, proven by Turing, states that there is no general algorithm that can determine whether any given Turing machine will halt on any given input tape, highlighting the limitations of computation.
- 🔢 The Busy Beaver function, denoted as Σ(n), is not computable, meaning there is no finite procedure to calculate it for all n, but specific values for small n can be determined through exhaustive analysis.
- 📈 The sequence of Busy Beaver numbers grows faster than any computable function, making it an extraordinary function that outpaces all others in terms of growth rate.
- 🏗️ The difficulty in calculating higher Busy Beaver numbers increases exponentially with the number of states in the Turing machines, making it a monumental task for mathematicians.
- 🧩 The Busy Beaver numbers are so complex that even determining Σ(5) involves analyzing trillions of machines, a task that remains beyond current mathematical capabilities.
- 🔮 Some Busy Beaver numbers are linked to unsolved problems in mathematics, such as the Goldbach conjecture and the Riemann hypothesis, suggesting that calculating these numbers could resolve longstanding mathematical mysteries.
- 💡 The Busy Beaver function challenges our understanding of computability and the foundations of mathematics, with implications that reach far beyond the function itself.
- 🌐 The Busy Beaver function is a boundary separating the computable from the uncomputable, serving as a stark reminder of the limits of what we can algorithmically determine.

### Q & A

### What is the Busy Beaver function and why is it significant?

-The Busy Beaver function, denoted as Σ(n), is a function that counts the maximum number of ones a Turing machine with n states can write on an initially all-zero tape before halting. It is significant because it represents a boundary between the computable and the uncomputable. No algorithm can keep pace with the growth rate of the Busy Beaver function, and computing its values for larger n involves resolving unsolved problems in mathematics.

### What is a binary Turing machine and how does it operate?

-A binary Turing machine is an abstract computational device that operates on an infinitely long tape consisting of zeros and ones. It has an internal state and reads a cell on the tape. Depending on its state and the value read, it writes a one or a zero, shifts left or right, transitions to a new state, and may eventually halt, completing the computation. The machine's behavior is represented using a state table.

### What is the Church-Turing thesis and its implication for Turing machines?

-The Church-Turing thesis suggests that any computation, which is any finite sequence of steps applied to some input to produce some output, is equivalent to the operations of some Turing machine. This means that all computations and algorithms can be thought of as Turing machines, and statements about Turing machines apply to all programs and algorithms.

### Why is the halting problem for Turing machines undecidable?

-The halting problem is undecidable because there is no general algorithm that can determine whether a Turing machine will halt on a given input tape. Some machines may run indefinitely, and there is no way to shortcut computation to know if they will ever halt without actually running them.

### What is the difference between a computable and a non-computable function?

-A computable function is one that can produce an output from an input through a finite sequence of steps. A non-computable function, like the Busy Beaver function, does not have a finite procedure that works for all inputs, as some machines may run forever, making it impossible to determine the output in a finite amount of time.

### What is theBusy Beaver machine and how is it related to the Busy Beaver number?

-The Busy Beaver machine is a specific Turing machine that achieves the maximum count of ones written on an all-zero tape before halting. The Busy Beaver number, Σ(n), is the maximum count of ones written by any Turing machine with n states that halts, and the Busy Beaver machine is the one that achieves this maximum.

### What is the significance of the Busy Beaver function surpassing other fast-growing functions?

-The Busy Beaver function surpassing other fast-growing functions indicates that its growth rate is beyond any computable function. This means that it grows faster than any function that can be computed in finite time, making it an extraordinary and unique function in the realm of mathematics and computation.

### How does the Busy Beaver function relate to unsolved problems in mathematics?

-The Busy Beaver function is connected to unsolved problems in mathematics because the computation of certain values of Σ(n) could potentially resolve long-standing conjectures like the Goldbach conjecture or the Riemann hypothesis. This is because some Turing machines designed to halt if and only if these conjectures are false have been proposed.

### What is the implication of the Busy Beaver function for the foundations of mathematics?

-The Busy Beaver function implies that there are true mathematical statements, such as specific values of Σ(n), that cannot be proven within our current axiomatic systems of mathematics. This suggests that beyond a certain point, mathematics loses the ability to make claims about these numbers, highlighting the limitations of our mathematical frameworks.

### Why is the name 'Busy Beaver' considered inappropriate by some?

-The name 'Busy Beaver' is considered inappropriate or unserious by some because it sounds like the title of a children's show rather than a significant concept in the study of computation and the foundations of mathematics.

### Outlines

### 🧠 Exploring the Busy Beaver Function

The speaker delves into the concept of the Busy Beaver function, which is a measure of the computational complexity of Turing machines. They recount their fascination with the function after watching videos about enormous numbers like Graham's number and Tree(3). The Busy Beaver function, denoted as Σ_n, is defined as the maximum number of ones a Turing machine with n states can write on an initially blank tape before halting. The speaker explains the concept of a binary Turing machine and how it operates, including its state table and the actions it takes based on its state and the value it reads. They also discuss the Church-Turing thesis, which posits that any computation can be simulated by a Turing machine, and Turing's proof that there is no general algorithm to determine whether a Turing machine will halt on a given input, highlighting the undecidability of the halting problem.

### 📈 The Growth and Complexity of Busy Beaver Numbers

This paragraph explores the growth and complexity of Busy Beaver numbers, starting with simple examples for two and three states, where the maximum number of ones written before halting are 4 and 6, respectively. The speaker then moves on to more complex examples, such as the four-state Turing machine that writes 13 ones, making Σ_4 equal to 13. They discuss the difficulty in calculating Σ_n for higher states due to the exponential growth in the number of possible Turing machines and the lack of a general solution to determine which machines will halt. The speaker emphasizes that the Busy Beaver function is not computable, as it cannot be determined by a finite sequence of steps for all n, but for specific n, it might be possible to find the answer through analysis. They also compare the growth rate of the Busy Beaver function to other fast-growing functions and attempt to construct a challenger function to illustrate the sheer scale of the Busy Beaver's growth.

### 🔮 The Uncomputability and Mathematical Implications of Busy Beaver Numbers

The speaker discusses the uncomputability of the Busy Beaver function and its profound implications for mathematics and the foundations of algorithms. They explain that while the function is not computable for all n, specific values of n can sometimes be computed through finite analysis. The speaker then attempts to create a fast-growing function to challenge the Busy Beaver, using exponential notations and recursive applications of these notations to construct an incredibly large number. However, they acknowledge that the Busy Beaver function surpasses their constructed function for sufficiently large n, due to the finite procedure used to define their function. The speaker also touches on the connection between Busy Beaver numbers and unsolved problems in mathematics, such as the Goldbach conjecture and the Riemann hypothesis, suggesting that computing certain Busy Beaver numbers could potentially resolve these long-standing questions. They conclude by pointing to resources for further exploration, such as Scott Aronson's blog and papers, and express gratitude to their patrons for supporting their content.

### Mindmap

### Keywords

### 💡Busy Beaver

### 💡Graham's number

### 💡Turing machine

### 💡Computable function

### 💡Halting problem

### 💡State table

### 💡Church-Turing thesis

### 💡Unsolved problems

### 💡Axiomatic systems

### 💡Growth rate

### Highlights

The Busy Beaver function is a mind-blowing concept that outpaces all algorithms in theory.

Busy Beaver numbers are so large that they cannot be computed by any known algorithm.

The concept of Busy Beaver numbers involves binary Turing machines operating on an infinitely long tape.

A Turing machine's behavior is represented by a state table with specific actions for each state and read value.

The Church-Turing thesis suggests that all computations can be thought of as Turing machines.

Turing proved that there is no algorithm to decide whether a Turing machine will halt on any given input tape.

The Busy Beaver function, denoted as Sigma n, is the maximum count of ones a Turing machine can write before halting.

Sigma 2 is 4, achieved by a specific two-state Turing machine configuration.

Sigma 3 is 6, and Sigma 4 is 13, with the corresponding Busy Beaver machines for each found through exhaustive analysis.

Calculating Sigma for five states and beyond is currently unattainable due to the exponential growth in the number of Turing machines.

Sigma n is not computable because it involves an infinite sequence of steps to determine the output.

The Busy Beaver function grows faster than any computable function, making it a boundary between the computable and uncomputable.

The Busy Beaver numbers have implications for unsolved problems in mathematics, such as the Goldbach conjecture and the Riemann hypothesis.

There are true mathematical statements about Busy Beaver numbers that cannot be proven within our current axiomatic systems.

The Busy Beaver function serves as a boundary separating the realm of what can be computed from what remains a mystery.

Efforts to understand and calculate Busy Beaver numbers contribute to the foundations of mathematics and algorithms.

Scott Aronson's blog and papers provide an excellent introduction to the complex world of Busy Beaver numbers.