# The Boundary of Computation

TLDRThe video explores the concept of the Busy Beaver function, a mathematical function that represents the maximum number of ones a Turing machine can write before halting. It delves into the complexity of calculating Busy Beaver numbers, which grows exponentially with the number of states, and highlights the function's significance as a boundary between the computable and the uncomputable. The video also touches on the implications of this function for the foundations of mathematics and the unsolved problems it may help solve, such as the Goldbach conjecture and the Riemann hypothesis.

### Takeaways

- 🧠 The Busy Beaver function is a mind-blowing concept that represents a boundary between the computable and the uncomputable.
- 🐻 Busy Beaver numbers are generated by binary Turing machines, which are abstract devices that manipulate an infinitely long tape of ones and zeros.
- 🔍 The function involves determining the maximum count of ones a Turing machine can write before halting when started on an all-zero tape.
- 🔢 The Busy Beaver number for a given number of states (n) is denoted as Σ(n), and it represents the maximum number of ones written by any n-state Turing machine that halts.
- 🚫 Turing proved that there is no general algorithm that can determine whether a Turing machine will halt on any given input tape.
- 🌐 The Church-Turing thesis suggests that any computation can be thought of as operations of a Turing machine, making the Busy Beaver function relevant to all algorithms.
- 📈 The sequence of Busy Beaver numbers grows faster than any computable function, outpacing even the fastest-growing functions imaginable.
- 🤔 Calculating Σ(n) for larger values of n is extremely difficult because it involves analyzing an exponentially growing number of Turing machines.
- 🔮 There are Busy Beaver numbers that, if computed, would imply solving long-standing unsolved problems in mathematics, such as the Goldbach conjecture or the Riemann hypothesis.
- 💡 Some true statements about Busy Beaver numbers cannot be proven within our current axiomatic systems of mathematics, indicating a limit to what mathematics can prove.
- 🌟 The study of Busy Beaver numbers has implications for the foundations of mathematics and the understanding of algorithms and computation.

### 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 can write on an initially all-zero tape before halting, given n states. It is significant because it represents an upper bound of computational complexity and is known to grow faster than any computable function, highlighting the boundary between the computable and the uncomputable.

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

-A binary Turing machine is an abstract computational model that operates on an infinitely long tape divided into cells, each of which can be in one of two states, typically represented as 0 or 1. The machine has an internal state and a read/write head that can move along the tape. Depending on its current state and the symbol it reads, the machine writes a symbol, moves the head left or right, and transitions to a new state, potentially halting.

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

-The Church-Turing thesis is the idea that any computation that can be performed by any finite sequence of steps applied to some input to produce some output is equivalent to the operations of some Turing machine. This implies that all algorithms can be thought of as Turing machines, and any statement about Turing machines applies to all programs and computations.

### What is the halting problem and why is it undecidable?

-The halting problem is the question of whether a Turing machine will halt on a given input or continue running indefinitely. It is undecidable because there is no general algorithm that can determine whether any given Turing machine will halt for any given input tape. This means there's no way to shortcut computation to know if it halts; sometimes you have to run it and wait.

### How is the Busy Beaver number calculated for a given number of states?

-To calculate the Busy Beaver number for a given number of states, one must consider all possible Turing machines with that number of states, run each on an all-zero tape, and then determine which machines halt. The nth Busy Beaver number, Σ(n), is the maximum count of ones written by any machine that halts.

### What is the current status of calculating the Busy Beaver function for five states?

-As of the script's knowledge cutoff, humanity has not been able to calculate the Busy Beaver function for five states. This is due to the exponential growth in the number of Turing machines that must be considered, which involves trillions of machines, each more complex than before.

### Why is the Busy Beaver function considered non-computable?

-The Busy Beaver function is non-computable because there is no finite procedure that can determine the function's value for all n. Some Turing machines may run forever, and thus, there's no finite computation that can be done to determine the function's value for every n.

### How does the Busy Beaver function relate to other large numbers like Graham's number?

-The Busy Beaver function grows faster than any computable function, including Graham's number, which is already considered a very large number. This means that beyond a certain point, the Busy Beaver function will surpass Graham's number in terms of growth rate.

### What are the implications of the Busy Beaver function for the foundations of mathematics and algorithms?

-The Busy Beaver function has profound implications for the foundations of mathematics and algorithms. It touches upon the limits of computation and the undecidability of certain problems. It also suggests that there are true mathematical statements that cannot be proven within our current axiomatic systems, indicating the depth and complexity of mathematical knowledge.

### What resources are available for further exploration of the Busy Beaver function?

-For those interested in further exploration of the Busy Beaver function, Scott Aronson's blog and papers are recommended starting points. They provide a deep understanding of the topic and reference other work related to the Busy Beaver numbers and their implications in mathematics and computation.

### Outlines

### 🧠 Exploring the Busy Beaver Function

The speaker delves into the concept of the Busy Beaver function, which is a function that measures the maximum number of ones a Turing machine can write on an initially all-zero tape before halting. They mention Graham's number and 'tree three' as examples of incomprehensibly large numbers and express fascination with the Busy Beaver function, which they describe as a boundary between the computable and the uncomputable. The explanation involves a binary Turing machine, an abstract computational model that operates on an infinitely long tape of zeros and ones, transitioning between states based on a predefined set of rules. The speaker introduces the concept of the Busy Beaver number, denoted as Σ_n, which is the highest number of ones written by any n-state Turing machine before halting, and illustrates this with examples for n=2, n=3, and n=4, noting the difficulty in calculating these numbers and the unsolved problems in mathematics they can potentially resolve.

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

The speaker discusses the incomputability of the Busy Beaver function, emphasizing that it cannot be determined by a finite procedure for all n, and thus, it grows faster than any computable function. They provide a detailed explanation of how the number of Turing machines grows exponentially with the number of states, making the calculation of Σ_n for higher n extremely challenging. The speaker also attempts to create a fast-growing function to challenge the Busy Beaver function but acknowledges that it is quickly outpaced. They further explain that the Busy Beaver function surpasses any computable function after a certain point, highlighting the profound implications of this for the foundations of mathematics and the limits of computation.

### 🔮 The Implications of Busy Beaver Numbers in Mathematics

The speaker explores the profound implications of Busy Beaver numbers in the field of mathematics, noting that the calculation of certain Busy Beaver numbers, such as Σ_27, could potentially resolve long-standing conjectures like the Goldbach conjecture and the Riemann hypothesis. They also touch upon the idea that there are true statements about Busy Beaver numbers that cannot be proven within our current axiomatic systems of mathematics, indicating a boundary to what can be mathematically known. The speaker encourages further exploration of the Busy Beaver function and its impact on the world of algorithms and mathematical foundations, pointing to Scott Aronson's blog and papers as valuable resources. They conclude with a humorous critique of the name 'Busy Beaver,' suggesting it sounds more like a children's show than a subject of serious mathematical inquiry.

### Mindmap

### Keywords

### 💡Busy Beaver function

### 💡Turing machine

### 💡Computable function

### 💡Unsolved problems

### 💡Halting problem

### 💡Binary Turing machine

### 💡State table

### 💡Graham's number

### 💡Church-Turing thesis

### 💡Axiomatic systems

### Highlights

The Busy Beaver function is a mind-blowing concept that represents the boundary between the computable and the uncomputable.

Busy Beaver numbers are so large that no algorithm can keep pace with their growth.

Calculating Busy Beaver numbers for small values can involve solving centuries-old unsolved problems in mathematics.

A binary Turing machine is an abstract device that operates on an infinitely long tape of ones and zeros.

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

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

The Busy Beaver function, denoted as σ_n, is the maximum count of ones written by n-state Turing machines that halt.

σ_2 is 4, achieved by a specific two-state Turing machine, making it a Busy Beaver machine.

σ_3 is 6, and σ_4 is 13, with these numbers growing at an exponential rate as the number of states increases.

Calculating σ_5 has not been achieved due to the exponential growth in the number of Turing machines to consider.

The Busy Beaver function is not computable, as it involves an infinite sequence of steps for some machines.

For specific n, the Busy Beaver function can potentially be computed by analyzing a finite set of machines.

The sequence of Busy Beaver numbers grows faster than any computable function.

The Busy Beaver function surpasses any conceivable fast-growing function, including those defined with exponential notations.

There exists a 27-state Turing machine that halts if and only if the Goldbach conjecture is false.

Calculating certain Busy Beaver numbers could potentially resolve long-standing mathematical conjectures.

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

The Busy Beaver function has implications for the foundations of mathematics and the understanding of algorithms.

Scott Aronson's blog and papers provide an excellent starting point for further exploration of the Busy Beaver function.