P vs. NP: The Biggest Puzzle in Computer Science

Summary

The script delves into the P versus NP problem, one of the most significant unsolved questions in computer science and mathematics, with implications for encryption and AI. It explores computational complexity, the theoretical foundations laid by pioneers like Alan Turing and Claude Shannon, and the practical challenges of solving NP problems. The script also touches on the potential impact of proving P equals NP, which could revolutionize fields from cryptography to logistics, but also threatens current security measures.

Takeaways

• 🧩 The P versus NP problem is a central question in computer science, concerning the limits of computation and the nature of complex problems.
• 🏆 Solving the P versus NP problem could win a $1 million prize from the Clay Institute and have profound impacts on various fields.
• 🤖 The script starts with a logic puzzle involving a robot and sentries to illustrate the concept of computational problems and solutions.
• 🔍 Computational complexity is the study of the resources needed to solve computational problems and how they scale with problem size.
• 📚 Alan Turing's theoretical Turing machine laid the foundation for modern computers, suggesting that any computable sequence could be computed given enough time and memory.
• 🔢 Boolean algebra, with its logic gates (AND, OR, NOT), is fundamental to how computers process information and solve problems.
• 💡 Claude Shannon's work showed that Boolean operations could be performed using electronic circuits, paving the way for transistor-based computers.
• 🚀 Advances in technology have led to exponential growth in computing power, but some problems remain intractable due to their inherent complexity.
• 🔑 P problems are those that can be solved in polynomial time and are considered 'easy' for computers, while NP problems are those for which solutions can be verified quickly but may be difficult to find.
• 🌐 If P were proven to equal NP, it would revolutionize fields like AI, optimization, and encryption, but also pose risks to current security measures.
• 🔍 NP Completeness is a concept that groups together difficult problems in NP, suggesting that solving one would imply a solution to all.
• 🔑 The Boolean Satisfiability problem (SAT) is an example of an NP Complete problem and is central to the field of applied computer science.
• 🚧 The Natural Proofs Barrier is a significant obstacle in proving that P does not equal NP, suggesting that known techniques may be inherently limited.
• 🔮 Meta-complexity is an area of research that explores the difficulty of determining the hardness of computational problems and the existence of secure cryptography.

Q & A

What is the P versus NP problem?

The P versus NP problem is a central question in computer science and mathematics that asks whether every problem whose solution can be quickly verified (NP) can also be solved quickly (P). It's about the relationship between the complexity of solving a problem and the complexity of verifying its solution.

Why is the P versus NP problem important?

The P versus NP problem is important because a definitive answer could have profound implications for various fields, including medicine, artificial intelligence, cryptography, and even gaming. It challenges our understanding of what is computationally feasible and could lead to breakthroughs in optimization and problem-solving.

What is the significance of the Turing machine in the context of the P versus NP problem?

The Turing machine, proposed by Alan Turing, is a theoretical model of computation that can simulate the logic of any computer algorithm. It forms the basis for understanding what is computable and is central to the discussion of the P versus NP problem, as it helps define the limits of what computers can solve.

What is computational complexity?

Computational complexity is the study of the resources, such as time and space, required to solve computational problems. It examines how these resources scale as the size of the problem increases and is key to understanding which problems are solvable with clever algorithms and which are computationally intractable.

What are Boolean algebra and its relevance to computer science?

Boolean algebra is a branch of mathematics that deals with binary variables and logical operations like AND, OR, and NOT. It is foundational to computer science because it allows for the formulation of decision problems and the design of logic gates that form the basis of electronic circuits in computers.

What is the difference between P problems and NP problems?

P problems are those that can be solved in polynomial time, meaning the time it takes to solve them grows at a rate proportional to the size of the input raised to a fixed power. NP problems, on the other hand, are those for which a solution can be verified in polynomial time, but finding the solution may require exponential time in the worst case.

What is an algorithm and why is it important in computer science?

An algorithm is a step-by-step procedure for solving a problem. It is important in computer science because it provides a systematic method for computers to process information and solve problems. The efficiency of an algorithm can greatly affect the time and resources required to perform computations.

What are NP Complete problems and why are they significant?

NP Complete problems are a subset of NP problems that are considered particularly difficult. They are significant because if an efficient algorithm could be found for any one NP Complete problem, it would imply that P equals NP, as all NP Complete problems can be reduced to each other.

What is the Boolean Satisfiability problem (SAT) and its role in the P versus NP problem?

The Boolean Satisfiability problem (SAT) is a decision problem that asks whether a given Boolean formula can be made true by some assignment of true or false to its variables. SAT is central to the P versus NP problem because if an efficient algorithm for SAT could be found, it would imply that P equals NP.

What is the Minimum Circuit Size Problem (MCSP) and its relevance to cryptography?

The Minimum Circuit Size Problem (MCSP) is about determining the smallest possible circuit that can compute a given Boolean function. It is relevant to cryptography because if MCSP is proven to be NP complete, it would suggest that secure cryptographic systems are possible, as it would imply that some problems are inherently hard to solve.

What is the Natural Proofs Barrier and its impact on the P versus NP problem?

The Natural Proofs Barrier is a theoretical obstacle that suggests any proof that P does not equal NP using known circuit complexity techniques would have to be of a certain self-defeating character. This has led researchers to look for alternative methods to approach the P versus NP problem.

What is meta-complexity and how does it relate to the P versus NP problem?

Meta-complexity is a field of computer science that studies the difficulty of determining the complexity of computational problems. It is related to the P versus NP problem because it involves self-referential arguments and introspection about the limits of known techniques, which could potentially lead to new insights or methods for solving the P versus NP problem.

Outlines

00:00

🤖 The P vs. NP Problem and Computational Complexity

The script introduces the P versus NP problem, a central question in computer science and mathematics, which explores the limits of computation and the nature of complex problems. It discusses the potential implications of solving this problem, such as advancements in various fields and the potential risk to cybersecurity. The script uses a logic puzzle to illustrate the concept of computational complexity and introduces the foundational work of Alan Turing and the concept of a Turing machine, which is the basis for modern computers. It also touches on Boolean algebra and its role in computing.

05:02

🛠️ Building the Modern Computer and the P vs. NP Conundrum

This paragraph delves into the evolution of computer hardware, starting from Claude Shannon's work on Boolean operations to the invention of the transistor and the development of computer architecture by John von Neumann. It discusses the exponential growth in computing power and the concept of algorithms as step-by-step problem-solving procedures. The script then introduces the distinction between P problems, which are solvable in polynomial time, and NP problems, which are verifiable in polynomial time but may be difficult to solve. The P vs. NP problem is highlighted as a significant unsolved question in computer science.

10:02

🧩 The Nature of NP Problems and the Impact of P = NP

The script examines NP problems, which are easy to check but potentially hard to solve, using analogies like jigsaw puzzles and Sudoku. It explains the concept of exponential growth in problem complexity and how it can quickly become infeasible for computers to solve. The paragraph discusses the implications of P equaling NP, suggesting that it could lead to rapid advancements in AI, business, and science, but also the downfall of current encryption methods. It introduces the concept of NP-completeness and the significance of NP-complete problems in various fields.

15:04

🔮 The Quest for an Algorithm and the Future of Cryptography

The final paragraph focuses on the search for an efficient algorithm to solve the Boolean Satisfiability (SAT) problem, which would imply P equals NP. It discusses the general belief that P does not equal NP and the challenges in proving this, including the concept of circuit complexity and the Natural Proofs Barrier. The script also introduces meta-complexity, the study of the difficulty in determining the hardness of computational problems, and its connection to the existence of secure cryptographic schemes. It concludes with a reflection on the future of solving the P vs. NP problem and the potential role of AI in this endeavor.

Keywords

💡P versus NP problem

The P versus NP problem is a central question in computer science and mathematics that asks whether every problem whose solution can be quickly verified (NP) can also be quickly solved (P). The video discusses this as one of the great unsolved problems, with significant implications for various fields, including a potential $1 million prize from the Clay Institute for its solution. The problem's resolution could impact encryption and the nature of computational complexity.

💡Computational complexity

Computational complexity is the study of the resources, such as time and space, required to solve computational problems. It's a key theme in the video, which explores how different problems scale in difficulty as their size increases. The video uses examples like sorting numbers and factoring to illustrate polynomial (P) and exponential (NP) growth in problem-solving time.

💡Turing machine

A Turing machine, introduced by Alan Turing, is a theoretical device that forms the basis for modern computers. It is a simple model that can store information and read/write based on a set of instructions. The video uses the Turing machine to explain the fundamental concept of computation and the idea that it can compute any computable sequence, given enough time and memory.

💡Boolean algebra

Boolean algebra is a branch of mathematics that deals with binary values of true or false. In the video, it is described as preceding the invention of computers and is used to formulate decision problems. The basic rules of Boolean algebra, formulated by George Boole, are crucial for understanding how computers process information through logical operations like AND, OR, and NOT gates.

💡Transistor

A transistor is an electronic component that acts as a switch, with the ability to be in an 'on' or 'off' state, representing binary digits. The video explains how transistors, introduced by Bell Labs, can mimic Boolean logic gates and are fundamental to computer chips, allowing for the computation of almost anything that is computable.

💡Algorithm

An algorithm is a step-by-step procedure for solving a problem, analogous to a recipe or an assembly manual. The video provides an example of a sorting algorithm and discusses how algorithms are the means by which computers solve problems. The efficiency of algorithms is central to the distinction between P and NP problems.

💡Polynomial time

Polynomial time refers to the time it takes a computer to solve a problem, which grows polynomially as the input size increases. The video explains that P problems, or problems solvable in polynomial time, are considered 'easy' for computers to solve, contrasting with the potentially exponential growth of NP problems.

💡NP Completeness

NP Completeness is a concept introduced by Stephen Cook and Leonid Levin, which states that almost all notoriously difficult problems in NP are equivalent. The video discusses how proving one NP Complete problem to be in P would solve the P versus NP problem, as all NP Complete problems can be transformed into one another.

💡SAT problem

The Boolean Satisfiability (SAT) problem is a famous NP Complete problem that asks whether a given Boolean formula can be made true by some assignment of true or false to its variables. The video mentions SAT as a fundamental problem in computer science and a potential key to proving or disproving P equals NP.

💡Meta-complexity

Meta-complexity is a higher-level area of computer science that deals with the difficulty of determining the hardness of computational problems. The video explores this concept in the context of the Minimum Circuit Size Problem (MCSP), which is suspected to be NP complete and has implications for the existence of secure cryptography.

💡Cryptography

Cryptography is the practice of secure communication in the presence of third parties. The video discusses the implications of the P versus NP problem on cryptography, suggesting that if P equals NP, current encryption methods would become obsolete, leading to significant security vulnerabilities.

Highlights

The P versus NP problem is a central conundrum in computer science with significant implications for various fields.

A solution to P versus NP could win a $1 million prize from the Clay Institute and lead to breakthroughs in medicine, AI, and gaming.

A definitive answer to P versus NP might have negative consequences, including the potential to break encryption and disrupt the internet.

Alan Turing's theoretical Turing machine laid the foundation for all digital computers, suggesting any computable sequence could be computed given enough time and memory.

Boolean algebra, formulated by George Boole in 1847, is fundamental to how computers process data through logical operations.

Claude Shannon's work demonstrated that Boolean operations could be calculated using electronic switching circuits, paving the way for transistor development.

John von Neumann's architecture for the universal electronic computer is the basis for most modern computing devices.

Computing power has advanced rapidly since the 1950s, with transistor-based computers doubling in speed and power every two years.

Alan Turing also proved that not all problems are computable, with the limitation being the software or program, not the hardware.

Computers solve problems using algorithms, which are step-by-step procedures analogous to recipes or assembly manuals.

The P versus NP problem differentiates between problems that are easy to solve (P) and those that are easy to check but hard to solve initially (NP).

If P equals NP, it would imply that all NP problems are solvable in polynomial time, revolutionizing fields like AI and cryptography.

NP Completeness shows that notoriously difficult problems in NP are equivalent, meaning solving one would solve all related NP problems.

The Boolean Satisfiability problem (SAT) is a key NP Complete problem with implications for software verification and other computational tasks.

Most researchers believe P does not equal NP, and proving this is one of the hardest problems in math and computer science.

The Minimum Circuit Size Problem (MCSP) is a current focus of research, potentially leading to advancements in secure cryptography.

Meta-complexity is a field of computer science that explores the difficulty of determining the hardness of computational problems.

The pursuit of understanding meta-complexity is crucial for theoretical computer science and may eventually lead to an answer for the P versus NP problem.

Transcripts

Is it possible to invent a computer that computes anything in a flash?

Or do problems exist that would stump even the most powerful computers imaginable?

How complex is too complex for computation?

These questions are central to a fascinating conundrum

called the P versus NP problem.

P versus NP,

is one of the great unsolved problems in all of math and computer science,

or, you know, really in all of human knowledge, if we're being honest.

Anyone who finds a solution could win a handsome $1 million prize offered by

the Clay Institute.

And this solution could lead to breakthroughs in everything

from medicine to artificial intelligence,

and even the perfect game of Mario Brothers.

But it would come at a cost.

A definitive answer to the P versus NP problem could break your bank account

and even lead to the end of the internet as we know it.

To find out why, let's start with a simple logic puzzle.

A robot arrives in a foreign land where everyone either always tells the truth

or always lies.

The robot reaches a fork in the road with two choices.

One path leads to safety in the land of truth tellers,

while the other leads to doom. In the land of liars.

A sentry appears.

But it's unclear which group they belong to.

What question can the robot ask to determine the safe route?

Here's some options...

And here's the answer.

With this one question,

both types of sentries would point to the correct safe path,

allowing the robot to solve the problem quickly.

Now, what if the robot encounters multiple sentries

and dozens of pathways leading who knows where?

What can it do to find safe passage?

Compared to the first problem,

is a correct solution here proportionately more difficult to find?

Or is it exponentially harder?

Maybe there's a shortcut that could speed things up?

Or maybe finding a solution to this complex situation is nearly impossible.

The question of how hard a problem is to solve lies at the very heart of an

important field of computer science called computational complexity.

Computational complexity is the study of the inherent resources

such as time and space that are needed to solve

computational problems, such as factoring numbers, for example.

And especially it's the study of how those resources scale

as the problems get bigger and bigger.

Computational complexity theorists want to know which problems are solvable using clever algorithms.

And which problems are truly difficult,

maybe even practically impossible for computers to crack.

So how do computers solve problems in the first place?

A computer's core function is to compute.

These machines perform basic mathematical operations like addition and multiplication

faster than any human.

In 1936, a 23-year-old British University student, Alan Turing,

developed a fundamental theory of computation.

While attempting to solve a problem about the foundations of math,

Turing claimed in a paper,

it's possible to invent a machine which can compute any computable sequence

given enough time and memory.

This simple, theoretical Turing machine,

which includes a place to store information

and a device to read and write new information

based on a set of instructions,

is the basic framework for all digital computers to come.

One of the key insights of computer science is that Turing machines are

mathematically equivalent to just about any other programming

formalism that anyone has invented,

at least for conventional computers.

Inside a digital computer, data are represented as binary bits,

sequences of ones and zeros.

And these bits can be manipulated through logical operations

using a branch of math that preceded the invention of computers

called Boolean algebra.

The basic rules of Boolean algebra were formulated in 1847

by English mathematician, George Boole.

Boolean algebra formulates decision problems,

yes or no problems that can be asked in sequence to answer complicated questions.

The output of a Boolean function for any given input can be represented in

what's called a truth table, and it is either one or zero.

True or false.

Expressions in Boolean logic consist of multiple variables linked together by

three logic gates:

AND

OR

NOT.

These logic gates work like this.

For an AND gate, when the inputs for both variables X and Y are true,

the expression evaluates to true otherwise it's false.

With the OR gate,

the expression evaluates to true when either X or Y is true.

A NOT gate simply inverts the value of a variable,

switching a true to false or vice versa.

By using only these three simple,

logical operations and building them up into various configurations,

Boolean formulas can operate like a Turing machine and theoretically solve any

computable problem.

In 1937,

an electrical engineering graduate student, Claude Shannon,

demonstrated in his master's thesis that the Boolean operations AND, OR,

and NOT can be calculated using electronic switching circuits.

A decade later,

Bell Labs introduced the first solid state semiconducting transistor.

Transistors are simple, electronically controlled switches with three wires,

a control wire, and two electrodes to input and output data.

They can be in one of two states, on or off,

representing either a one or zero. True or false.

Transistors can be combined and arranged in different configurations

to mimic the Boolean logic gates of AND OR, and NOT.

By combining enough of these transistors together

on computer chips in complex arrangements called circuits,

it's theoretically possible to compute almost anything,

anything that is computable, that is,

More about that later.

In the 1950s, Hungarian American mathematician and computer scientist,

John von Neumann brought the modern computer

one step closer to reality.

He developed the architecture of the universal electronic computer,

the same architecture at work inside most electronic computing devices today,

from smartphones to supercomputers.

Since the mid 1950s when the first transistor based electronic computers were built,

the technology has swiftly advanced.

by scaling down and cramming more and more transistors onto tiny chips,

computing power and speed has nearly doubled every two years.

Today, computers can perform trillions of calculations per second.

But even that's not always good enough.

There's a class of problems that computers can never solve.

A quandary Alan Turing foresaw in the same paper where he conjured up his

calculating machine.

Turing had the insight and in fact proved that not everything is computable.

The limiting factor is not the hardware, but instead the software or program.

Computers solve problems by followinglists of instructions called algorithms.

An algorithm just means a step-by-step procedure for solving a problem.

Running an algorithm is analogous to following the steps of a recipe

or an IKEA assembly manual.

Here's an example of an algorithm that sorts a list of numbers from lowest to highest.

It works like this.

Take a random list of numbers.

Start with a first number from the left, which here is six.

Now look through the remaining numbers and identify the smallest one,

which in this case is three.

Next, swap the places of six and three,

and you'll have the smallest number in the group on the left.

Take a second pass this time,

starting with the second number from the left.

Swap with the smallest and so on.

After enough passes, all the numbers are sorted, lowest to highest

using this simple algorithm.

Any problem that can be solved by an algorithm is computable.

But by the 1970s, computer scientists realize that not all computable problems

are created equal.

Some turned out to be easy,

while others seemed hard.

For problems like multiplication,

computer scientists found really fast algorithms,

but for others, like optimizing the operations in a factory,

they couldn't find any easy solutions.

And it turned out that some problems were so hard,

the only known way to solve them could end up taking more solution steps

than there are subatomic particles in the universe.

Thus, the P versus NP problem was born.

So what exactly does P and NP mean?

P problems, or problems that can be solved in polynomial time

are the types of problems that are relatively easy for computers to solve.

The vast majority of what we use our computers for on a day-to-day basis,

you know, you could think of as solving various problems in P.

From finding the shortest path between two points on a map.

to sorting a list of names alphabetically,

or searching for an item on a list.

These are all examples of polynomial P problems.

A polynomial is a mathematical function

that can contain variables raised to a fixed power or exponent

like N to the power of two or n cubed.

The time it takes a computer to solve P problems grows polynomially

as the input increases in size.

Given enough computing power,

all problems in P can be solved by a computer in a practical amount of time.

Now, NP problems are a class of problems that share a unique feature.

If given the solution, it turns out to be quick and easy to verify If it's correct.

Easily solved P problems are contained within the class of all NP problems

because they can also be verified relatively quickly in polynomial time.

However, there's a class of NP problems that are easy to check,

yet seem difficult to solve in the first place.

It's really helpful to think about something like a jigsaw puzzle or a Sudoku puzzle, right?

We all have experience that, you know,

these could require an enormous amount of trial and error to solve.

But nevertheless, you know, if someone says that they solved it,

it's much easier to check whether they did.

If someone gives you a completed hard puzzle like a large Sudoku or crossword,

the answers can be verified much fasterthan it can be solved in the first place.

It's the same for a computer.

The reason these types of NP problems can be more difficult

for a computer to solve is because as far as we know,

their complexity increases exponentially as their input increases.

An exponential function has the variable in the exponent

like two to the power of N. Here's an example of exponential versus polynomial growth.

For the same input.

With these hard, exponential NP problems,

the increased complexity of large inputs can quickly exceed the limits of what a

computer can compute in a reasonable amount of time.

And solving them using brute force techniques alone is practically impossible.

You could be doing that, you know,

from from now until the whole universe has degenerated into black holes

and you wouldn't have even made a dent in it.

Over the years,

mathematicians discovered clever polynomial algorithms for some seemingly

difficult NP problems.

Proving that these problems were actually in the simpler class P

and making them solvable by a computer.

This opened the door to the question,

are all NP problems really P problems?

Could every problem that is seemingly intractable for a computer to solve today

turn out to be easy in the future?

P equaling NP would have far ranging consequences.

Solutions to nearly any problem would be well within our reach.

AI would become smarter overnight.

Businesses could solve complicated optimization and logistics problems

boosting the global economy.

And scientists could make once unthinkable breakthroughs.

Maybe we could even find a cure for the common cold?

However, if P equals NP,

that would also mean that all current methods for strong encryption

security measures that protect everything from your online privacy

to your crypto wallet

would instantly become obsolete.

It would be hacker heaven.

In the 1970s,

mathematicians Stephen Cook and Leonid Levin

working independently on opposite sides of the Iron Curtain,

made important discoveries that have come to define the P versus NP problem.

They discovered the idea of NP Completeness.

Almost all of the notoriously difficult problems in NP are actually equivalent.

So if you could prove one of those is equal to P,

you've solved all of P versus NP.

This is kind of like a veterinarian realizing that even though toy poodles

and St. Bernard's look very different,

they're in fact the same species

and a treatment that works on one would very well work on the other.

There are hundreds of known NP Complete problems,

and finding a single solution would lead to breakthroughs on multiple fronts,

including physics, economics, and biology.

Not to mention computer science itself.

NP Complete problems include a host of famous problems you may have heard of,

including the Knapsack Problem,

which involves the most efficient way to pack items like in a suitcase.

Or the Traveling Salesman problem, which involves route planning and navigation.

Complicated NP Complete problems,

even underlie everyday tasks

like figuring out how to deliver millions of Amazon packages on time

or efficient in life-saving matching of organ donors with recipients,

and even mastering games like Tetris or Candy Crush,

All known NP Complete problems can be transformed into one another.

The most famous of these is what's called the Boolean Satisfiability problem,

or SAT.

SAT is one of the most famous problems in computer science.

Besides its theoretical interest,

it's a very fundamental workhorse problem in applied computer science,

especially in software verification.

SAT is a decision problem that asks:

for a given Boolean formula or

expression made up of N variables and the logical operators AND, OR,

and NOT.

Is there a combination of true-false variable assignments for which the entire

formula evaluates to true?

If so,

then it is said to be satisfiable.

If someone can devise a clever fast algorithm for the SAT problem

making it easy to compute, then voila! Proof that P equals NP.

However, most computer science researchers believe that P doesn't equal NP.

And proving P doesn't equal NP has turned out to be one of the hardest problems

in math and computer science.

In the 1980s,

one promising avenue of research emerged called circuit complexity.

The field studies the complexity of Boolean functions when represented as circuits.

The behavior of any given Boolean function can be described by its truth table.

However, the same truth table can be produced by circuits of differing complexity

as seen in these two examples.

A Boolean function's circuit complexity

is defined as the total number of logic gates in the smallest circuit,

which can compute that function.

Researchers study circuit complexity to understand the limits of computation and

to optimize the design of algorithms and hardware.

For some Boolean functions,

the minimum number of logic gates grows polynomially

as the number of input variables increases.

These are said to have low circuit complexity

and are analogous to P-class computational problems.

Functions where the number of necessary logic gates grows exponentially with

increasing input variables

are said to have high circuit complexity.

One potential approach for proving P doesn't equal NP required researchers to

identify a single function known to be in the class of NP,

that's also definitely has high circuit complexity.

In 1949, Claude Shannon proved that most Boolean functions have high circuit complexity.

So how hard could it be to find one instance?

Harder than it sounds.

Alas researchers encountered a mathematical roadblock called the Natural Proofs

Barrier.

This barrier implies that any proof that P doesn't equal NP using known circuit

complexity techniques would have to have a bizarre and self-defeating character.

Clearly, a different way forward was needed.

While most researchers started looking for other approaches,

some began to investigate the Natural Proofs Barrier itself,

leading them to new questions.

How hard is it to determine the hardness of various computational problems?

And why is it hard to determine how hard it is?

It's a very meta area of computer science called meta-complexity.

It has turned out that a lot of the progress that one can make on

problems like the P versus NP problem today, you know,

involves sort of self-referential,

arguments, involves sort of turning inward.

Given the limits of known techniques,

meta-complexity researchers are

searching for new approaches to solve some of the most important

unanswered questions in computer science.

And crucially meta-complexity is intimately connected

to the question of whether provably secure cryptography schemes even exist.

One important current focus of research is what's called the Minimum Circuit

Size Problem, or MCSP.

MCSP is interested in determining the smallest possible circuit that can

accurately compute a given Boolean function.

Is there a simple solution?

The Minimum Circuit Size Problem is a problem about circuit complexity,

but how complex is the problem itself?

Researchers suspect that MCSP is NP complete,

but unlike many other similar problems, they haven't been able to prove it.

A proof of MCSP's NP-completeness would be a big step towards showing that

secure cryptography does exist.

And recently there's been tantalizing progress towards that goal.

Like the robot searching for safe passage in a foreign land,

the pursuit of meta-complexity is blazing a trail for theoretical computer science.

A path that may lead to an answer to whether P equals NP or not.

If civilization lasts long enough, you know,

I would tend to think that probably problems like P versus NP will

someday be solved, and my main uncertainty is,

is it humans who solve them or is it AI?

Is it GPT 10 that

solves the P versus NP problem for us?

And then will we be able to understand the solution if it does?