The Peterman Pod
Turing Award Winner: P vs NP, Zero-Knowledge Proofs, Quantum Computation | Avi Wigderson
01 Jun 2026
Transcript generated automatically by AI and may contain errors.
Chapter 1: What is the main topic discussed in this episode?
If tomorrow somebody finds even a classical factoring algorithm in polynomial time, I think there will be chaos in the world.
Chapter 2: What is the P vs NP problem and why is it significant?
This is Avi Wigderson, Turing Award and Abel Prize winner, and I interviewed him all about his field.
The question of p versus mp is whether we can solve all the problems we really want to solve. The quality of randomness is in the eye of the beholder, or in the computational power of the beholder. Here's the full episode.
In one of your lectures, you mentioned that P equals NP is philosophically about the fundamental limits of human knowledge. What is P equals NP, and how does it relate to human knowledge?
Well, in the simplest way, and I will talk at the high level, We are interested in various problems in life, in all sorts of problems, mathematical problems, scientific problems, medical problems, personal problems, intellectual problems. We are in the business in computer science of solving problems by computers. Now, we would like to somehow classify those problems that we can solve
To know what we can know. The problems we can solve are the things we will know, we will understand. So there are all these problems that we want to understand.
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Chapter 3: What happens if we relax correctness in problem-solving?
There are many of them. And there are the problems that we can understand, can solve. In a very basic way, the first class is NP. All the problems we want to solve are really NP problems. So it's a class of problems. A subset of it is the problems we can solve.
There are the problems we can currently solve, but we are interested in all the problems we can solve in principle, we can ever really solve. The question of P versus NP is whether we can solve all the problems we really want to solve, whether we can know everything we want to know. This is basically about the limits of our knowledge.
Of course, I have to justify why these classes which are mathematically defined actually are captured by this intuitive intuitive meanings I'm giving them.
Chapter 4: Why are NP-complete problems considered equivalent?
P is very easy to justify. What we can solve is what we can solve in our lifetime using whatever we have, let's say all computers that we have, using efficient algorithms. Problems we can solve are problems you see in our apps in our phones.
If there's a navigation app, it means somebody invented an algorithm that's efficient enough to solve any shortest path problem between any two places in any kind of map. Okay.
Chapter 5: How do space and time complexity affect problem-solving?
Why are all the problems we want to solve are NP problems? What is NP? NP is a class of problems, the mathematical definition, are problems so that whether they are easy or hard to solve, we don't know. But if somebody hands us a solution, then we can easily check that indeed it is a good solution. Now why are all problems humanity is interested in of this nature?
You can think there's no limit to what we may want to know. But I claim that essentially in any human endeavor, if you are embarking on any problem, if you really seriously want to solve a problem, the least you want to know is that when you hit upon a solution, you recognize it as a solution.
I mean, why would you ever start on looking for something if you will never recognize it as what you looked for? So you think mathematicians are looking for proofs of theorems? Certainly if somebody gives a proof, you know, wrote a paper about it, we can verify. If a scientist wants to explain data, you know, about the planets or about, I don't know,
bacteria or whatever, want to develop a theory, so that's what you're searching for in this case. Again, scientists write papers about their theories.
Chapter 6: Why do people use SAT solvers in computational problems?
They have to be consistent with the data. We have a way of recognizing that this is a good solution.
Chapter 7: How is randomness utilized as a resource in computation?
Engineers are usually tasked with creating something, bridge or phone or something, under some constraints. It can be monetary constraints, physical constraints, all sorts of, you can use this but not this, and so on. Anyway, if an engineer designs something, whoever gave them their task can look at it and say, violated some constraints, or no, it's good.
You really delivered what we were looking for. So I'm giving you examples. I mean, I can give many more. Detectives are supposed to solve crimes and, you know, we know from detective books, you know, they eventually provide that. These are examples of very fundamental human endeavors that encompass almost everything I can think of.
In all of them, what people are searching for have the property that when a solution is found, it is easily recognized. repeating in the beginning, NP are all problems that we can honestly say we really want to solve. NP are those that we can solve.
If they are equal, then everything we want to know, cure for cancer, anything you imagine, can be just the very fact that a solution can be easily recognized would imply that it can be efficiently found. That means that we can know everything we ever want to know. It's a fundamental question about human knowledge.
So I know this is one of those Millennium Prize problems. There's a million dollar prize if you can provide a proof for this. I can imagine you could prove that they're equivalent. You could also prove that they're not. If you had to guess when and if a proof comes, which direction would your intuition say it goes and why?
I think that my intuition, and that probably holds for almost all members of the theoretical computer science community, is that they are different. I think there are several reasons for that, and I will already tell you now that they are not that convincing. One is the intuition that most of us have that finding something is really hard than just checking that it's there.
If you lost your keys or your phone and you have no idea where to look, but somebody points out, oh, you left it on the windowsill or something like this, you say, ah, yeah, I did. So, we have this experience from life that finding is typically a harder task than just checking that it is what we look for. Another is that real NP problems occur all over the place.
They occur in optimization, in logic, in other aspects, in mathematics. verifying that algorithms and protocols and security systems actually work, all sorts of. And in this generality, these optimization problems that are NP problems, maybe NP-hard problems, People try to solve for completely egocentric reasons to make their company richer.
There are thousands of these and they manifest themselves in many different forms. Over the maybe 50, 60, 70 years, people seriously looked for algorithms for these very different ones and couldn't find any. this may seem like there's no such solution. And getting a bit more technical, NP problems, NP complete problems in particular, seem to require search over an exponentially large space.
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Chapter 8: What are zero-knowledge proofs and their significance?
And somehow P versus NP is about having a general method for cutting down this exponential research space into searching over something much smaller in a clever way. And this should apply to all these problems. So that would be equal NP and people don't think that there is such a way. But as I said, you know, these are, yeah, these are intuitions that are, you know, maybe pretty strong.
And yeah, I think that's the only, you know, if tomorrow somebody finds an algorithm for, an efficient algorithm for NP-complete problems using some new idea that nobody ever thought of, you know, what do you do?
That would change the world. In software engineering, we use algorithms all the time and they often have these very satisfying properties where the solution, you know, we use space or some algorithm that can take something where the brute force is something much larger to down to something that's polynomial time or something very reasonable.
But in all these NP problems, it's pretty unsatisfying that the best we could do is brute force. Am I understanding that correctly?
Yeah, you're understanding it perfectly correctly. I think that maybe what you are getting at is that NP-complete problems, a particular NP-complete problem, is really a family of instances, right? I mean, you want to find a Hamiltonian tour, some traveling settlement tour through some map. There's this map and that map and this network and other network. There are many, many instances.
When we talk about an algorithm for them, typically we say an algorithm is efficient if it's efficient and correct on all of them. In software engineering and many aspects of optimization, in many real-world problems, the instances is a much more restricted set. They come from trying to verify a particular protocol or protein folding, if you want to think about a biological example.
The way it is phrased as an NP, our problem is You want to minimize the energy of a system under some various constraints which have to relate to the chemistry of the various molecules and atoms that appear in the protein. This kind of optimization problem one can easily prove is NP-hard. Nevertheless, our body faults our proteins all the time. extremely efficiently. Now, why is that?
Of course, I don't know really why is that, but it would seem that evolution designed proteins in such a way that this, you know, folding them, only them, we don't have that many proteins. There are not exponentially many proteins in the body. There are only a few that maybe are more prone to efficient energy minimization.
And more generally, I think in software engineering, or at least in verification and testing, you often want to check that something meets the specifications. You usually translate it into a satisfiability question of Boolean formulas. The formulas that you get, the instances that you get out of these have some structure.
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