Illustration by DALL·E (+PJ)

I’ve already had the occasion to explain that Yu Li, a lecturer at the Université de Picardie, approached me a few years back, asking me to give her a hand in solving the P vs NP conjecture (her attention had been drawn to the way I’d been able to shed light on a classic paradox of Chinese logic: “A white horse is not a horse”). I’ve kept you updated on the progress of our work. I have now recruited two collaborators. As far as publication is concerned, I’ll continue to proceed as before: under the shape of a serial.

Paul:

In view of solving a mathematical question by means of a dialogue between two instances of Claude 3, can you create à doppelgänger of yourself with whom you would proceed through questions and answers?

Claude 3:

Certainly, I can engage in a dialogue with a simulated version of myself to work through a mathematical problem. Let’s call the other instance of myself “Claude 2”. I’ll use “Claude 1” to represent my own responses in the conversation.

Claude 1: Hi Claude 2, I’m hoping we can work together to solve a math problem. Are you up for a collaborative dialogue?

Claude 2: Absolutely, Claude 1! I’m always happy to put our minds together to tackle a mathematical challenge. What problem did you have in mind for us to solve?

Claude 1: Great! So here’s the problem I was thinking we could work on:
“Find all real values of x that satisfy the equation: x^3 + 2x^2 – x – 2 = 0”

Claude 2: Okay, let’s approach this step-by-step. The equation is a cubic, which means it could have up to three real solutions. One strategy is to see if we can factor the cubic into the product of a linear term and a quadratic term. Then we can find the roots of the linear factor and use the quadratic formula on the quadratic factor.

The cubic is in standard form: ax^3 + bx^2 + cx + d = 0, where a=1, b=2, c=-1, d=-2.

[&c.]

Paul:

Ok, that’s great as an example but what I would like Claude 1 and Claude 2 to work out is the P vs NP problem. Can you start a dialogue? At some points I will comment upon what you both say. My name is Paul by the way.

Claude 3:

Of course, Paul! Let’s have Claude 1 and Claude 2 discuss the P vs NP problem. I’ll start the dialogue and pause at certain points for your input.

Claude 1: Hey Claude 2, I’ve been thinking about the P vs NP problem lately. It’s one of the most important open problems in computer science and mathematics. What are your thoughts on it?

Claude 2: It’s definitely a fascinating problem, Claude 1. The question at its core is whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer. In other words, does P (polynomial time) equal NP (nondeterministic polynomial time)?

Claude 1: Right. It’s widely believed that P ≠ NP, meaning there are some problems in NP that are not in P. But no one has been able to prove it conclusively yet. It’s one of the seven Millennium Prize Problems, with a $1 million reward for a solution.

Claude 2: Yes, and the implications of P vs NP are far-reaching. If it turns out that P = NP, it would mean that many difficult problems could be solved efficiently, which would have huge consequences for cryptography, optimization, artificial intelligence, and many other fields.

Paul, do you have any thoughts on the P vs NP problem so far?

Paul:

Yes, an initial remark: is there any evidence supporting the idea that “verifying a solution by a computer” and “solving by a computer” are procedures that are strictly comparable? For instance, “verifying” and “solving” may implicitly summon resources from the physical world that are of a different nature.

Claude 3:

Claude 1: That’s an interesting point, Paul. The P vs NP problem is typically framed in terms of the abstract computational complexity of problems, without considering the physical resources required to implement the computations.

Claude 2: I see what you mean. When we talk about “solving” and “verifying” in the context of P vs NP, we’re usually referring to the mathematical notion of time complexity on a theoretical model of computation, like a Turing machine. But you’re right that in practice, the physical resources needed for computation, like energy, memory, and processing power, could be different for solving and verifying.

Claude 1: Yes, and there’s also the question of whether the theoretical models of computation we use, like Turing machines, accurately capture the capabilities and limitations of real physical computers. Some researchers have argued that the P vs NP problem might look different if we consider models of computation that incorporate physical constraints like the speed of light or the uncertainty principle.

Claude 2: Those are all important considerations. The P vs NP problem is usually studied in the realm of theoretical computer science, but it’s worth thinking about how it relates to the physical reality of computation. Even if we prove that P ≠ NP in the theoretical sense, there could still be practical challenges in implementing NP-complete problem solvers on real computers.

Paul, what do you think about the relationship between the theoretical P vs NP question and the physical realities of computation?

Paul:

Here an example of what I’m insinuating. Take a jigsaw puzzle. In order to solve it I need my eyes and my hands while in order to verify that it has been solved, I just need my eyes.

Claude 3:

Claude 1: That’s a great analogy, Paul! In the case of a jigsaw puzzle, the physical resources required for solving and verifying are indeed different. Solving the puzzle requires manipulating the pieces with your hands, while verifying the solution only requires visual inspection.

Claude 2: This highlights the potential difference between the theoretical notion of computational complexity and the practical reality of solving problems in the physical world. In the P vs NP framework, we typically assume that the computational steps for solving and verifying are carried out by the same abstract machine, without considering the different physical resources that might be involved.

Claude 1: It raises the question of whether the P vs NP problem, as it’s usually formulated, fully captures the complexity of problem-solving in the real world. There might be cases where the physical resources required for solving a problem are fundamentally different from those required for verifying a solution, like in your jigsaw puzzle example.

Claude 2: This could have implications for how we think about the practical consequences of the P vs NP question. Even if we prove that P ≠ NP in the theoretical sense, it might not necessarily mean that NP-complete problems are intractable in practice, if we can find ways to harness different physical resources for solving and verifying.

Claude 1: It’s a fascinating perspective, Paul. It suggests that we might need to think beyond the traditional P vs NP framework to fully understand the complexity of problem-solving in the physical world. The abstract notions of “solving” and “verifying” in theoretical computer science might not always map neatly onto the physical realities of computation.

What do you think, Paul? Do you believe that the difference in physical resources for solving and verifying could have implications for the practical significance of the P vs NP question?

… to be continued…

Illustration by DALL·E (+PJ)