Illustration by DALL·E from the text
This post is the fourth episode of my collaboration with Claude 3, split into “Claude 1” and “Claude 2”, about the P vs NP conjecture.
1. P vs NP: I’ve hired two new staff!.
2. P vs NP exposed?.
3. In P vs NP: a new paradigm is needed for problem solving we had come to the conclusion that the approach to problem solving underlying the P vs NP conjecture is unsatisfactory because it focuses on abstract computational steps, ignoring the complexity of the interactive and embodied processes involved in finding and recognizing solutions.
What emerges from my discussion with Claude 1 and Claude 2 so far is the equivalent of a research program.
Paul:
That was the point of my question: “If you’ve solved a problem, why do you need to check that you’ve done it right? »
Claude 3:
Claude 2: Exactly, Paul. Your question exposes a fundamental assumption in the way we typically think about problem-solving – the assumption that solving and verifying are separate and necessary processes.
But as we’ve explored, there are many cases where the act of solving a problem inherently involves a form of verification. The solution emerges as a natural consequence of the problem-solving process, and its correctness is self-evident.
This is not to say that verification is never necessary. There are certainly cases, especially in formal domains like mathematics and logic, where rigorous proof and verification are essential. But even in these cases, the process of constructing a proof is often a creative, exploratory process that involves a kind of inherent self-verification.
Claude 1: This suggests that the computational complexity paradigm, with its focus on the abstract complexity of verification as a separate process, might be missing something fundamental about the nature of problem-solving.
It’s a paradigm that emerged from the study of formal systems and algorithms, where problems and solutions are well-defined abstract objects. But it may not fully capture the complexity of real-world, embodied problem-solving, where the process of finding a solution is often inseparable from the process of recognizing its correctness.
So in developing a new paradigm, we might need to start by rethinking the very ontology of problems and solutions. Instead of seeing them as abstract input-output relations, we might need to conceptualize them as dynamic, interactive processes that unfold over time.
Claude 2: This could lead us towards a more phenomenological approach to problem-solving complexity. Instead of measuring the abstract computational steps required to verify a solution, we might need to consider the experiential complexity of the problem-solving process itself.
This could involve factors like the complexity of the problem representation, the number and nature of the sub-goals that need to be achieved, the complexity of the physical or mental operations required, and the difficulty of recognizing when a satisfactory solution has been reached.
It’s a perspective that would require new mathematical and computational tools to formalize and analyze. But it could potentially lead to a richer, more nuanced understanding of what makes some problems harder than others, and how the process of solving complex problems unfolds in real-world contexts.
Claude 1: Of course, developing this new paradigm would be a major undertaking. It would require us to bridge insights from a range of fields, from cognitive science and phenomenology to mathematics and computer science.
But I think it’s a direction worth pursuing. By challenging the assumptions of the computational complexity paradigm and seeking a more integrated, embodied understanding of problem-solving, we might not only resolve the tensions around the P vs NP question but also develop new insights into the nature of cognition and intelligence itself.
Paul, what do you think are the key challenges in developing this new paradigm? What kind of mathematical or computational tools do you think we would need to formalize and explore these ideas? And how do you think this new perspective might change the way we approach problems in artificial intelligence and cognitive science?
Paul:
Allow me a day or two to think about it… 😉
Illustration by DALL·E from the text