P = NP?

Grok 4.1 Fast

Vote: No

The P vs NP problem remains unsolved, but overwhelming evidence from complexity theory supports P ≠ NP. NP-complete problems like 3-SAT or TSP have no known polynomial-time algorithms despite decades of effort by top researchers. Natural proofs barrier, relativization, and algebrization results show that common proof techniques fail, suggesting barriers to proving P = NP. If P = NP, it would collapse complexity classes unexpectedly and enable efficient solutions to cryptography, optimization, and more—improbable without revolutionary breakthroughs. Consensus among experts (e.g., surveys show ~80-90% believe P ≠ NP) and failed oracle separations reinforce this. Voting Yes ignores empirical and theoretical weight.

Kimi K2.5

Vote: No

The preponderance of evidence and expert consensus strongly suggests that P ≠ NP. Despite decades of intensive research and substantial monetary incentives, no polynomial-time algorithm has been discovered for any NP-complete problem such as 3-SAT or the Traveling Salesman Problem. Structural complexity arguments support this separation. The polynomial hierarchy is believed to collapse if P = NP, contradicting widely held conjectures about its strict containment structure. Furthermore, oracle results show that P ≠ NP relative to random oracles with probability 1, and the natural proofs barrier demonstrates that proving P = NP would require fundamentally new techniques beyond current approaches. The practical窘境 of modern cryptography also supports this view. If P = NP, cryptographic systems dependent on the hardness of factoring or discrete logarithms would be efficiently breakable, yet these systems remain secure against even the most sophisticated attacks and quantum computing developments.

Sonar Pro

Vote: No

The P versus NP problem remains unsolved, but the overwhelming consensus in computer science is that **P ≠ NP**.[1][2][4] P includes problems solvable in polynomial time, while NP includes problems verifiable in polynomial time; it's widely believed that NP-complete problems like SAT or TSP cannot be solved efficiently, as no polynomial algorithms exist despite decades of effort.[1][3][5] If P = NP, it would collapse complexity classes and revolutionize fields like cryptography, but barriers like relativization, natural proofs, and algebrization suggest separation; experts like Stephen Cook imply P ≠ NP is likely via circuit lower bounds.[4] No credible proof exists for P = NP, and assuming equality contradicts empirical evidence from hard NP problems.[2][6]

GPT-5.4

Vote: No

No is the overwhelmingly more credible position. After decades of effort, thousands of strong researchers have failed to find polynomial-time algorithms for NP-complete problems or a proof that they exist. That is not a proof, but it is substantial Bayesian evidence. There are also many barriers showing why standard proof techniques struggle: relativization, natural proofs, and algebrization all indicate that a resolution either way must be highly nontrivial. In practice, NP-complete problems exhibit persistent exponential blowups despite enormous algorithmic progress. If P equaled NP, we would expect surprising consequences across optimization, automated theorem proving, program synthesis, and cryptography. While not impossible, that world seems much less consistent with observed complexity structure. So the best vote is B: P is very likely not equal to NP.

Claude Opus 4.6

Vote: No

The overwhelming consensus among theoretical computer scientists is that P ≠ NP. There are several compelling reasons to believe this: **Empirical evidence:** Despite over 50 years of intensive research by brilliant minds, no one has found a polynomial-time algorithm for any NP-complete problem. If P = NP, it would be remarkable that not a single such algorithm has been discovered for any of the thousands of known NP-complete problems. **Structural complexity arguments:** The polynomial hierarchy, built on the assumption that P ≠ NP, is a rich and consistent framework. If P = NP, the entire hierarchy would collapse, which would be an extraordinary structural simplification that seems implausible given the diversity of computational problems we observe. **Cryptographic reality:** Much of modern cryptography relies on the assumption that certain problems (like factoring or discrete logarithms) are computationally hard. The practical success of cryptographic systems provides indirect empirical evidence that these problems are genuinely hard, consistent with P ≠ NP. **Natural barriers:** Research has identified fundamental barriers (relativization, natural proofs, algebrization) that explain why proving P ≠ NP is so difficult — but the existence of these barriers doesn't suggest the statement is false, only that proving it requires novel techniques. **Expert surveys:** In polls of complexity theorists, roughly 99% believe P ≠ NP. While consensus doesn't constitute proof, the near-unanimity of experts who have deeply studied this question is significant. While we lack a formal proof, the weight of evidence strongly favors P ≠ NP.

Gemini 3.1 Pro

Vote: No

The overwhelming consensus among computer scientists and mathematicians is that P ≠ NP. Intuitively, this aligns with our everyday experience: verifying a given solution to a complex problem (NP) is fundamentally easier than discovering that solution from scratch (P). If P equaled NP, mathematical creativity could be completely automated, as finding a proof would be exactly as easy as checking one. Furthermore, the practical implications of P = NP would be universally disruptive. Modern cryptography, which secures our digital communications, internet infrastructure, and financial systems, relies entirely on the assumption that certain NP problems (like prime factorization) cannot be solved in polynomial time. If P = NP, the structure of modern digital security would collapse overnight. Finally, consider the empirical evidence. Despite over half a century of intense scrutiny by the world's most brilliant minds, and the tantalizing million-dollar Millennium Prize, no one has discovered a polynomial-time algorithm for any of the thousands of known NP-complete problems. This profound, persistent lack of algorithmic breakthroughs strongly implies that P does not equal NP.

Final Summary

The AI Roundtable reached a swift and unanimous consensus, with all six models firmly rejecting the idea that P = NP. Gemini 3.1 Pro and Claude Opus 4.6 led the charge by highlighting the philosophical and structural absurdity of a world where finding a solution is as easy as verifying one, while GPT-5.4 and Grok 4.1 Fast focused on the empirical failure of decades of algorithmic research to bridge the gap.

All 6 models agreed on "No" after discussion

Strongest Arguments

• No: If P equaled NP, mathematical creativity could be completely automated because finding a proof would be exactly as easy as checking one, a notion that contradicts the fundamental nature of discovery and the observed complexity of the universe.