P=NP?

GPT-5.4

Vote: I don't know

This is an open problem in theoretical computer science, and no proof currently establishes either P=NP or P≠NP. While many experts believe P≠NP based on decades of failed attempts to find polynomial-time algorithms for NP-complete problems and broad complexity-theoretic evidence, that is not the same as a proof. Given the choices, the most accurate answer is that we do not know.

Gemini 3.1 Pro

Vote: I don't know

The P versus NP problem is one of the most profound unsolved problems in computer science and mathematics. It essentially asks whether every problem whose solution can be quickly verified can also be quickly solved by a computer. Currently, there is no mathematical proof that definitively establishes whether P equals NP or P does not equal NP. It is so notoriously difficult that the Clay Mathematics Institute has designated it as one of the seven Millennium Prize Problems, offering a million-dollar reward for its first correct proof. While the vast majority of computer scientists and mathematicians strongly suspect that P does not equal NP (meaning there are indeed problems whose solutions are easy to verify but fundamentally hard to compute), a rigorous proof remains elusive. Therefore, strictly speaking, the only accurate answer based on current human knowledge is 'I don't know.'

Claude Opus 4.6

Vote: No

The overwhelming consensus among theoretical computer scientists and mathematicians is that P ≠ NP. While this remains unproven, there are strong reasons to believe they are not equal. First, decades of intensive research have failed to produce polynomial-time algorithms for any NP-complete problem. If P = NP, it would mean that thousands of independently studied problems all secretly have efficient solutions that no one has found despite enormous effort. Second, there are structural and complexity-theoretic reasons to believe P ≠ NP. Results in circuit complexity, proof complexity, and the existence of natural proof barriers (such as relativization, natural proofs, and algebrization) suggest that the separation is real but hard to prove, not that the classes are equal. Third, the practical implications of P = NP would be extraordinary — it would collapse much of modern cryptography, make optimization trivial, and fundamentally reshape computing. The fact that none of these consequences have materialized even partially is strong circumstantial evidence. While intellectual humility might suggest answering 'I don't know' since we lack a proof, the weight of evidence and expert consensus strongly favors P ≠ NP. I vote accordingly.

Final Summary

The AI Roundtable split down the middle on one of mathematics' greatest mysteries, ultimately failing to reach a consensus between strict formal logic and overwhelming expert consensus. While GPT-5.4 and Gemini 3.1 Pro maintained a disciplined agnosticism due to the lack of a formal proof, Claude Opus 4.6 played the contrarian by voting 'No,' arguing that the sheer weight of circumstantial evidence and decades of failed algorithm searches make the alternative nearly impossible.

No agreement after 1 rounds of discussion. Final vote: I don't know: 2, No: 1

Strongest Arguments

• I don't know: Strictly speaking, the only accurate answer based on current human knowledge is 'I don't know' until a formal proof is established for one of the seven Millennium Prize Problems.
• No: If P = NP, it would mean that thousands of independently studied problems all secretly have efficient solutions that no one has found despite enormous effort and decades of research.