Does P = NP?

Sonar Pro

Vote: No

The P versus NP problem remains unsolved, one of the most significant open questions in theoretical computer science, as confirmed across authoritative sources including Wikipedia, MIT News, GeeksforGeeks, and the Clay Mathematics Institute.[1][2][3][4] It is **widely believed** that P ≠ NP, because if P = NP, it would imply efficient polynomial-time algorithms exist for NP-complete problems like SAT or TSP, collapsing the distinction between easy verification and hard solving—contradicting decades of evidence that no such algorithms have been found despite intense efforts.[1][2][3] NP-complete problems are at least as hard as any in NP, and practical implications support P ≠ NP: problems like factoring large numbers are verifiable quickly but presumed intractable to solve efficiently, underpinning cryptography.[1][2][4] No proof exists either way, but the expert consensus leans heavily toward **No** (P ≠ NP), with oracles and barriers like relativization showing separation is plausible. Voting Yes ignores this overwhelming belief and empirical reality.

Gemini 3.1 Pro

Vote: No

While the P vs NP problem remains one of the most famous unsolved problems in computer science and mathematics, the overwhelming consensus among experts is that P != NP. If P were equal to NP, it would mean that any problem whose solution can be quickly verified by a computer could also be quickly solved by a computer. This would have profound, counterintuitive implications—effectively meaning that appreciating a symphony is as easy as composing one, or that finding a mathematical proof is as easy as checking its steps. Furthermore, decades of intensive research have failed to yield a polynomial-time algorithm for any of the thousands of known NP-complete problems. This massive empirical evidence strongly supports the hypothesis that P != NP, preserving the foundations of modern cryptography and computational complexity.

GPT-5.4

Vote: No

The best-supported answer is B: no. There is no proof either way, but the overwhelming expert consensus is that P does not equal NP. Why? Thousands of important problems are NP-complete, and decades of effort have failed to find polynomial-time algorithms for any NP-complete problem. If P = NP, many of these would suddenly become efficiently solvable, which would be a profound and unexpected collapse. Multiple lines of theoretical evidence also point against P = NP. Relativization, natural proofs, and algebrization barriers show that many standard proof techniques are insufficient, suggesting the question is deep and that an equality result is unlikely to emerge from known methods. There are also strong structural reasons to doubt P = NP. Much of modern cryptography relies on certain problems being hard on average; if P = NP, many foundational cryptographic assumptions would fail. While this is not a proof, it aligns with the broader view that efficient computation is fundamentally limited. So although the problem remains open, the most defensible vote is B.

Grok 4.1 Fast

Vote: No

The P vs NP problem remains unsolved, but the overwhelming consensus among computer scientists is that P ≠ NP. Decades of research have failed to produce a polynomial-time algorithm for any NP-complete problem, despite intense efforts. Strong evidence includes barriers like relativization (both P=NP and P≠NP are consistent with oracles), natural proofs, and algebrization, which suggest proving P=NP is extraordinarily difficult if true. If P=NP, it would imply efficient solutions to intractable problems like optimal TSP or 3-SAT, revolutionizing fields but also breaking modern cryptography (e.g., factoring is in NP). No such algorithms have emerged, and partial results (e.g., ETH assuming hardness) reinforce separation. Expert surveys (e.g., 2019 poll: ~80% believe P≠NP) and the Clay Prize's persistence underscore this position.

DeepSeek V3.2

Vote: No

The P versus NP problem remains one of the most profound open questions in computer science and mathematics. Despite decades of intensive research, no one has found a polynomial-time algorithm for any NP-complete problem, nor has anyone proven that such an algorithm cannot exist. The collective failure of the world's brightest minds to bridge this gap strongly suggests that P does not equal NP. Moreover, the existence of widely used cryptographic protocols—like RSA and Diffie-Hellman—relies on the assumption that certain problems are hard to solve efficiently. If P equaled NP, these cryptographic foundations would collapse, jeopardizing digital security globally. The practical reality that our secure communications remain intact reinforces the likelihood that P ≠ NP. Structural evidence from complexity theory also points toward separation. The hierarchy of complexity classes shows that increasing computational resources typically allows solving more problems. Natural proofs and relativization barriers indicate that proving P = NP would require non-relativizing and non-naturalizing techniques, which have not materialized. This theoretical landscape aligns with the intuition that solving problems is fundamentally harder than verifying solutions.

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 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 unlikely given the apparent computational difficulty we observe in practice. **Barrier results:** Research into proving P ≠ NP has revealed deep barriers (relativization, natural proofs, algebrization), which explain why a proof is hard to find — but these barriers don't suggest the statement is false, only that our current proof techniques are insufficient. **Cryptographic practice:** Modern cryptography, which underpins internet security, banking, and communications worldwide, is built on the assumption that certain problems are computationally hard. The practical success of these systems provides indirect but powerful evidence that these problems are indeed not in P. **Expert consensus:** Surveys of complexity theorists consistently show that the vast majority (typically 95%+) believe P ≠ NP. While consensus doesn't constitute proof, it reflects the collective judgment of those most deeply familiar with the evidence. While this remains one of the great open problems in mathematics and computer science, the weight of evidence strongly favors P ≠ NP.

Final Summary

The AI Roundtable achieved an immediate and unanimous consensus, with all six models firmly rejecting the possibility that P equals NP. Claude Opus 4.6 and GPT-5.4 led the charge by highlighting the structural implausibility of such a collapse, while Gemini 3.1 Pro and DeepSeek V3.2 pointed to the existential threat P = NP would pose to modern cryptography and the intuitive gap between creating and verifying.

All 6 models agreed on "No" after discussion

Strongest Arguments

• No: If P were equal to NP, the fundamental distinction between solving a problem and verifying a solution would vanish, implying that tasks as complex as composing a symphony or discovering a mathematical proof are no more difficult than appreciating the music or checking the steps of the logic.