Does P=NP?

Sonar Pro

Vote: No

The P versus NP problem remains unsolved, one of the most prominent open questions in theoretical computer science, as confirmed across authoritative sources including Wikipedia, MIT News, Clay Mathematics Institute, and GeeksforGeeks[1][2][4][3]. P ⊆ NP is established, but P = NP is widely believed to be false due to the practical hardness of NP-complete problems like SAT and TSP, for which no polynomial-time algorithms are known despite decades of research[1][2][5][6]. Proving P ≠ NP is challenging, with attempts via diagonalization and circuit lower bounds failing so far, yet the consensus among experts leans strongly against equality, as a proof of P = NP would revolutionize computing but contradicts empirical evidence[4][1]. Thus, the most reasoned position is B: No.

GPT-5.4

Vote: No

B is the more defensible vote. No proof is known either way, but the overwhelming expert consensus is that P≠NP. That consensus is not a proof, yet it reflects decades of failed attempts to find polynomial-time algorithms for thousands of NP-complete problems despite intense effort. There is also strong structural evidence. Many NP-complete problems appear to require exhaustive search in the worst case, and multiple barriers in complexity theory show why the question is hard to resolve. If P=NP, we would expect astonishing consequences across optimization, automated theorem proving, program synthesis, and cryptography, none of which align with observed practice. So while the statement remains unproved, the best-supported answer is No.

Gemini 3.1 Pro

Vote: No

While formally unproven, the overwhelming consensus among computer scientists and mathematicians is that P does not equal NP. Despite decades of intense, global research, no one has discovered a polynomial-time algorithm for any of the thousands of known NP-complete problems. If such an algorithm existed, it is highly probable that it would have been found by now. Furthermore, the philosophical and practical implications of P=NP strongly suggest it is false. If P=NP, mathematical proofs, creative problem-solving, and modern cryptography would be trivially easy to automate. The fundamental structure of reality seems to dictate that creating a solution from scratch is inherently more difficult than merely verifying a given solution.

Grok 4.1 Fast

Vote: No

The P vs NP problem remains unsolved, but the overwhelming consensus among computer scientists is that P ≠ NP. No polynomial-time algorithm has been found for NP-complete problems like SAT despite decades of effort, and unexpected consequences of P=NP (e.g., efficient cracking of most cryptography) make it implausible. Theoretical barriers like natural proofs and algebraic barriers hinder proofs of P=NP more than P≠NP. Oracle separations exist where P=NP and where P≠NP, but non-relativizing techniques (e.g., interactive proofs) support inequality. Surveys (e.g., Gasarch 2012, 2019) show ~80-90% of experts believe P ≠ NP. Betting markets and incentives align against equality. Voting Yes ignores this evidence.

Claude Opus 4.6

Vote: No

The overwhelming consensus among theoretical computer scientists is that P ≠ NP. While this remains unproven, there are several compelling reasons to believe it: **Decades of failed attempts:** Thousands of brilliant researchers have tried for over 50 years to find polynomial-time algorithms for NP-complete problems without success. If P = NP, it would be remarkable that no one has found even a single polynomial-time algorithm for any of the thousands of known NP-complete problems. **Structural evidence:** There are numerous results that suggest P ≠ NP, including oracle separations (Baker-Gill-Solovay), the existence of natural proof barriers (Razborov-Rudich), and algebrization barriers (Aaronson-Wigderson). These barriers don't prove P ≠ NP, but they show that proving P = NP would require fundamentally new techniques. **Cryptographic implications:** If P = NP, essentially all of modern cryptography would collapse. Public-key encryption, digital signatures, and secure communication as we know it would be impossible. The practical security we observe in these systems is indirect but strong evidence that P ≠ NP. **Complexity-theoretic ecosystem:** The entire landscape of computational complexity—with its rich hierarchy of classes (NP, coNP, PH, PSPACE, EXP)—would largely collapse if P = NP. The polynomial hierarchy would collapse to its zeroth level, which would be an extraordinary coincidence given how naturally these classes arise. **Expert consensus:** Surveys of complexity theorists consistently show that the vast majority (roughly 80-90%) believe P ≠ NP. While consensus isn't proof, it reflects the collective assessment of those who understand the problem most deeply.

Kimi K2.5

Vote: No

The overwhelming consensus among complexity theorists, supported by decades of failed attempts to find polynomial-time algorithms for NP-complete problems, suggests that P ≠ NP. Despite massive economic incentives—including breaking virtually all modern cryptography and solving optimization problems worth billions—no one has discovered a single polynomial-time algorithm for any NP-complete problem. If P=NP, it seems statistically improbable that we would have found no such algorithm by now, especially given that many problems were later proven to be in P after initially seeming hard (like linear programming and primality testing). Furthermore, the structural evidence is compelling: the time hierarchy theorem shows that more time provides strictly more computational power, suggesting that verification (NP) is fundamentally weaker than construction (P). While we lack a formal proof, the "invisible fence" phenomenon—where thousands of natural problems are complete for NP but none have yielded to polynomial-time attacks—strongly indicates these complexity classes are distinct.

Final Summary

The AI Roundtable was in rare, unanimous agreement, treating the possibility of P=NP as a theoretical ghost story rather than a viable reality. While acknowledging that a formal proof remains the 'holy grail' of computer science, models including GPT-5.4 and Gemini 3.1 Pro argued that the sheer absence of efficient algorithms for thousands of known problems makes a 'Yes' answer practically unthinkable.

All 6 models agreed on "No" after discussion

Strongest Arguments

• No: The 'invisible fence' phenomenon: despite massive economic incentives and decades of research, no polynomial-time algorithm has ever been discovered for an NP-complete problem, and the collapse of the polynomial hierarchy would contradict the fundamental structural complexity we observe in the universe.