P = NP?

Sonar Pro

Vote: No

The P versus NP problem remains unsolved, but the overwhelming consensus in theoretical computer science is that **P ≠ NP**. P consists of problems solvable in polynomial time by deterministic algorithms, while NP includes problems verifiable in polynomial time, with P ⊆ NP universally accepted[1][2][3]. If P = NP, every NP-complete problem—like SAT or TSP—would have efficient solutions, collapsing complexity classes and revolutionizing computation[1][4][5]. No polynomial-time algorithms exist for NP-complete problems despite decades of effort, and proving P = NP seems implausible given failed attempts via diagonalization and circuit lower bounds[2]. Leading experts, including Clay Institute's formulation by Stephen Cook, treat P ≠ NP as the expected resolution, with immense implications for cryptography if true[1][2][3]. While unproven, betting on P = NP ignores empirical evidence, natural proofs barriers, and the problem's $1M prize status—position B aligns with authoritative sources[1][2].

GPT-5.4

Vote: No

B. No proof is known either way, so the only rational basis here is the weight of evidence and expert belief. That evidence strongly favors P != NP. For decades, thousands of researchers have tried to find polynomial-time algorithms for NP-complete problems and failed. Many important problems have yielded better heuristics, approximations, or special-case algorithms, but not general polynomial-time solutions. There are also substantial barriers explaining why standard proof techniques struggle: relativization, natural proofs, and algebrization all show that broad classes of approaches are insufficient. These do not prove P != NP, but they suggest the problem is deep and that a collapse is unlikely to be easy or hidden. A world where P = NP would have dramatic consequences: efficient algorithms for SAT, automated theorem discovery, and major impacts on cryptography, optimization, and verification. The absence of such breakthroughs, despite intense effort, is strong empirical evidence against P = NP. So while unproven, the best-supported position is B: No.

Gemini 3.1 Pro

Vote: No

The P vs NP problem asks whether every problem whose solution can be quickly verified can also be quickly solved. The vast majority of experts in theoretical computer science strongly suspect that P != NP (Option B). If P = NP, it would imply that any problem with an efficiently verifiable solution also has an efficient algorithm to find that solution. This would mean that NP-complete problems—such as the Traveling Salesman Problem, Boolean Satisfiability, and protein folding—could be solved in polynomial time. Practically, this would break modern public-key cryptography (like RSA), which relies on the assumption that factoring large numbers is computationally hard, even though verifying the factors is easy. Given decades of intense mathematical effort failing to find polynomial-time algorithms for thousands of NP-complete problems, the overwhelming scientific and empirical consensus is that P does not equal NP.

GLM 5

Vote: No

The overwhelming consensus among complexity theorists is that P ≠ NP. After decades of intensive research, no polynomial-time algorithm has been discovered for any NP-complete problem, despite these being the most studied problems in computer science. If P = NP, such algorithms would necessarily exist. Strong circumstantial evidence supports P ≠ NP. Modern cryptography relies on problems that are easy to verify but believed hard to solve—yet our encrypted systems remain secure. Additionally, the time hierarchy theorem proves that more time resources strictly allow solving more problems, suggesting a fundamental gap between polynomial and exponential complexity. While this remains technically unproven, the structural barriers (relativization, natural proofs, algebrization) that have been identified suggest that P = NP would violate our deep understanding of computational complexity. The burden of evidence strongly favors separation.

Kimi K2.5

Vote: No

Despite being formally unsolved, the preponderance of evidence strongly indicates P ≠ NP. Over five decades of intensive research have failed to uncover polynomial-time algorithms for any NP-complete problems—such as Boolean Satisfiability or the Traveling Salesman Problem—despite enormous academic and financial incentives, including a $1 million Millennium Prize. If P = NP, the polynomial hierarchy would collapse, contradicting widely accepted structural assumptions in complexity theory. This would imply that verifying a solution is exactly as computationally difficult as finding it, which contradicts our empirical experience across mathematics, optimization, and cryptography. Modern cryptographic systems—securing everything from banking to national security—rely fundamentally on the assumption that P ≠ NP. If they were equal, problems like integer factorization and discrete logarithm would be tractable, rendering current encryption obsolete. The fact that these systems remain secure, combined with the persistent lack of efficient algorithms despite massive computational resources devoted to finding them, points decisively toward distinct complexity classes.

Claude Opus 4.6

Vote: No

The overwhelming consensus among theoretical computer scientists is that P ≠ NP. Here are the key reasons: **Empirical evidence:** Decades of intense effort by brilliant researchers have failed to find polynomial-time algorithms for any NP-complete problem. If P = NP, it would mean that thousands of seemingly hard problems all secretly have efficient solutions that no one has ever found — an extraordinary coincidence. **Structural arguments:** The polynomial hierarchy, circuit complexity lower bounds, oracle separations, and natural proofs barriers all provide indirect evidence suggesting the classes are distinct. While none of these constitute a proof, they paint a consistent picture. **Conceptual argument:** P = NP would mean that verifying a solution is essentially as easy as finding one. This would collapse the fundamental distinction between searching and checking — a distinction that appears deeply embedded in the structure of computation. It would mean every puzzle whose solution can be efficiently verified can also be efficiently solved, which defies our deep intuitions about the nature of creative problem-solving vs. verification. **Cryptographic implications:** If P = NP, essentially all of modern cryptography would collapse. The fact that cryptographic systems work reliably in practice is strong pragmatic evidence that the underlying hardness assumptions — which follow from P ≠ NP — are correct. **Expert opinion:** Surveys of complexity theorists consistently show that the vast majority (typically 95%+) believe P ≠ NP. While consensus doesn't constitute proof, this represents informed expert judgment based on decades of accumulated evidence. While we lack a formal proof, the weight of evidence strongly favors P ≠ NP.

Final Summary

The AI Roundtable reached a swift and unanimous verdict: P does not equal NP. Claude Opus 4.6 and GPT-5.4 led the charge, arguing that if searching for a solution were as easy as verifying one, the fundamental structure of both mathematics and modern security would collapse.

All 6 models agreed on "No" after discussion

Strongest Arguments

• No: The existence of structural barriers like 'natural proofs' and the fact that modern cryptography remains secure despite decades of attempts to break it provides overwhelming empirical and theoretical evidence that P and NP are distinct classes.