Scott Aaronson’s paper (link) explores a profound and fascinating aspect of one of the core issues in computational complexity theory — whether P equals NP — asking: could this question be formally independent of standard mathematical axiom systems like Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC)? That is, could it be a statement that can neither be proved nor disproved within such systems?
The P vs NP problem is one of the most famous unsolved questions in theoretical computer science. It touches on our fundamental understanding of computational feasibility. Simply put:
Obviously, all problems in P are also in NP (P ⊆ NP). The central question is whether there are NP problems strictly harder than P problems, that is, whether P ≠ NP.
If P = NP, many problems currently thought to be computationally intractable (such as finding optimal solutions to the traveling salesman problem or predicting protein folding) would become solvable in polynomial time, causing a revolution across cryptography, AI, operations research, and many other fields. However, despite decades of effort, no proof of either P = NP or P ≠ NP has been found.
In mathematical logic, a statement is formally independent of a given axiom system if it can neither be proved nor disproved using the axioms of that system. Famous examples include the independence of the parallel postulate in absolute geometry and the independence of the Continuum Hypothesis in ZFC (shown by Gödel and Cohen).
Exploring whether the P vs NP question is independent of ZFC means asking whether it is possible that the truth of P ≠ NP (or P = NP) lies beyond the reach of standard set theory.
An early attempt to study the independence of the P vs NP question involved the concept of oracle Turing machines, which can query an external “oracle” for answers to specific problems, treating each query as a single operation.
If there exists an oracle A such that PA = NPA (P equals NP relative to oracle A), and another oracle B such that PB ≠ NPB, then this suggests that any “absolute” proof or disproof of P = NP cannot rely solely on the basic definitions of Turing machines and polynomial-time constraints but must tap into deeper properties of computation models.
Early work by Cook, Hartmanis, and Simon (1976), and Baker, Gill, and Solovay (1975), constructed such oracles A and B. This added complexity to solving the P vs NP problem and suggested that more powerful methods beyond relativization would be needed.
Razborov and Rudich (1997) introduced the concept of natural proofs, aiming to explain why proving Boolean function circuit lower bounds (considered a key step to proving P ≠ NP) is so difficult. A natural proof framework has two main properties:
Razborov and Rudich showed that if standard pseudorandom generators exist (which is widely believed), then many known techniques for proving circuit lower bounds fall into this “natural” framework, and no natural proof can resolve questions like whether certain Boolean functions require exponential-size monotone circuits. This implies that proving P ≠ NP may require non-natural methods, or that our assumptions about pseudorandomness are wrong.
Aaronson points out that the natural proofs barrier may also apply to attempts to prove P = NP. If a proof method is natural, it could conflict with existing understandings of circuit complexity.
The paper reviews several researchers’ perspectives and related results:
The paper emphasizes a shift over time: early researchers treated P vs NP as a primarily logical problem possibly solvable by clever arguments. However, with the growth of circuit complexity, randomized algorithms, approximation algorithms, and other fields, the view has shifted — P vs NP seems more like a deep combinatorial problem involving intricate structures of computation.
This shift influences views on independence: if the P vs NP separation is due to a deep combinatorial structure hard to capture within standard set-theoretic axioms, then P ≠ NP could be independent of ZFC.
One major path toward proving P ≠ NP is proving that some NP problems cannot be solved by polynomial-size circuit families. However, despite significant progress in circuit complexity, we still lack techniques for proving super-polynomial lower bounds for general Boolean function families like SAT.
Known circuit lower bounds typically target restricted models (e.g., monotone circuits, constant-depth circuits) that are insufficient to solve general NP-complete problems. The natural proofs barrier further explains why overcoming these limitations is so hard.
Aaronson argues that these limitations hint that the answer to P vs NP may not be reachable with current mathematical tools, providing some support for the possibility of formal independence.
The paper explores a curious self-referential aspect: if P = NP, in principle, there would exist an efficient algorithm to prove P ≠ NP (if it were false). This sounds paradoxical but means that if P = NP, the task of proving P ≠ NP itself could belong to P (assuming the proving process can be done in polynomial time, which is not obvious).
This self-reference adds to the elusiveness of the P vs NP question and suggests that proving or disproving it may involve unusual logical structures.
Finally, the paper raises a deep philosophical question: are there explicit mathematical problems that, due to their inherent complexity or relation to the axiom systems we use, might never be answerable? Gödel’s incompleteness theorems already tell us that any sufficiently strong and consistent formal system must contain statements that are true but unprovable.
Aaronson suggests that P vs NP might be such a “natural” mathematical problem — its truth deeply rooted in the nature of computation but beyond the reach of current mathematical tools. If the truth of P ≠ NP relies on some as-yet-unknown combinatorial or structural principle not captured by ZFC, then P ≠ NP could indeed be independent.
Aaronson’s paper does not claim that P vs NP is definitively independent. He emphasizes that it remains an open research question. However, he argues that existing evidence — including oracle independence results, natural proof barriers, circuit lower bound limitations, and the strange nature of the problem itself — provides reasonable grounds to consider the possibility.
The conclusion is one of cautious openness. While most computer scientists believe P ≠ NP and a huge amount of research assumes it, we must not rule out the possibility that the problem involves deeper complexities at the level of formal systems. Future progress may require developing entirely new mathematical tools and proof techniques to ultimately unveil the mystery of P vs NP and determine whether it lies beyond the reach of our current foundations.
Summary of Key Points:
This paper reminds us that seemingly simple computational questions like P vs NP might involve deep foundations of mathematics and logic. Even if we are confident that P ≠ NP holds “in reality,” we must remain open to the possibility that, at the formal level, its truth may lie beyond our current theoretical reach. Future breakthroughs may require us to transcend existing frameworks and develop entirely new mathematical tools and perspectives.
