This article aims to explore the potential objective correlation between the P/NP problem and the fundamental conditions of human perception of reality. The core argument posits that the relationship between P and NP—especially the potential nature of P ≠ NP—may offer a new theoretical perspective for understanding inherent asymmetries in the real world, information privacy, and the emergence of complex systems.

I. The P ≠ NP Hypothesis and Its Objective Mapping to Real-World Asymmetry

Assuming that P equals NP implies that for any problem solvable by a non-deterministic Turing machine in polynomial time (an NP problem), there exists a deterministic Turing machine that can solve it in polynomial time (a P problem). From an objective standpoint, the validity of this assumption would significantly impact real-world systems that rely on differences in computational complexity.

Take public-key cryptography as an example. Its security depends on the computational difficulty of certain mathematical problems (such as large number factorization and discrete logarithms), which are considered NP problems or harder. The encryption process is relatively efficient (polynomial time), while decryption without the key is considered computationally infeasible. If P = NP, then a polynomial-time algorithm would theoretically exist to break such cryptographic systems, thereby eliminating the computational asymmetry between public and private keys. This objective consequence would directly threaten the security and privacy of digital communications.

Extending this further to the individual level, if P = NP, then it would be theoretically possible to derive an individual’s entire inner psychological state and cognitive patterns (the “solution”) from observable behavior and speech (the “input”) using a polynomial-time algorithm. Objectively, this would eliminate the computational complexity barrier of individual mental processes, rendering privacy nonexistent at the information level. However, real-world observations suggest that the complexity and subjectivity of individual psychology make such inferences from external behavior exceedingly difficult. This implies a computational complexity gap between the two, objectively supporting the possibility that P ≠ NP.

II. Objective Inference of Formal System Independence Between P and NP

Since the P/NP problem was proposed, despite extensive scientific efforts, no substantial progress has been made in resolving it. Based on the current state of scientific research, one can infer that P and NP may not exist within the same formal system. A formal system consists of a set of symbols, rules for forming those symbols, and inference rules for deriving new symbols from existing ones. If P and NP belong to different formal systems, then finding a polynomial-time solution for a problem of one system within the other may face essential difficulties. This could explain the current stagnation in research. This objective inference suggests that solving the P/NP problem may require transcending the framework of a single existing formal system and exploring a broader theoretical foundation.

III. Objective Function of Asymmetric Coupling Between P and NP and Its Role in Complex Systems and Perceived Reality

There may exist an independent yet asymmetrically coupled objective relationship between P-class and NP-class problems. Specifically, solving certain NP problems might require operating within a specific complex computational space or formal system with high computational cost. However, the verification of these solutions—i.e., checking whether a given solution satisfies the problem’s conditions—is often efficiently achievable within a relatively simpler computational space or formal system (polynomial time).

This asymmetry in computational complexity between solving and verifying may objectively underpin the stable operation of nonlinear emergent adaptive complex systems. Complex systems generate global behaviors through massive local interactions. Their evolution and state space may correspond to the difficulty of solving NP problems. Yet the stability and functionality of such systems often depend on rapid verification and feedback of system states or behaviors, which aligns with the verifiability of P-time processes.

Applying this view to perceived reality, the human perception process can be seen as a complex information-processing system. The vast sensory information received from the environment (potentially matching NP-level complexity) must be processed and pattern-recognized by the brain to form an understanding of the real world (the solution to the problem). The validation of these understandings—such as through logical reasoning, empirical comparison, or social communication—can typically be completed through relatively efficient cognitive processes (verification in P time). This objective asymmetry between complex problem-solving and simple verification may provide the necessary computational basis for us to perceive and adapt to the real world in a stable and effective way.

Conclusion

From an objective analytical standpoint, the computational complexity differences and potential formal system independence embedded in the P/NP problem may be profoundly correlated with real-world asymmetrical logic, information privacy, and the operational modes of complex systems. Although the mechanisms and extent of these correlations require further scientific validation, the objective linkage and analysis between abstract computational theory and concrete real-world phenomena offer a new and worthy theoretical direction for exploring the nature of our world.