Solving the P/NP Problem under Intrinsic Uncertainty

Heisenberg’s uncertainty principle states that it is not possible to compute both the position and momentum of an electron with absolute certainty. However, this computational limitation, which is central to quantum mechanics, has no counterpart in theoretical computer science. Here, I will show that we can distinguish between the complexity classes P and NP when we consider intrinsic uncertainty in our computations, and take uncertainty about whether a bit belongs to the program code or machine input into account. Given intrinsic uncertainty, every output is uncertain, and computations become meaningful only in combination with a confidence level. In particular, it is impossible to compute solutions with absolute certainty as this requires infinite run-time. Considering intrinsic uncertainty, I will present a function that is in NP but not in P, and thus prove that P is a proper subset of NP. I will also show that all traditional hard decision problems have polynomial-time algorithms that provide solutions with confidence under uncertainty.

💡 Research Summary

The paper attempts to resolve the P versus NP question by importing the notion of “intrinsic uncertainty” from quantum physics, specifically the Heisenberg uncertainty principle, into computational complexity theory. The author argues that a bitstring stored on a Turing‑machine tape can be partitioned into two subsets, S₁ and S₂, without a clear distinction between which part encodes the program and which part encodes the input. This ambiguity is termed intrinsic uncertainty, and the author proposes to quantify it using entropy and a confidence measure proportional to the fraction of bits interpreted as code.

Under this framework every computation yields a result together with a confidence value; higher confidence corresponds to a larger presumed code segment. To increase confidence the author suggests recursively applying the same procedure—essentially running the machine on its own output (self‑computation T(T(S)))—which supposedly expands the code portion and thus raises confidence at the cost of larger tape usage.

The central claim is that, given such a model, one can construct a function that belongs to NP but not to P, thereby proving P ≠ NP. The construction relies on the assumption that at least one of the two possible interpretations implements an NP‑complete function f. The opposite interpretation is allowed to run arbitrarily long or even diverge, yet the existence of a correct interpretation guarantees a correct answer with some confidence. Because confidence can be amplified arbitrarily through self‑computation, the author concludes that the function is efficiently verifiable (NP) but cannot be solved deterministically in polynomial time (outside P).

Additionally, the paper claims that all traditionally hard decision problems admit polynomial‑time algorithms when uncertainty is tolerated, because the algorithm can always fall back on the higher‑confidence interpretation. No concrete algorithms, error‑rate analyses, or rigorous reductions are provided to substantiate this sweeping statement.

Critically, the paper conflates physical measurement uncertainty with computational indistinguishability of code versus data, a notion that has no established counterpart in standard Turing‑machine models. In classical complexity theory the program (the transition function) is fixed and finite; only the input length varies. Introducing code size as a variable contradicts the foundational definitions of P and NP. Moreover, the confidence metric is defined informally, and the relationship between confidence and correctness is never mathematically formalized. The self‑computation argument amounts to a trivial recursion without demonstrating any lower bound that would separate P from NP.

Consequently, while the idea of blending quantum‑inspired uncertainty with algorithmic analysis is conceptually intriguing, the paper lacks rigorous definitions, fails to provide a valid proof of a language in NP \ P, and does not meet the standards required for a credible resolution of the P versus NP problem.