[Paper Review] Solving the P/NP Problem under Intrinsic Uncertainty

Stefan Jaeger|ArXiv.org|Nov 4, 2008

Numerical Methods and Algorithms9 references3 citations

TL;DR

This paper proposes a novel resolution to the P/NP problem by introducing intrinsic uncertainty into computational models, treating program code and input as indistinguishable at the bit level. By modeling uncertainty via entropy and confidence thresholds, it proves that P ≠ NP by constructing a function in NP but not in P, while showing all traditional hard decision problems remain in P due to polynomial-time approximation under uncertainty.

ABSTRACT

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.

Motivation & Objective

• To resolve the P/NP problem by introducing intrinsic uncertainty into computational models.
• To challenge the classical assumption of hard decisions in Turing machines by embedding uncertainty from the start.
• To demonstrate that P ≠ NP by constructing a function computable in NP but not in P under intrinsic uncertainty.
• To show that all traditional hard decision problems remain in P when solved with confidence thresholds under uncertainty.
• To establish a new complexity hierarchy where soft decisions (with entropy thresholds) distinguish P and NP, while hard decisions remain in P.

Proposed method

• Introduces an intrinsic uncertainty principle where bits cannot be definitively classified as program code or input, leading to binary ambiguity.
• Models this uncertainty using standard entropy, with uncertainty measured as H = -p log p - (1-p) log(1-p), where p is the probability of a bit being code.
• Defines a confidence level via a sigmoid-like function o' = 1 / 2^L(S), linking input size L(S) to observational precision.
• Uses self-computation to generate confidence: T_e'(L(S))^NP(S) = T_e'(L(T_e'(L(S))^NP(S)))^NP(T_e'(L(S))^NP(S)), creating recursive uncertainty.
• Applies entropy thresholds to distinguish soft decisions (with confidence) from hard decisions (no uncertainty), leading to a modified complexity hierarchy.
• Proves that any implementation of a self-computing Turing machine requires exponential time or space due to uncertainty, making it impossible to be in P.

Experimental results

Research questions

• RQ1 Can the P/NP problem be resolved by introducing intrinsic uncertainty into the computational model?
• RQ2 Does the indistinguishability of program code and input lead to a fundamental computational limitation that separates P and NP?
• RQ3 Can a function be constructed that is verifiable in polynomial time (NP) but not computable in polynomial time (not in P) under intrinsic uncertainty?
• RQ4 Do traditional hard decision problems remain in P when computed with confidence thresholds under uncertainty?
• RQ5 How does the uncertainty principle from quantum mechanics analogously constrain classical computation when applied to Turing machines?

Key findings

• A function exists that is in NP but not in P under intrinsic uncertainty, proving P ≠ NP.
• Self-computation under uncertainty requires exponential time or space, making such computations impossible in polynomial time.
• All traditional hard decision problems, including NP-complete problems, are in P when solved with confidence thresholds under uncertainty.
• The confidence level o' = 1 / 2^L(S) ensures that input occupies at most 1% of the input to a self-computation, leading to exponential complexity.
• The intrinsic uncertainty principle leads to a modified complexity hierarchy where P is a proper subset of NP, with soft decisions distinguishing the classes.
• The paper concludes that P ≠ NP, but all traditional decision problems are solvable in polynomial time with confidence, not certainty.