Exposure Time and the Structural Trilemma of P versus NP
Authors/Creators
Description
Description
This repository contains the full version of RP21, a structural study of the P versus NP problem based on the notion of exposure time.
Rather than attempting to prove a separation between P and NP or to establish new super-polynomial lower bounds, this work identifies a reduction-preserving structural obstruction that explains why deterministic search procedures cannot be significantly compressed.
The central contribution is the formalization of exposure time, a measure of the minimal temporal depth required for decisive constraints to become observable in a search process.
We show that any deterministic transcript of length L can certify the elimination of at most O(L) candidates, even under weak and unconditional regularity assumptions. This limitation is captured by the Exposure Time No-Mass-Determination Theorem, whose proof relies on exact counting arguments and communication-complexity techniques, without assuming independence, cryptographic hardness, or model-specific gadgets.
The results are invariant under polynomial-time reductions and therefore describe a structural barrier rather than a problem-specific artifact. While the framework is motivated by concrete structured examples, such as layered CNF families, the statements and proofs are formulated independently of any particular NP-complete problem.
This paper should be viewed as a structural companion to more technical works developing quantitative lower bounds. Its purpose is to clarify the obstruction landscape surrounding P versus NP, explain the persistent failure of compression-based techniques, and delineate what any successful separation proof must ultimately overcome.
This version is released to enable open verification, discussion, and reuse by the theoretical computer science community.
Abstract
We propose a unifying structural framework for the P versus NP problem based on the notion of exposure time: the minimal temporal depth required for decisive constraints to become observable in a search process.
Rather than establishing new lower bounds or separation results, this paper articulates a necessary-condition perspective on why any polynomial-time decision procedure must confront irreducible exploration over time.
Our framework identifies a structural trilemma underlying P versus NP: any proof of separation must necessarily resolve at least one of three obstacles—(i) time-structured emergence of constraints, (ii) limits of certified elimination per transcript, or (iii) reduction-preserving non-compressibility of search.
We show that these obstacles are tightly interconnected and cannot be bypassed simultaneously without a genuinely new principle.
The role of exposure time is to formalize the intuition that certain constraints are not statically detectable but arise only through temporal composition, independent of specific problem encodings.
This perspective clarifies why known techniques—relativization, algebrization, and natural proofs—systematically fail to address the core difficulty of P versus NP.
Importantly, this paper does not claim new complexity-theoretic lower bounds.
Concrete quantitative results and fully formalized proofs concerning structural non-compressibility of search are developed separately in a companion work.
Here, we aim instead to provide a conceptual map of the obstruction landscape, offering a coherent target for future technical advances.
Keywords
P vs NP computational complexity search complexity exposure time structural lower bounds
communication complexity non-compressibility reduction preservation proof complexity
theoretical computer science
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【RP21】P vs NP PAPER Vol3.pdf
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- Alternative title
- RP No.21