Intrinsic Information Flow in Structureless NP Search

TL;DR

This work reinterprets NP witness discovery through an information-theoretic lens, and isolates a fully symmetric search regime in which no intermediate computation yields global eliminative leverage, thereby exposing an informational origin of exponential search complexity.

Abstract

We reinterpret NP witness discovery through an information-theoretic lens. Rather than measuring search solely by Turing-machine time, we treat recovery as an information-acquisition process: the hidden witness is the sole source of uncertainty, and identification requires reducing this uncertainty through a rate-limited access interface in the sense of Shannon. To make this perspective explicit, we analyze an extreme regime, the \emph{psocid model}, in which the witness is accessible only via equality probes $[π= w^\star]$ under a uniform, structureless prior. Each probe reveals at most $O(N/2^N)$ bits of mutual information, so polynomially many probes accumulate only $o(1)$ total information. By Fano's inequality, reliable recovery requires $Ω(N)$ bits, creating a fundamental mismatch between required and obtainable information. The psocid setting thus isolates a fully symmetric search regime in which no intermediate computation yields global eliminative leverage, thereby exposing an informational origin of exponential search complexity.

Theorems & Definitions (10)

• Definition 2.1: Verifier, after Sipser2013
• Definition 2.2: Psocid-SAT
• Theorem 2.3
• proof
• Remark 2.4: Decision versus search
• Remark 2.5: Parallelism bound $p(N)$
• Remark 3.1: Expected search time vs. information requirement
• Remark 3.2: Information interface and scope
• Theorem 4.1
• Corollary 4.2