Information Redistribution Under Reductions in NP Search

TL;DR

This paper explores how reductions from structured P-matrix violation search to NP-complete problems reconfigure hidden witness information, affecting local inferability and information accessibility.

Contribution

It introduces a conceptual framework viewing reductions as mechanisms that redistribute and expose hidden witness information through representational expansion.

Findings

Reductions can expose globally encoded witness information to local inference.

Auxiliary variables and consistency structures may enhance local inferability.

Representational expansion preserves the need to recover the original witness information.

Abstract

Using reductions from structured P-matrix violation search to classical NP-complete formulations such as 3-SAT and Subset Sum, we examine the relationship between representational expansion, auxiliary variables, local inferability, and information accessibility. Rather than viewing reductions purely as computational transformations, we interpret them as mechanisms that redistribute hidden witness information across enlarged representations. From this perspective, reductions, gadgets, and auxiliary structures may expose globally encoded witness information to local propagation and inference, while search algorithms act as decoding procedures attempting to recover the original hidden witness. The resulting observations suggest that representational expansion may improve local inferability by introducing auxiliary variables and consistency structures, while preserving the need to recover the underlying witness information. This work is exploratory in nature and proposes a conceptual framework for understanding how reductions reshape information accessibility in NP search.