Beyond the Static Barrier for Ordinary Dynamic Approximate Membership

2026-08-23

AI Summary

本文证明了静态和普通动态近似成员资格在空间使用上的严格分离，通过引入一种新的传输引理和联合后验KL边界方法解决了此问题。

Abstract

We prove a strict space separation between static and ordinary dynamic approximate membership at every fixed error rate. For each fixed $\varepsilon\in(0,1)$, a capacity-$n$ ordinary dynamic filter over a universe of size $u$, with zero false negatives, pointwise false-positive probability at most $\varepsilon$, arbitrary history dependence, a free public random tape, and at most $H$ bits of persistent state, satisfies \[ H\ge \bigl(\log_2(1/\varepsilon)+a_\varepsilon^{\rm c}\bigr)n-o(n), \] under only $u/n\to\infty$. The constant $a_\varepsilon^{\rm c}$ is an explicit variational threshold obtained by preserving the dependence between the parent accepted mass and the successor reservoir. The structural step is a common-continuation transport lemma. A joint posterior KL bound gives a branch-specific survivor support; the same legal delete--insert word transports that support to one successor state, forcing an accepted reservoir. We then keep the parent outside mass $1-X$ in the conditional-entropy argument instead of replacing it by $1-\varepsilon$. This yields a two-variable analytic envelope, with no selected thresholds, dyadic witnesses, or numerical assumptions.

Problem

• static
• dynamic approximate membership
• space separation
• error rate

Innovation

• dynamic approximate membership
• space separation
• variational threshold
• common-continuation transport lemma
• joint posterior KL bound