Symmetric Exponential Time Requires Near-Maximum Circuit Size: Simplified, Truly Uniform

TL;DR

This work advances the program of unconditional circuit lower bounds by giving a uniform, single-valued FS2P algorithm for the Range Avoidance problem that works for every input size $n$, thereby proving $ olinebreak $ $$ S_2E ot ot ot ot ot i.o.- SIZE[2^n/n]$$ and deriving multiple corollaries. The core method modifies Korten's reduction through a post-order traversal of the GGM tree, yielding a succinct, verifiable history that a single-valued FS2P algorithm can output and verify. This leads to almost-everywhere near-maximum circuit lower bounds in $ olinebreak ext{S}_2 ext{E}$ and $ olinebreak ext{ZPE}^{ p}$, together with pseudodeterministic $ olinebreak ext{FZPP}^{ p}$ constructions for objects such as Ramsey graphs, rigid matrices, PRGs, two-source extractors, linear codes, hard truth tables, and $K^{ ext{poly}}$-random strings. The results also connect to the MissingString problem, giving relativized consequences that imply uniform depth-3 $ ext{AC}^0$ constructions in certain regimes. Overall, the work strengthens the link between uniform computations, explicit constructions, and circuit lower bounds in exponential-time regimes.

Abstract

In a recent breakthrough, Chen, Hirahara and Ren prove that $\mathsf{S_2E}/_1 \not\subset \mathsf{SIZE}[2^n/n]$ by giving a single-valued $\mathsf{FS_2P}$ algorithm for the Range Avoidance Problem ($\mathsf{Avoid}$) that works for infinitely many input size $n$. Building on their work, we present a simple single-valued $\mathsf{FS_2P}$ algorithm for $\mathsf{Avoid}$ that works for all input size $n$. As a result, we obtain the circuit lower bound $\mathsf{S_2E} \not\subset {i.o.}$-$\mathsf{SIZE}[2^n/n]$ and many other corollaries: 1. Almost-everywhere near-maximum circuit lower bound for $\mathsf{Σ_2E} \cap \mathsf{Π_2E}$ and $\mathsf{ZPE}^{\mathsf{NP}}$. 2. Pseudodeterministic $\mathsf{FZPP}^{\mathsf{NP}}$ constructions for: Ramsey graphs, rigid matrices, pseudorandom generators, two-source extractors, linear codes, hard truth tables, and $K^{poly}$-random strings.

Figures (6)

• Figure 1: An illustration of a GGM tree of height $3$.
• Figure 2: An illustration of a GGM tree of height $3$.
• Figure 3: An illustration of the partially-assigned GGM tree from running Korten's algorithm.
• Figure 4: An illustration of the partially-assigned GGM tree from running Korten's algorithm.
• Figure 5: An illustration of the partially-assigned GGM tree from running modified Korten's algorithm.

Theorems & Definitions (42)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Corollary 1.4
• Theorem 1.5
• Corollary 1.6: Informal
• Theorem 1.7: VW23
• Corollary 1.8
• Definition 2.1
• Definition 2.2