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An MDL-Style Cost Functional KC, Distribution-Preserving Reductions ($A2^d$), and an $AC^0$+log Lower Bound for 3SAT via Balanced 3XOR

Marko Lela

TL;DR

A model-agnostic MDL-style cost functional $K_C$ for resource-bounded classifiers is introduced and the first explicit $K_C$-reading of such size-aware bounds under a $\delta=0$ measure-preserving reduction in small-depth circuit lower bounds is yielded.

Abstract

We introduce a model-agnostic MDL-style cost functional $K_C$ for resource-bounded classifiers and prove a Total-Variation stable reduction lemma ($A2^d$) for distribution-preserving many-to-one reductions. On a balanced distribution of random 3XOR instances (with co-rank $t'=Θ(n)$) we obtain a size-aware lower bound against P-uniform AC^0+log models: $\Pr[M=χ] \le \frac{1}{2} + s(N)\exp(-α_d m^{c/d})$ with an absolute $c \in (0,1)$ (e.g., $c=1/3$ gives $β_d=1/(3d)$). A deterministic, injective 3XOR->3SAT translation (four 3-clauses per XOR, no auxiliaries) is $δ=0$ measure-preserving on its image window; by $A2^d$ the bound transfers to 3SAT. This yields, to our knowledge, the first explicit $K_C$-reading of such size-aware bounds under a $δ=0$ measure-preserving reduction in small-depth circuit lower bounds. We provide artifacts (generator -> DIMACS -> verification) with match-rate 1.0.

An MDL-Style Cost Functional KC, Distribution-Preserving Reductions ($A2^d$), and an $AC^0$+log Lower Bound for 3SAT via Balanced 3XOR

TL;DR

A model-agnostic MDL-style cost functional for resource-bounded classifiers is introduced and the first explicit -reading of such size-aware bounds under a measure-preserving reduction in small-depth circuit lower bounds is yielded.

Abstract

We introduce a model-agnostic MDL-style cost functional for resource-bounded classifiers and prove a Total-Variation stable reduction lemma () for distribution-preserving many-to-one reductions. On a balanced distribution of random 3XOR instances (with co-rank ) we obtain a size-aware lower bound against P-uniform AC^0+log models: with an absolute (e.g., gives ). A deterministic, injective 3XOR->3SAT translation (four 3-clauses per XOR, no auxiliaries) is measure-preserving on its image window; by the bound transfers to 3SAT. This yields, to our knowledge, the first explicit -reading of such size-aware bounds under a measure-preserving reduction in small-depth circuit lower bounds. We provide artifacts (generator -> DIMACS -> verification) with match-rate 1.0.

Paper Structure

This paper contains 39 sections, 7 theorems, 14 equations, 3 tables.

Table of Contents

  1. Keywords.
  2. Introduction
  3. Conventions.
  4. IECZ Core: KC, Measure, and A2d (brief)
  5. General form on distributions.
  6. Advantage notation.
  7. Advice model and uniformity.
  8. Calibration: $K^{2\mathrm{SAT}}_{C_{\mathrm{alg}}}(n,\alpha,0)=O(1)$.
  9. TV contraction (1-Lipschitz).
  10. Reduction robustness (A2d).
  11. Overhead note.
  12. Balanced 3XOR (structure, balance) and size-aware AC0+log hardness
  13. Label balance (theory vs. artifacts).
  14. Sampling of $u$ (distributional balance).
  15. Co-rank scaling at $m=(1+\gamma)n$.
  16. ...and 24 more sections

Key Result

Lemma 1

For any deterministic transducer $T$ and distributions $\mu,\nu$ on the domain, $\|\;T_\#\mu - T_\#\nu\;\|_{\mathrm{TV}} \le \|\;\mu-\nu\;\|_{\mathrm{TV}}$.

Theorems & Definitions (10)

  • Lemma 1: Deterministic push-forward contracts TV
  • proof : Proof of A2d
  • Lemma 2: Random projection to a surviving parity
  • proof : Proof of Lemma \ref{['lem:proj-one-parity']}
  • Theorem 1: Size-aware AC0+log hardness on Balanced 3XOR
  • Lemma 3: Injectivity and $\delta{=}0$
  • proof
  • Theorem 2: Size-aware AC0+log lower bound for 3SAT in the balanced window
  • Corollary 1: Reading as a KC statement
  • Lemma 4: Permutation concentration for free support