GANs promise indistinguishability, logic explains it. We put the two on a budget: a discriminator that can only ``see'' up to a logical depth $k$, and a generator that must look correct to that bounded observer. \textbf{LOGAN} (LOGical GANs) casts the discriminator as a depth-$k$ Ehrenfeucht--Fra\"iss\'e (EF) \emph{Opponent} that searches for small, legible faults (odd cycles, nonplanar crossings, directed bridges), while the generator plays \emph{Builder}, producing samples that admit a $k$-round matching to a target theory $T$. We ship a minimal toolkit -- an EF-probe simulator and MSO-style graph checkers -- and four experiments including real neural GAN training with PyTorch. Beyond verification, we score samples with a \emph{logical loss} that mixes budgeted EF round-resilience with cheap certificate terms, enabling a practical curriculum on depth. Framework validation demonstrates $92\%$--$98\%$ property satisfaction via simulation (Exp.~3), while real neural GAN training achieves $5\%$--$14\%$ improvements on challenging properties and $98\%$ satisfaction on connectivity (matching simulation) through adversarial learning (Exp.~4). LOGAN is a compact, reproducible path toward logic-bounded generation with interpretable failures, proven effectiveness (both simulated and real training), and dials for control.

翻译：生成对抗网络（GANs）追求不可区分性，而逻辑学则为其提供解释。本研究将二者置于有限资源框架下：判别器仅能“观察”至逻辑深度$k$，而生成器必须在该有限观察者面前呈现正确样本。\textbf{LOGAN}（逻辑生成对抗网络）将判别器设定为深度-$k$的埃伦费斯特-弗赖塞（EF）博弈\emph{挑战方}，专门搜寻微小且可解释的缺陷（如奇环、非平面交叉、有向桥），而生成器则扮演\emph{构建方}，生成能与目标理论$T$进行$k$轮匹配的样本。我们提供了最小化工具包——包含EF探针模拟器和MSO风格图检验器，以及四项实验（含基于PyTorch的真实神经GAN训练）。除验证外，我们通过混合有限EF轮次抗扰性与低成本证书项的\emph{逻辑损失函数}评估样本，实现了深度上的实用课程学习。框架验证显示：通过模拟实验（实验3）达到92\%–98\%的属性满足率；真实神经GAN训练在对抗学习（实验4）中，针对挑战性属性实现5\%–14\%的性能提升，连通性属性满足率达98\%（与模拟结果一致）。LOGAN为逻辑有界生成提供了一条紧凑、可复现的路径，其特点包括：可解释的失败模式、经证明的有效性（涵盖模拟与真实训练）以及可控调节机制。