Logical GANs: Adversarial Learning through Ehrenfeucht Fraisse Games

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. LOGAN (LOGical GANs) casts the discriminator as a depth-k Ehrenfeucht--Fra"iss'e (EF) Opponent that searches for small, legible faults (odd cycles, nonplanar crossings, directed bridges), while the generator plays 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 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.