Logical GANs: Adversarial Learning through Ehrenfeucht Fraisse Games

TL;DR

This paper introduces LOGAN, a logical GAN framework that uses Ehrenfeucht--Fra"issé games to enforce logical properties in generated data, providing interpretable failure modes and improved property satisfaction.

Contribution

It presents a novel logical discriminator based on EF games and a generator that matches logical theories, enabling logic-bounded generative modeling with practical tools and validation.

Findings

Achieved 92-98% property satisfaction in simulations

Improved property satisfaction by 5-14% in neural GAN training

Demonstrated practical effectiveness with real neural network experiments

Abstract

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.