Discrete World Models via Regularization

World models aim to capture the states and dynamics of an environment in a compact latent space. Moreover, using Boolean state representations is particularly useful for search heuristics and symbolic reasoning and planning. Existing approaches keep latents informative via decoder-based reconstruction, or instead via contrastive or reward signals. In this work, we introduce Discrete World Models via Regularization (DWMR): a reconstruction-free and contrastive-free method for unsupervised Boolean world-model learning. In particular, we introduce a novel world-modeling loss that couples latent prediction with specialized regularizers. Such regularizers maximize the entropy and independence of the representation bits through variance, correlation, and coskewness penalties, while simultaneously enforcing a locality prior for sparse action changes. To enable effective optimization, we also introduce a novel training scheme improving robustness to discrete roll-outs. Experiments on two benchmarks with underlying combinatorial structure show that DWMR learns more accurate representations and transitions than reconstruction-based alternatives. Finally, DWMR can also be paired with an auxiliary reconstruction decoder, and this combination yields additional gains.

Lifted Inference beyond First-Order Logic

Weighted First Order Model Counting (WFOMC) is fundamental to probabilistic inference in statistical relational learning models. As WFOMC is known to be intractable in general ($\#$P-complete), logical fragments that admit polynomial time WFOMC are of significant interest. Such fragments are called domain liftable. Recent works have shown that the two-variable fragment of first order logic extended with counting quantifiers ($\mathrm{C^2}$) is domain-liftable. However, many properties of real-world data, like acyclicity in citation networks and connectivity in social networks, cannot be modeled in $\mathrm{C^2}$, or first order logic in general. In this work, we expand the domain liftability of $\mathrm{C^2}$ with multiple such properties. We show that any $\mathrm{C^2}$ sentence remains domain liftable when one of its relations is restricted to represent a directed acyclic graph, a connected graph, a tree (resp. a directed tree) or a forest (resp. a directed forest). All our results rely on a novel and general methodology of "counting by splitting". Besides their application to probabilistic inference, our results provide a general framework for counting combinatorial structures. We expand a vast array of previous results in discrete mathematics literature on directed acyclic graphs, phylogenetic networks, etc.