Does A World Model Learn A Global Understanding?

AI systems often feel frustratingly brittle, fragmented, and poorly patched together. A large language model may correctly explain a concept but fail to reason with it or follow safety instructions in one context but not another. This behavior suggests a lack of global understanding. To gain fundamental insight into "understanding", we frame it as learning and propagating constraints and investigate it by constructing monoid learning tasks. A monoid, a set with an associative binary operation and identity element, contains the minimal structure to study controlled dynamics and entity relations without the reversibility assumptions that groups require. Monoid theory enables us to read off and control a specific class of constraints, equality constraints over action sequences, captured by algebraic equations in the action quotient. We construct monoid worlds where generalization to unseen monoid transitions, (state, action, next state), relies on a model's ability to learn and propagate constraints like inverses, commutativity, composition, and periodicity, key to understanding space and semantics. We find that next-state prediction, directly learning monoid transitions, fails to generalize across all constraints and architectures tested (e.g. transformers, state-space models). We introduce compositional path training which learns compositions of transitions and successfully generalizes with 63% accuracy across architectures and constraints (transformers often achieve $\geq$ 90% accuracy). We measure the complexity of an unseen transition by its proof depth $d$, the length of its shortest derivation, and find that generalization decreases geometrically with proof depth (often with rates $\leq 0.5$). We find, however, that increasing the composition length $T$ improves generalization at higher proof depths, e.g. increasing $T$ from 2 to 4 improved transformer $d=2$ generalization from 41% to 71%. These results provide a formal way to investigate global understanding in language and world models and demonstrate that compositional training promotes information propagation and integration.