Constraints Are Graphs, Not Chains: Exact Decoding for Diffusion Language Models

Diffusion language models (dLLMs) predict masked positions in arbitrary order, but their exact constrained decoders still encode constraints as sequential languages, whose state must track every unresolved dependency between positions. For relational constraints this encoding grows exponentially: for same-order copy, every finite automaton needs $4^k$ states, deterministic or nondeterministic, and every context-free grammar has size $2^{Ω(k)}$, while the factor graph of the same relation has size $O(k)$ and a 16-entry peak table. We introduce FactorDLM, a training-free decoder that represents finite-domain relations as a factor graph and, at each denoising step, conditions the model's mea...

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