Chain-of-thought (CoT) traces are used across language model prompting, training, test-time inference, and interpretability, yet they are often modeled in task-specific ways. We propose treating CoT traces as discrete probabilistic programs and introduce Copper, a language in which reasoning steps are stochastic variables, control flow encodes dependencies, and answers are program outputs. Its finite reachability semantics reduces CoT-related quantities to probabilistic queries over Copper programs, placing CoT analysis within tractable probabilistic inference. This formulation recovers standard trace-likelihood quantities while exposing limitations of likelihood-only scoring. It also supports differentiable operators for composing likelihoods, confidence estimates, and feedback over trace structure. We present the language and its tractable semantics, establish several basic formal correspondences with standard CoT inference, and outline ongoing case studies using the framework to derive new test-time inference strategies and verification techniques.

@inproceedings{RichardsonTPM26,
  title     = {CoTs as Tractable Probabilistic Programs},
  author    = {Richardson, Kyle and Feng, Yu and Garg, Poorva and Cheng, Junyan and Van den Broeck, Guy and Roth, Dan},
  booktitle = {Proceedings of the UAI Workshop on Tractable Probabilistic Modeling (TPM)},
  url       = "https://starai.cs.ucla.edu/papers/RichardsonTPM26.pdf",
  month     = 7,
  year      = {2026},
  keywords  = {workshop}
}