Defeasible and Deontic Probabilistic Circuits: Semantic Topology for Belief Revision and Normative Audit

A model that learns from what people do can predict what they will do next, but it cannot tell whether what they do is allowed or not. Nor does it have a principled way to retract a conclusion when an exception arrives without disturbing everything else it believes. To address this, we introduce Defeasible Probabilistic Circuits (DPCs) for structured belief revision and Deontic Probabilistic Circuits (DePCs) for normative auditing. These models designate selected gated sum nodes of a conditional probabilistic circuit as defeasible or deontic (audited) nodes, preserving exact, linear-time inference. To support defeasible reasoning—where previous conclusions can be overridden by new evidence—a DPC structures a node so one branch is the default and the rest are prioritized exceptions. It compiles any learned softmax gate exactly into this ordered-rule form without retraining. A new exception can then be inserted without altering higher-priority weights, accompanied by an evidence-specific certificate bounding its effect. To support deontic reasoning—evaluating what is allowed versus forbidden—a DePC designates branches as either compliant or violating. It pairs the learned descriptive distribution with a policy-supplied normative reference, exactly computing severity-weighted departures and using a gate-wise audit budget to localize policy violations. Across two physical-reasoning and two clinical environments, DPCs and DePCs improve update and action ranking, respectively, over comparably sized conditional circuits. Although trained solely from observations and without ranking supervision, they also outperform GPT-5.6 Sol, Qwen3.8-27B, and Qwen3-32B under zero-shot evaluation.