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CSP failures semantics augments visible traces with refusal information. In the stable failures model, a pair $(s,X)$ means that after visible trace $s$, the process can reach a stable state with no internal transition and no offered event in $X$. It records a possible refusal, not a requirement that every state reached after $s$ refuse $X$.^[ref]

External choice between offering $a$ and offering $b$ cannot initially refuse either offered event. Internal choice between the same branches may select the $b$ branch and refuse $a$. Their visible trace sets can agree even though these refusals differ. Failures-divergences semantics additionally records possible infinite internal activity; it is a distinct model.

A CSP refusal set is a set $X$ of visible CSP events that a process can decline together in a stable state. Stability excludes an enabled internal transition. Pairing $X$ with a preceding trace gives a failure observation; different internal resolutions after the same trace can produce different refusal sets.^[ref]

For an event alphabet $\{a,b\}$, a state offering only $a$ can refuse $\{b\}$ but cannot refuse $\{a\}$. Every subset of a possible refusal is also possible. In a model without a separate successful-termination event, refusing the entire event alphabet signals deadlock. Refusals therefore distinguish a missing offer from an event merely not selected by the environment.