We use the notion of oracle machines and reductions from computability theory to formalise different Human-in-the-loop (HITL) setups for AI systems, distinguishing between trivial human monitoring (i.e., total functions), single endpoint human action (i.e., many-one reductions), and highly involved human-AI interaction (i.e., Turing reductions). We then proceed to show that the legal status and safety of different setups vary greatly. We present a taxonomy to categorise HITL failure modes, highlighting the practical limitations of HITL setups. We then identify omissions in UK and EU legal frameworks, which focus on HITL setups that may not always achieve the desired ethical, legal, and sociotechnical outcomes. We suggest areas where the law should recognise the effectiveness of different HITL setups and assign responsibility in these contexts, avoiding human "scapegoating". Our work shows an unavoidable trade-off between attribution of legal responsibility, and technical explainability. Overall, we show how HITL setups involve many technical design decisions, and can be prone to failures out of the humans' control. Our formalisation and taxonomy opens up a new analytic perspective on the challenges in creating HITL setups, helping inform AI developers and lawmakers on designing HITL setups to better achieve their desired outcomes.
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