Definition. Interactive realization of a task [ftip-006T]
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    For a task \(\mathsf T=(\mathcal X_{\mathsf T},\mathcal O_{\mathsf T}, \operatorname {Adm}_{\mathsf T})\) from Definition [ftip-001O], an interactive realization is

    \[ \mathsf {Env}_{\mathsf T} =\left (\mathcal S_{\mathsf T},\mathcal O^{\rm obs}_{\mathsf T}, \mathcal A_{\mathsf T},T_{\max }, (\rho _0^x)_x,(K_t^x)_{x,t},\operatorname {out}_{\mathsf T}\right ). \]

    For each \(x\in \mathcal X_{\mathsf T}\), the law \(\rho _0^x\) is on \(\mathcal S_{\mathsf T}\times \mathcal O^{\rm obs}_{\mathsf T}\), and \(K_t^x\) maps a state and action to the next state-observation law for \(0\leq t<T_{\max }\). The outcome map sends a stopped trajectory \(Z\) to \(\operatorname {out}_{\mathsf T}(x,Z)\in \mathcal O_{\mathsf T}\) and satisfies \(\operatorname {Adm}_{\mathsf T}(x,\operatorname {out}_{\mathsf T}(x,Z))\). For one fixed realization, write its three spaces as the unadorned \(\mathcal S,\mathcal O,\mathcal A\) of Notation [ftip-001X].

    The agent policy and stopping rule are not environment fields. They are combined with this realization only when an interaction law is formed.