Definition. Uniform abstraction of the finite history model [ftip-00M9]

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Use the finite controller model of Definition [ftip-00M5]. For each time, choose a finite nonempty state set \(S_t\) and a map \(\phi _t:\mathcal H_t\to S_t\). Require finite nonempty action sets \(A_t(s)\) such that \(A_t(h)=A_t(\phi _t(h))\) at every history. The state retains residual budgets and enough information for the same hard admission rule. The controller still observes the full history.

Let \(Q_t(\cdot \mid h,a)\) be the image of the fixed history kernel under \(\phi _{t+1}\), and let \(\widehat P_t(\cdot \mid s,a)\) be a proposed state transition law. Choose finite errors \(\epsilon _t(s,a)\geq 0\) so that, for every permitted history and action,

This condition includes histories unvisited by a chosen policy: the kernel there is part of the model. For finite functions \(\widehat q:S_H\to \mathbb R\) and \(e_H:S_H\to \mathbb R_+\), require

Write \(\nu (s)=\mu \{h:\phi _0(h)=s\}\) for the initial state law. Low average prediction error on sampled histories does not establish these uniform inequalities. The erased-bit example in Example [ftip-00M7] shows why policy-relevant information cannot simply be averaged away.