Definition. Economically conditioned potential [ftip-00O0]
Definition. Economically conditioned potential [ftip-00O0]
For any admitted class \(\mathcal Z\) of joint executions and acquisition score \(Q_{\rm acq}(P)\in [0,1]\), define
\[\Phi (\Xi ,\Omega ,T;\mathcal Z)= \sup _{z\in \mathcal Z}Q_{\rm acq}(P_z).\]Take the supremum of an empty class to be zero in the ordered interval \([0,1]\). For a nonnegative counted compute rate \(u_z\), define the hard resource envelope
\[\overline B(T;\mathcal Z)= \sup _{z\in \mathcal Z}\operatorname *{ess\,sup} \int _0^T u_z(t)\,dt,\]with zero for an empty class. The essential supremum is over the execution's declared randomness. A claim about expected expenditure alone does not bound this quantity. The compute unit is fixed by the operational semantics and must agree with any subsequent learning lower bound.
Class inclusion gives \(\Phi (\mathcal Z_1)\leq \Phi (\mathcal Z_2)\) and \(\overline B(\mathcal Z_1)\leq \overline B(\mathcal Z_2)\) whenever \(\mathcal Z_1\subseteq \mathcal Z_2\), with other arguments fixed. Indeed every value in the first supremum also occurs in the second. A supremum need not be attained: \(\Phi \geq \tau \) does not by itself provide a procedure with score at least \(\tau \).