Definition. Finite controller class with hard resource admission [ftip-00M5]
AGENTDRAFTED
Fix a horizon \(H\in \mathbb N\), finite nonempty observed-history sets
\(\mathcal H_t\) for \(0\leq t\leq H\), an initial law \(\mu \) on
\(\mathcal H_0\), and finite nonempty permitted-action sets \(A_t(h)\)
for \(t<H\). An environment kernel \(K_t(\cdot \mid h,a)\) is a probability
law on histories extending \(h\) by action \(a\) and its observed outcome.
The initial law and all kernels are fixed before choosing a controller.
A controller \(\pi \) assigns a probability law
\(\pi _t(\cdot \mid h)\) on \(A_t(h)\). It may depend on every observed
component of \(h\), including retained traces, generated programs, and
previous updates. Random choices may be recorded in the history. The class
\(\Pi (\mathbf B)\) consists of all such controllers under the declared
hard resource vector \(\mathbf B\in \mathbb R_+^m\). It does not require
a fixed prompt, a memoryless policy, or independent worker outputs.
Full-history access is allowed in this mathematical class; it may enlarge
the class of executable controllers with limited memory or computation.
Each history records accumulated nonnegative cost
\(\mathbf c(h)\preceq \mathbf B\). Every permitted non-stop action has a
declared worst-case increment \(\bar {\mathbf c}_t(h,a)\) satisfying
\[
\mathbf c(h)+\bar {\mathbf c}_t(h,a)\preceq \mathbf B.
\]
Every successor in the kernel's support must have realized increment
between zero and this declared bound, componentwise. Proposal work,
controller computation, worker calls, communication, verification, failed
attempts, and persistent updates must be included in whichever resource
coordinates are bounded; the units remain those of Definition [ftip-005H].
This is the enforced-contract assumption of Theorem [ftip-00CK].
A stop action is always permitted: it commits the current terminal
artifact and pads the remaining steps with no further cost or change in
its evaluated quality. At the horizon a fixed evaluator assigns
\(q(h_H)\in [0,1]\) to the committed artifact recorded in the terminal
history. If the evaluator has random outcomes, include them in the
history law. Evaluation work must be charged before the zero-cost padding
begins. Define
\[
J(\pi )=\mathbb E_\pi [q(h_H)],
\qquad V(\mathbf B)=\sup _{\pi \in \Pi (\mathbf B)}J(\pi ).
\]
The artifact identity, evaluator, observations, kernels, and action
sets are part of the mathematical problem. A training procedure or new
tool belongs to this frontier only if it is among the permitted actions
and its effects and costs are represented. Finiteness is an explicit
restriction on histories, representations, and horizon; a theorem for
this class does not bound an unrestricted agent that can extend them.
For an executable system, an upper bound applies only after its
observations, actions, outcomes, and charged costs are represented by this
model. Conversely, a mathematical policy supplies an executable lower bound
only when it has an implementation respecting the stated resource cap;
an arbitrary history-to-action table does not establish that fact.