← Mathematical compendium

Published equation contexts

P=πplan(o0, g),at=πexec(P, ot)P = \pi_{\text{plan}}(o_0,\ g), \qquad a_t = \pi_{\text{exec}}(P,\ o_t)

Why this formula appears here

Formally, this replaces the single-loop policy with two functions and one artifact: P=πplan(o0, g),at=πexec(P, ot)P = \pi_{\text{plan}}(o_0,\ g), \qquad a_t = \pi_{\text{exec}}(P,\ o_t). The plan P is computed against the belief state available at time zero and then held fixed; the executor applies it against a stream of later observations without necessarily invoking the planning policy again. That is precisely the assumption a classical open-loop plan makes, and it is a specific instance of the belief-state fragility decision theory already names: a policy computed once from a belief state is only as good as that belief state stays accurate, and nothing in the equation above notices when the two have quietly come apart [ 11 ] .

Read the full article-specific guide →

Read the representative guide

πplan\pi_{\text{plan}}

Symbol pi_plan

pipi_plan is an input to the expression that computes the quantity on the left.

Read this term in its guide →
πexec\pi_{\text{exec}}

Symbol pi_exec

piei_exec is an input to the expression that computes the quantity on the left.

Read this term in its guide →

How to interpret it

Read it with the definitions, units, and assumptions supplied by the article.

Research cited beside this formula

Learn the mathematics behind it

Published contexts (1)

A symbol can carry a different meaning in another article. Each occurrence keeps its own guide and term definitions.

P=πplan(o0, g),at=πexec(P, ot).P = \pi_{\text{plan}}(o_0,\ g), \qquad a_t = \pi_{\text{exec}}(P,\ o_t).

Equation 6 · AI Agents & Systems

Comparing the Main Approaches to AI Agent Architecture

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

Formally, this replaces the single-loop policy with two functions and one artifact: P=πplan(o0, g),at=πexec(P, ot)P = \pi_{\text{plan}}(o_0,\ g), \qquad a_t = \pi_{\text{exec}}(P,\ o_t). The plan P is computed against the belief state available at time zero and then held fixed; the executor applies it against a stream of later observations without necessarily invoking the planning policy again. That is precisely the assumption a classical open-loop plan makes, and it is a specific instance of the belief-state fragility decision theory already names: a policy computed once from a belief state is only as good as that belief state stays accurate, and nothing in the equation above notices when the two have quietly come apart [ 11 ] .

Meanings in this article

Equation guide → · Article →