Equation 6 · Comparing the Main Approaches to AI Agent Architecture

What does this equation mean?

P = \pi_{\text{plan}}(o_0,\ g), \qquad a_t = \pi_{\text{exec}}(P,\ o_t).

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Inputs and operationspi_plan(o_0, g), qquad a_t = pi_exec(P, o_t)

Result or conditionP

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This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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P

Symbol P

the plan.

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\pi_{\text{plan}}

Symbol pi_plan

p $i_p$ lan is an input to the expression that computes the quantity on the left.

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o_0

Symbol o_0

$o_0$ is an input to the expression that computes the quantity on the left.

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g

Symbol g

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

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a_t

Symbol a_t

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

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\pi_{\text{exec}}

Symbol pi_exec

p $i_e$ xec is an input to the expression that computes the quantity on the left.

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o_t

Symbol o_t

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

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=

The expressions on both sides represent the same quantity under the stated assumptions.

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subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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How to interpret it

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What the article says around this equation

Formally, this replaces the single-loop policy with two functions and one artifact: $P = \pi_{\text{plan}}(o_0,\ g), \qquad a_t = \pi_{\text{exec}}(P,\ o_t)$ . 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 ] .

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Sources cited in the surrounding passage

[11] Planning and Acting in Partially Observable Stochastic Domains ↗

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