Equation 6 · From Autocomplete to Delegation: The Technical History Behind OpenAI Codex

What does this equation mean?

a_t\sim\pi_\theta(a\mid o_{\le t},h_t,g), \qquad o_{t+1}=E(s_t,a_t),

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Inputs and operationsE(s_t,a_t)

Result or conditiona_tsimpi_θ(amid o_ ≤ t,h_t,g), qquad o_t+1

How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states a bound: one expression must stay on the indicated side of the other 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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a_t

Symbol a_t

$a_t$ is part of the quantity the equation computes from the expression on the right.

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\pi_\theta

Symbol pi_θ

pi_θ is part of the quantity the equation computes from the expression on the right.

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a

Symbol a

a is part of the quantity the equation computes from the expression on the right.

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o_{\le t}

Symbol o_ ≤ t

o_ ≤ t is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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h_t

Symbol h_t

$h_t$ is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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g

Symbol g

g is an argument of the function-like quantity on the left; its role is set by that function’s stated inputs.

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o_{t+1}

Symbol o_t+1

$o_t$ +1 is part of the quantity the equation computes from the expression on the right.

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E

Symbol E

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

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s_t

Symbol s_t

the environment e executes it against state.

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=

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

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≤

Less than or equal to.

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addition

Add the term after the plus sign to the term or group before it.

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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

An agent trajectory can be represented as $a_t\sim\pi_\theta(a\mid o_{\le t},h_t,g), \qquad o_{t+1}=E(s_t,a_t)$ . where the model policy chooses an action, environment E executes it against state $s_t$ , and the result becomes the next observation. This closes a feedback loop absent from one-shot completion.

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Sources cited in the article section

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