Equation 1 · Evaluating Trajectories, Not Answers

What does this equation mean?

\tau = \big(s_0,\ (a_1, o_1),\ \dots,\ (a_n, o_n),\ s_n\big)

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 operationsbig(s_0, (a_1, o_1), dots, (a_n, o_n), s_nbig)

Result or conditionτ

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

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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\tau

Symbol τ

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

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s_0

Symbol s_0

$s_0$ is an input to the expression that computes the quantity on the left.

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a_1

Symbol a_1

$a_1$ is an input to the expression that computes the quantity on the left.

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o_1

Symbol o_1

$o_1$ is an input to the expression that computes the quantity on the left.

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a_n

Symbol a_n

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

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o_n

Symbol o_n

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

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s_n

Symbol s_n

$s_n$ 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

A language model evaluation is, structurally, a function from a prompt to a string. Grading the string may be very hard, but the object being graded is small, static, and fully observed. An agent produces something categorically larger. The natural unit of observation is a trajectory: $\tau = \big(s_0,\ (a_1, o_1),\ \dots,\ (a_n, o_n),\ s_n\big)$ . an initial environment state, an ordered sequence of actions with the observations they returned, and a terminal state. Everything that separates a competent run from a lucky one lives in the middle terms. A conventional success indicator reads only the last one.

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