$$\mathcal C_t = (\pi_{\theta,t}, H_t, \mathcal T_t, P_t, E_t, V_t, U_t)$$

Equation 1 · AI Agents & Systems

OpenAI Codex as a Software-Engineering System: Models, Harnesses, and Evidence

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

A useful abstraction is $\mathcal C_t = (\pi_{\theta,t}, H_t, \mathcal T_t, P_t, E_t, V_t, U_t)$. where $\pi_{\theta,t}$ is the model policy at time t ; $H_t$ is the harness and its context/state machinery; $\mathcal T_t$ is the available tool set; $P_t$ is the permission and sandbox policy; $E_t$ is the execution environment; $V_t$ is the collection of verification functions; and $U_t$ represents user and institutional authority. The time index is essential. OpenAI can update the model, CLI, documentation, or cloud runtime without changing the product name.

Meanings in this article

• $\pi_{\theta,t}$: the model policy at time t.
• $H_t$: the harness and its context/state machinery.
• $T_t$: the available tool set.
• $P_t$: the permission and sandbox policy.
• $E_t$: the execution environment.
• $V_t$: the collection of verification functions.
• $U_t$: user and institutional authority.

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