$$a_t = \pi_\theta(h_t \mid \mathcal{T}_t), \qquad o_t = \mathrm{Call}(a_t), \qquad h_{t+1} = h_t \oplus (a_t,\, o_t)$$

Equation 5 · AI Infrastructure

How Tool Protocols and the Model Context Protocol Actually Work

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

It helps to write the whole loop down once, because the assumption buried in it is the one the rest of this article is about: $a_t = \pi_\theta(h_t \mid \mathcal{T}_t), \qquad o_t = \mathrm{Call}(a_t), \qquad h_{t+1} = h_t \oplus (a_t,\, o_t)$. Here $\mathcal{T}_t$ is the tool set most recently returned by tools/list , each member carrying its own schema; $\pi_\theta$ is the model, which is expected — but not forced by anything outside its own training — to emit a call $a_t$ whose name and arguments validate against some tool in $\mathcal{T}_t$ ; $\mathrm{Call}$ is dispatch through the client to the server and back, exactly as described above; and $h_{t+1}$ is the context the model reasons over on the next step, built by appending the call and its result to what came before. The assumption is in…

Meanings in this article

• $\pi_\theta$: the model, which is expected — but not forced by anything outside its own training — to emit a call $a_t$ whose name and arguments validate against some tool in $\mathcal{T}_t$.
• $o_t$: concatenated into the model’s context and fed straight back into $\pi_\theta$ with no protocol-level check on what it contains.
• $h_{t+1}$: the context the model reasons over on the next step, built by appending the call and its result to what came before.

Equation guide → · Article →