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Equation 5 · How Tool Protocols and the Model Context Protocol Actually Work

What does this equation mean?

at=πθ(ht∣Tt),ot=Call(at),ht+1=ht⊕(at, ot)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)

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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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ata_t

Symbol a_t

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

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πθ\pi_\theta

Symbol pi_θ

the model, which is expected — but not forced by anything outside its own training — to emit a call ata_t whose name and arguments validate against some tool in Tt\mathcal{T}_t.

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hth_t

Symbol h_t

hth_t is one of the signed contributions combined to compute the quantity on the left.

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Tt\mathcal{T}_t

Symbol T_t

TtT_t is one of the signed contributions combined to compute the quantity on the left.

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oto_t

Symbol o_t

concatenated into the model’s context and fed straight back into πθ\pi_\theta with no protocol-level check on what it contains.

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ht+1h_{t+1}

Symbol 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.

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=

=

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

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addition

addition

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

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subscript

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

Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

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: at=πθ(ht∣Tt),ot=Call(at),ht+1=ht⊕(at, ot)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 Tt\mathcal{T}_t is the tool set most recently returned by tools/list , each member carrying its own schema; πθ\pi_\theta is the model, which is expected — but not forced by anything outside its own training — to emit a call ata_t whose name and arguments validate against some tool in Tt\mathcal{T}_t ; Call\mathrm{Call} is dispatch through the client to the server and back, exactly as described above; and ht+1h_{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…
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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: at=πθ(ht∣Tt),ot=Call(at),ht+1=ht⊕(at, ot)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 Tt\mathcal{T}_t is the tool set most recently returned by tools/list , each member carrying its own schema; πθ\pi_\theta is the model, which is expected — but not forced by anything outside its own training — to emit a call ata_t whose name and arguments validate against some tool in Tt\mathcal{T}_t ; Call\mathrm{Call} is dispatch through the client to the server and back, exactly as described above; and ht+1h_{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 that last term: oto_t is concatenated into the model’s context and fed straight back into πθ\pi_\theta with no protocol-level check on what it contains. Whatever a tool returns becomes part of what the model reads next, on exactly the same footing as the user’s own words. Once that is stated plainly, the entire security section below is really just an inventory of what happens when oto_t , or the tool descriptions inside Tt\mathcal{T}_t , are not what they claim to be.

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Sources cited in the article section

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