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Equation 2 · Typed Tools: Protocols as an Agent's Action Alphabet

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At\mathcal{A}_t

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the argument type : what shapes of input the tool accepts at all. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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At\mathcal{A}_t

Symbol A_t

the argument type : what shapes of input the tool accepts at all.

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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

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What the article says around this equation

At\mathcal{A}_t is the argument type : what shapes of input the tool accepts at all. The map is partial — the harpoon rather than the arrow — because for some arguments and some world-states the tool has no defined behaviour. Those are preconditions : a file must exist, a booking must not already be cancelled, a handle must not have expired. A precondition that is checkable before execution can reject a bad call cheaply. A precondition that is only discoverable during execution turns into an error, and possibly into damage.

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