Equation 15 · Comparing the Main Approaches to AI Agent Architecture

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G = (V, E), \qquad v \to v' \text{ fires iff } \gamma(v, v') = 1,

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Inputs and operations(V, E), qquad v to v' fires iff gamma(v, v') = 1

Result or conditionG

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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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G

Symbol G

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

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V

Symbol V

either a deterministic function or an invocation of some policy $\pi_v$ .

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E

Symbol E

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

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v

Symbol v

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

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\gamma

Symbol gamma

an explicit gate on the edge — a test, a validation check, or a human approval — that must pass before the transition is permitted.

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=

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

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What the article says around this equation

Formally, this pattern replaces the time-indexed loop with execution over a graph: $G = (V, E), \qquad v \to v' \text{ fires iff } \gamma(v, v') = 1$ . where each node v $\in$ V is either a deterministic function or an invocation of some policy $\pi_v$ , and $\gamma$ is an explicit gate on the edge — a test, a validation check, or a human approval — that must pass before the transition is permitted. That is the structural reason human approval is a first-class primitive in this pattern rather than a wrapper bolted around it from outside: approval is just one kind of $\gamma$ , attached to one particular edge, rather than a blanket condition on the whole run.

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