Flattened form in WAM

Question

The WAM: A Tutorial Reconstruction states that a query, p(Z, h(Z,W), f(W)), needs to be flattened using the following principles:

That being said, the query flattened form is:

X3=h(X2, X5), X4=f(X5), X1=p(X2, X3, X4);

I am lost with the definition of external variable, consider the following:

p(Z, h(Y, a(K, C), b(W)), f(W)).

Is Y an external variable? How should be the flattened form for this? From my understanding this would be the construction:

X1 = p(X2, X3, X4)
X2 = Z
X3 = h(X5, X6, X7)
X4 = f(X8)
X5 = Y
X6 = a(X7, X8)
X7 = K
X8 = C
X9 = b(X5)

But I am not sure, starting at X4 I got confused, should I have assigned the h inner values first?

Answer 1

You have the order the wrong way around: You are building terms before you have built their arguments. The text says to build the arguments before you build the outer terms. For example, you must build a(K, C) before you can build h(..., a(K, C), ...), and you must build that before you can build p(..., h(..., a(K, C), ...), ...). Here is one legal order:

X7 = K
X8 = C
X6 = a(X7, X8)
X5 = Y
X9 = b(X5)
X2 = Z
X3 = h(X5, X6, X7)
X4 = f(X8)
X1 = p(X2, X3, X4)