Though now I’m wondering whether the top-down traversal is intractable for any interesting reasons. In particular, because certain operations like unification only make sense as top-down traversals. And the traversal order also gets at the differences in formal power between deterministic and nondeterministic CFGs.

f z < x
============
z f~
x —–> f z ——> z
x ——> gx ——> z
g >=

f~ : converse of f
readed as x > fz and fz apply by f~ got z

; is the reverse relation composition , that is R ; S = S . R

so > ; f~ == g ; >=
(> ; f~ ) / >= == g

/ is the adjoin of right composition .

Hope this is clear.

Regards.


mss xs = snd . foldr step (0, max xs) xs
where step x (p,s) = (0 ↑ (x+p), ((x+p) ↑ s)


    z * 3 ≤ 9  ≡  z ≤ 9 / 3

the variable z is not 9 divided by 3, the value we aim to compute — 9 / 3 is. Variable z is merely a universally quantified dummy variable.

Pretend that we do not know what the value of 9 / 3 is. We must pick a value for it such that z * 3 ≤ 9 ≡ z ≤ 9 / 3 always evaluates to either true ≡ true or false ≡ false. Let’s try:
z = 0: 0 * 3 ≤ 9, thus we shall have 0 ≤ 9 / 3;
z = 1,2: (omitted);
z = 3: 3 * 3 ≤ 9, thus we shall have 3 ≤ 9 / 3
z = 4: 4 * 3 ≰ 9, thus we shall have 4 ≰ 9 / 3
Thus we conclude that 3 / 9 could only be 3.

This way it is still hard to see where maximality comes in (if 9/3 were not the largest, the cases above for z = 2 or 3, for example, would not be true. But such an explanation still feels like an accident). To see how maximality is captured, I’d still prefer the explanation in the post: look at this direction:

    z * 3 ≤ 9  ⇒  z ≤ 9 / 3

if any z satisfies z * 3 ≤ 9, it must be the case that z ≤ 9 / 3. For any z such that z ≰ 9 / 3, it cannot be the case that z * 3 ≤ 9. So 9 / 3 itself is the largest value z could be while still making z * 3 ≤ 9 true.

I’ve altered the text a bit to mention that z is a dummy variable. It might avoid some confusion had I explicitly quantify z.


    (∀ z :: z * 3 ≤ 9  ≡  z ≤ 9 / 3)

Functions f and g form a Galois connection iff:

    (∀ x z :: f z ⊑ x   ≡   z ≤ g x)