[Rusky]

In fact, those union-find trickeries come from the same paper that presented algorithm W, where they were named algorithm J. W was known from the start to be more useful for proofs than implementation:

> As it stands, W is hardly an efficient algorithm; substitutions are applied too often. It was formulated to aid the proof of soundness. We now present a simpler algorithm J which simulates W in a precise sense.

https://doi.org/10.1016/0022-0000(78)90014-4

[derdi]

Isn't the union-find trickery "just" standard almost-linear unification as discovered by Paterson-Wegman and Martelli-Montanari around 1976? Or is there more to it in the setting of Algorithm J?

I'm actually a bit surprised that it took so long to discover these formulations of unification. I wonder what Prolog systems were doing at the time, given the importance of efficient unification.

[Rusky]

Paterson-Wegman and Martelli-Montanari are worst-case linear, but Algorithm J just uses the earlier almost-linear union-find approach: unification variables are represented as mutable pointers to either a) nothing, or b) another type (which may itself be the "parent" unification variable in the union-find structure). Unification is then just a recursive function over a pair of types that updates these pointers as appropriate- and often doesn't even bother with e.g. path compression.