When I wrote "Pratt Parsing and Precedence Climbing Are the Same Algorithm", multiple people replied and suggested that the Shunting Yard algorithm is also the same, with the equivalence you suggest.
In particular the author of the repo Jean-Marc Bourguet also thought that. Then he implemented it and realized it wasn't true.
See this section of the README:
I had the mistaken impression that they were "just" replacing an explicit stack with the call stack of the implementation language but that was not clear enough from Andy's code. It was not clear because that's not the case. Compare Pratt's algorithm with recursive_operator_precedence.py, which is a recursive implementation of operator precedence, to see the difference. Pratt and operator climbing are really top down techniques: they call semantic actions as soon as possible, they don't delay the call until the whole handle is seen as operator precedence does. My current characterization is that they are to LL parsing and recursive descent what operator precedence is to LR and shift-reduce
I think we had a discussion about an analogy LL vs. LR. Those are obviously not equivalent algorithms -- despite both being linear time, they recognize different languages.
I think in the Shunting Yard algorithm, when you parse 1 + 2 / 3, the 2 / 3 tree is made first. I guess that is somewhat true of Pratt parsing too for natural reasons of tree building. You build the tree when you bubble out of the recursion.
But it DECIDES that it has "led" for + before it decides it has "led" for /.
So in summary, Shunting Yard "decides" 2 / 3 first, where as Pratt parsing "decides" 1+... first. That's what I think anyway. I think you could probably make this rigorous, but I'm not sure.
I think of Pratt Parsing as LL(2) parsing for an operator grammar, although that's not technically correct because the precedence table is an extra piece of info outside the "grammar". You either lookahead 1 or 2 tokens, for null and led respectively.
So basically I think you could translate Pratt parsing to use an explicit stack -- but then you'd still be using Pratt parsing. The Shunting Yard algorithm is something else.
E.g. Pratt is basically writing a generic recursive descent component ("parse left; parse operand; parse right") and then passing down a precedence where you terminate the subtree parse. That's a large part of the appeal - it slots seamlessly into a recursive descent parser without looking "alien", and it was what I started with. It's nice because it is top down.
The naive way of making the stack explicit here is to "push" a fake stack frame when you would have recursed, and pop when you would have exited. But to do all tree of the parse_left/parse_operand/parse_right triple this way you then need to push additional frames to run those steps one after each other, and you end up with something really ugly. But if you trace the processing of those additional frames, what you will see then is quickly that you can drop them by just splitting the stack into operands and operators. Then you have shunting yard.
So you're sort of right that a naive translation of Pratt that retains the top-down nature of recursive Pratt is distinct from shunting yard, but once you simplify it the before/after distinction evaporates, as there's nothing in between to make the decision before/after.
I think the distinction is large enough that it makes sense to treat them as different algorithms, but the transformation is still mostly a mechanical exercise. Each step is fairly straightforward.