Question

Can you implement the shunting yard algorithm in terms of regular expressions?

Answer 1

I believe this has been answered here: Can the shunting yard algorithm parse POSIX regular expressions?

I will say that the answer to your question is "no, you cannot
implement the shunting yard algorithm using a regular expression."
This is for the same reason you cannot parse arbitrary HTML using
regular expressions. Which boils down to this:

Regular expressions do not have a stack. Because the shunting yard
algorithm relies on a stack (to push and pop operands as you convert
from infix to RPN), then regular expressions do not have the
computational "power" to perform this task.

This glosses over many details, but a "regular expression" is one way
to define a regular language. When you "use" a regular expression, you
are asking the computer to say: "Look at a body of text and tell me
whether or not any of those strings are in my language. The language
that I defined using a regular expression." I’ll point to this most
excellent answer which you and everyone reading this should upvote
for more on regular languages.

So now you need some mathematical concept to augment "regular
languages" in order to create more powerful languages. If you were to
characterize the shunting yard algorithm as an realization of a model
of computational power, then you might say that the algorithm would be
described as a context-free grammar (hey what do you know, that
link uses an expression parse tree as an example.) A push-down
automata. Something with a stack.

If you are less-than-familiar with automata theory and complexity
classes, then those wikipedia articles are probably not that helpful
without explaining them from the ground up.

The point being, you may be able to use regex to help writing shunting
yard. But regex are not very good at doing operations that have an
arbitrary depth, which this problem has. So I would not spend too much
time going down the regex avenue for this problem.