I would say that any function that implicitly favors a single number must be explicitly stated, and thus, if used for this game, be the number 2. So all uses of the radical must state which root (2). Dirac's solution then wouldn't work because the use of 2 is O(n).
Logs would also need to state the base. No implicit use of e or 10, and lg wouldn't be allowed in place of log2.
I haven't said much other than logs and roots are binary operators with one of the operands usually implicit in the notation, so if we don't have special notation for powers and exponentiation, then we shouldn't allow the same for their inverse operations.
Why is it ok to use "22" = 2 * 10^1 + 2 (when it could be a number in base 3 — 2 * 3^1 + 2 = 8 decimal — or any other base)? This implies base 10, just like root implies base 2, or ln means e.
As I said, this is a game, and trying to imply certain artificial constraints will be really hard with how abstract maths is.
Again, mention of successor function is apt: everything else is built from 1, succ() and another axiom, definition or so. So everything else can be reduced to this.
I said that this implicit use of 10 or some other number shouldn't be allowed. So log, ln, lg (i.e base 2) shouldn't be allowed, but log_b(x) where b and x are states is OK, just as 10^x, e^x, and 2^x require you to explicitly expose the base (and for this puzzle, disallow 10 and e since neither is a 2).
Successor is essentially s(n) = n + 1, so that shouldn't be allowed either.
FWIW, "successor" is not really n + 1: you've got that the other way around.
Successor simply "is" (it's a relation that satisfies a number of conditions), and summation is defined in terms of successor function.
My point is that you can really define everything in terms of these primitive definitions, which means that there won't be any single use of a non-2 digit for any function, or you'll be going with a set of arbitrary allowances.
But the whole point should be: what are those arbitrary constraints that make the game fun? And once you clear that bar, it's ok to open up the next one (this does not make them non-arbitrary though).
Basically, I am saying your take at those arbitrary decisions is not a very fun one ;-)
Logs would also need to state the base. No implicit use of e or 10, and lg wouldn't be allowed in place of log2.
I haven't said much other than logs and roots are binary operators with one of the operands usually implicit in the notation, so if we don't have special notation for powers and exponentiation, then we shouldn't allow the same for their inverse operations.