TITLE: What is the intuition behind the negative exponential '?' in linear logic?
QUESTION [12 upvotes]: The positive exponential $!$ has a very satisfying interpretation in terms of the standard resource interpretation of linear logic. Given a resource $a$, we know that $!a$ means an infinite supply of $a$. Or, stated more concretely in terms of the connectives of linear logic: $!a \equiv !a \otimes a$. My question is, under this same interpretation of atomic propositions $a$ in linear logic as resources, how are we to interpret $?a$?
The best interpretation I've seen so far seems to be that $?a$ "consumes $a$'s", but what does that mean concretely in terms of the logic? Is there an analogue of the formula $!a = !a \otimes a$ for $?$?
Unfortunately, the usual sequent rules for linear logic are not very helpful to my intuition. I think the two-sided sequent presentation of linear logic is the most intuitive so far — $!$ reintroduces weakening/contraction on the left side of a sequent, and $?$ allows for the same on the right side of a sequent (as evidenced by the rules $!W, ?W, !C, ?C, !D,$ and $?D$ in the presentation here). But what are we to make of the rules $?L$ and $!R$?
So far, I have not been able to find an adequate or complete explanation of either the resource interpretation of $?$ in general, or even the meaning of the rules $?L$ and $!R$, whether it be with regards to the resource interpretation, or the original motivations of linear logic by Girard. And I am curious whether anyone could point me towards better explanations or resources than I have been able to find.

REPLY [5 votes]: The way I think of $!A$ is as a choice of how many $A$'s you want to receive, rather than simply an infinite number of them. You could write it as an additive disjunction like so:
$$!A \approx 1\&(A)\&(A\otimes A)\&(A\otimes A\otimes A)\& ...$$
The dual of that would be:
$$\DeclareMathOperator{\par}{\unicode{8523}}
?A = (!A^\bot)^\bot \approx \bot\oplus(A)\oplus(A\par A)\oplus(A\par A\par A)\oplus ...$$
So $?A$ is sort of similar to an unknown number (zero or more) of $A$'s that must be used in parallel in that way that $\par$ requires.

Edit: This interpretation also explains a lot of the exponential sequents, and their rules start to look very similar to those for the additives. I'll just show the right-handed sequents, as described by the Stanford Encyclopedia of Philosophy:
Weakening: If you can get $\Gamma$, you can also produce some number (i.e. zero) of $B$.
$$\cfrac{\Delta\vdash\Gamma}{\Delta\vdash?B,\Gamma}$$
Dereliction: If you can get $B$, you can instead produce some number (i.e. one) of $B$.
$$\cfrac{\Delta\vdash B,\Gamma}{\Delta\vdash?B,\Gamma}$$
Contraction: If you have an unknown quantity of $B$ and another unknown quantity of $B$, you can join them into one unknown quantity.
$$\cfrac{\Delta\vdash?B,?B,\Gamma}{\Delta\vdash?B,\Gamma}$$
Promotion: If you can use unlimited $\Delta$ to produce some number of $\Gamma$, and you can also produce a single $B$, then you can produce any amount of $B$ that you want.
$$\cfrac{!\Delta\vdash B,?\Gamma}{!\Delta\vdash!B,?\Gamma}$$