ActivityPub's PublicInbox is used to route all messages. This works for "small-world networking" where you're in a tight-knit community exchanging messages with known or friend-of-friend actors, but doesn't work well for "big-world networking" with folks you don't know and don't necessarily want to share all or even most of your messages with (a pretty big pain point right now in the Fediverse.)
ATProtocol claims to differentiate between these cases by introducing the concept of indexers which can apply different algorithms to order feeds, but it's unclear to me how exactly that will work.
This is not right. First of all, what do you mean by "PublicInbox"? Are you referring to the optional sharedInbox attribute, which allows delivering a payload to a whole server at once instead of iterating over the personal inbox of every actor on the server? Or are you referring to the inbox in general? The inbox is just a mechanism to deliver a payload from point A to point B. It is in fact used to receive messages from folks you don't know, but there is no obligation to share all or even most of your messages with anyone in particular. You can deliver posts to your followers, but you don't have to, protocol-wise, and you can choose, person by person, who to deliver to. The concept of indexers doesn't seem foreign to ActivityPub either. PeerTube runs sepia.search, which is exactly that kind of indexer that powers search for their video platform.
Sorry, yeah I meant "sharedInbox" vs "PublicInbox". My understanding is that the AP client (assuming you aren't using C2S ActivityPub) sends messages to your AP server which then either sends messages to the sharedInbox of the destination AP instance or directly to the inbox at that other AP instance. I realize the visibility of each message is up to the discretion of each AP implementation but as practiced now, messages are often seen by multiple parties before getting to the final destination.
As far as indexers are concerned, I realize it's entirely possible to use indexers with AP. It's wholly unclear to me how AT Protocol is using indexers or scoping any of their objects so I'm not sure it's worth belaboring the point.
ATProtocol claims to differentiate between these cases by introducing the concept of indexers which can apply different algorithms to order feeds, but it's unclear to me how exactly that will work.