Minor drive-by state res clarifications (#1042)

* Opinionated rewrite of unconflicted state map

* Define auth chain

changelogs/server_server/newsfragments/1042.clarification

@@ -0,0 +1 @@
+Clarify the meaning of "unconflicted state map" and "auth chain" in state res v2.

content/rooms/fragments/v2-state-res.md

@@ -30,11 +30,22 @@ the `membership` is `leave` or `ban` and the `sender` does not match the
 might remove someone's ability to do something in the room.
 
 **Unconflicted state map and conflicted state set.**
-The *unconflicted state map* is the state where the value of each key
+The keys of the state maps *S<sub>i</sub>* are 2-tuples of strings of the form
-exists and is the same in each state *S*<sub>*i*</sub>. The *conflicted
+*K* = `(event_type, state_key)`. The values *V* are state events.
-state set* is the set of all other state events. Note that the
+The key-value pairs (*K*, *V*) across all state maps *S<sub>i</sub>* can be 
-unconflicted state map only has one event per `(event_type, state_key)`,
+divided into two collections.
-whereas the conflicted state set may have multiple events.
+If a given key *K* is present in every *S<sub>i</sub>* with the same value *V* 
+in each state map, then the pair (*K*, *V*) belongs to the *unconflicted state map*.
+Otherwise (*K*, *V*) belongs to the *conflicted state set*.
+
+Note that the unconflicted state map only has one event for each key *K*,
+whereas the conflicted state set may associate multiple events to the same key.
+
+**Auth chain.**
+The *auth chain* of an event *E* is the set containing all of *E*'s auth events,
+all of *their* auth events, and so on recursively, stretching back to the
+start of the room. Put differently, these are the events reachable by walking
+the graph induced by an event's `auth_events` links.
 
 **Auth difference.**
 The *auth difference* is calculated by first calculating the full auth
@@ -111,9 +122,9 @@ the auth event is not rejected.
 
 The *resolution* of a set of states is obtained as follows:
 
-1.  Take all *power events* and any events in their auth chains,
+1.  Select all *power events* that appear in the *full conflicted set*. Compute
-    recursively, that appear in the *full conflicted set* and order them
+    the union of their auth chains, including the power events themselves.
-    by the *reverse topological power ordering*.
+    Sort the union using the *reverse topological power ordering*.
 2.  Apply the *iterative auth checks algorithm*, starting from the
     *unconflicted state map*, to the list of events from the previous
     step to get a partially resolved state.