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On the old Cube-Lovers list, the terms Starts-with and Ends-with were defined as follows.  For a cube position x, StartsWith(x)=S(x) is the set of all moves with which a minimal maneuver can start and EndsWith(x)=E(x) is the set of all moves with which a minimal maneuver can end.

The concept is much older than Cube-Lovers, of course.  It's obvious that from any position except for Start itself, there must be at least one move which takes the Cube closer to Start.  The set of all such moves is simply the set of inverses of E(x).