Piece mobility is defined as the number of squares you control. I love the simplicity of this definition since it is just a matter of counting. It says nothing about the quality of those squares though. That's why I think that piece activity is a better word to use for the things I'm talking about.
But what is the activity of a piece?
Just counting the owned squares isn't enough.
A rook at a1 can cover your whole backrank. Still it can be a very inactive piece. So activity has something to do with the enemy camp.
Just counting your squares at the enemy side of the board, seems to be an arbitrary choice.
A bishop can be good, bad or active.
A bishop is considered good when there are no central pawns which obstructs its activity.
A bishop is considered bad when the central pawns stand on its color thus blocking it.
An active bishop can be either good or bad; it is called active just because it performs some active function. Often the difference is if it is in- or outside the pawnchain.
A piece bound to defense is very restricted. Although it may cover a lot of squares, it cannot go there without losing material. A rook that attacks a pawn which is defended by a rook is called active, since it can go everywhere. The defending piece is called passive, since it hasn't that freedom. The moment the attacking rook moves somewhere else, the passive rook is released.
So the difference in mobility is decided by just one tempo. Hence the difference between active and passive can be decided by just one tempo.
A knight outpost is good when the knight is in the neighbourhood of the enemy king. So the activity seems to have to do something with targets in the hostile camp. It are not quite the same targets as we are used to in tactical problems. The king is a target, weak pawns are targets. Enemy pieces are just too volatile to be targets by non tactical means.
So in general: sitting ducks are the targets.
Sometimes, if black has a bishop at f5 and a rook at c8, black can penetrate via c2. Which becomes a new home for a black piece. So active pieces are strong when they work together.
In general: an active piece can have a host of potential new homes along it's open line from where it can be even more active.
Can there be an algorithm which expresses the activity of a piece?
Today's random chess quote at chesshere:
Michael Stean: The most important feature of the chess position is the activity of the pieces. This is absolutely fundamental in all phases of the game (opening, middlegame and especially endgame). The primary constraint on a piece's activity is the Pawn structure.