But when the weaknesses it has require looking 20 ply into the game, can anyone exploit those weaknesses? And furthermore, if the computer itself is able to see 20 ply into the game, then it can spot its own weaknesses and you need to look even further, making the question of whether it's really a weakness.

Weaknesses are only relative to capabilities of the opponent to exploit them. If a tank has a weak spot that rockets can hit, but it's being opposed by humans on horseback, is it really a weakness in that context?

The weaknesses that it has will be of the form that it has wrong opinions about certain kinds of positions. In the case of chess, those weaknesses showed up in closed positions where the program owned the center and large amounts of space. In the case of AlphaGo, the weaknesses will be much more subtle, but will be discoverable and exploitable in time.

Additionally AlphaGo has the advantage that it started with a database of human play, so it has some ideas what kinds of positions humans miscalculate.

As for your tank vs horseback analogy, that's flawed at the moment. AlphaGo is probably reasonably close in strength to the human facing him. Improved human knowledge could tip the balance.

However in the future it will become an apt analogy. Computers are going to become so good that knowing the relative weaknesses in their style of play may reduce the handicap you need against them, but won't give you a chance of becoming even with them. That happened close to 20 years ago in chess, and is now only a question of time in Go.

I wonder if AlphaGo has some specialized knowledge to handle ladders, where stones can have an effect at a distance that might only come into play after 20 moves.

>I wonder if AlphaGo has some specialized knowledge to handle ladders

Yes. A representation of ladders is among the input features of its neural networks.

https://gogameguru.com/i/2016/03/deepmind-mastering-go.pdf

Stone colour 3 Player stone / opponent stone / empty

Ones 1 A constant plane filled with 1

Turns since 8 How many turns since a move was played

Liberties 8 Number of liberties (empty adjacent points)

Capture size 8 How many opponent stones would be captured

Self-atari size 8 How many of own stones would be captured

Liberties after move 8 Number of liberties after this move is played

Ladder capture 1 Whether a move at this point is a successful ladder capture

Ladder escape 1 Whether a move at this point is a successful ladder escape

Sensibleness 1 Whether a move is legal and does not fill its own eyes

Zeros 1 A constant plane filled with 0

Player color 1 Whether current player is black

(The number is how many 19x19 planes the feature consists of.)

20 moves may sound like a huge number of variations but when you prune things early, it can be quite manageable. The alphabeta algorithm in chess does pruning quite a lot.