Tuesday, August 28, 2007

Levels of abstraction.

Warning: theoretical ranting ahead!

Before our house, along the water there used to stand about 20 pollard willows. Once every year the branches are truncated. The guys who do this have no idea what they are doing, so they cut all the life away. Even a strong tree like a willow cannot withstand such bad treatment forever. After 7 years 17 trees died. When I confronted the workmen with their incompetence, they were very surprised that there could be a relation between their treatment of the trees and the deceasing of the trees. They had never even thought of it. They thought it was an act of nature or something.

Just as stupid is my way of looking at a chessposition. I tend to see the position in my games as given facts, bearing no relation with my previous moves. Opportunities seem to materialize out of the blue.

When I tried to comment on Blue Devils comment on my latest post, a lot of new thoughts arose which I like to share. For your convenience I repeat his comment here:

Each time you explain it, it gets more useful and clear.

I have found that in my self-explanations of solutions there is a kind of optimal level of abstraction. I am getting to the point now where it sort of "clicks" when I am at the right level for a particular problem. Once I hit that optimal level it makes me more likely to remember the solution to that problem. Just memorizing moves is not enough: that is too concrete (even for mate in one, there are all sorts of features of the position to think about more generally: escape squares, the mating pattern used, etc).

It isn't that focusing on the solution isn't enough. Rather, a priori it is tricky to know the appropriate level of abstraction at which to explain the solution (or in your language, for the narrative). There is often a kind of cognitive "sweet spot", a level of abstraction that just feels right. But sometimes it takes a really long time to find it, like with this problem.

Plus, there probably isn't an objectively best level of generality. It depends how much you already know. E.g., I can use the concept of a 'trebuchet' as a basic explanatory device, but two years ago this would have made no sense.

I like to brainstorm about if it is a good idea to limit the study of a solution to a certain level of abstraction. If I take a look at the willow-killers, they saw trees each year as new. As something that had no relation with the trees of the previous year. Hence they couldn't see the relationship between their actions and the decease of the willows a year later. How weird it may sound, their surprise was genuine when I told them so.

It shows how sequential our consciousness work. We have difficulty to see things parallel. We tend to seperate the events in the tunnel of time. We see every situation as new. Take a look at a comic strip. The hero is present at 25 pictures at the same time. Yet we look at it as one hero going through the tunnel of time. We look at one picture at the time, forgetting the previous ones, seeing the next one as new.

If you step back, take some distance, zoom out, you get more overview. You start to see the relationships between topics. You see causes and consequenties. You will learn new things you didn't know before while being a prisoner of the sequential tunnel.

Scientific research showed that the amateur sees every chess position as new. His brainscans reveal that he uses a part of the brain that is suitable for solving new complex problems. He makes heavy use of his short term memory.

And that is exactly how I experience my own games. If I have a rook ending, it appears out of the blue. I don't relate that to my own actions before. If I take the position from Blue Devil of my last post, I look at it as new. Having quite forgotten that I studied a similar position for a week about 1.5 year ago.

In order to overcome this problem it looks logical to build a meaningfull framework of a chessgame. If you can give every position you work on a place in this framework, it will become easier to retrieve information from previous positions.

Generalisation of a solution of a problem is the way to go. When you formulate higher abstraction levels, you zoom out. Which lessens the amount of visible details, but it increases the overview and consistency of patterns. Every higher level of abstraction reveals new facts.
If I take the position of Blue Devil from yesterday, there are two new topics that I had to formulate:
  • When there are two pawns of opposite color on the board, at least one of them will fall.
  • The distance of the kings to the enemy pawn is all important.
These two facts are new to me. That makes it mentally demanding to generalise solutions. It takes a lot of effort to find these new facts. Once found, they are of a head-slapping triviality. Why hadn't I thought of this before?

It is hard to think of a reason why you should stop at a certain level of abstraction. Of course the more you zoom out, the more lack of detail there is. But I think that it is usefull to have a framework that covers the whole game. Since it helps you to give every chess experience a place. A framework should give you more cues to retrieve similar positions from memory.

Let me try in a crude way to formulate a branch of the total chesstree in relation to yesterdays problem.

Level 0. Win the game.
Level 1. There are two ways to win the game:
  • Go after the enemy king
  • Queen a pawn and go after the king.
Level 2. Queen a pawn. In order to queen a pawn you must:
  • Penetrate in the opponents position.
  • Conquer the blockader of your pawn.
  • Free the road to promotion.
Level 3. Techniques of penetration.
  • Walk with your king to the center.
  • Attack the blockading pawns from behind with your pieces.
  • Attack two weaknesses at the same time.
. . .

Level n-1.
  • When there are two blocked pawns of opposite color on the board, one of them is going to fall.
  • The king that is closest to the enemy pawn is paramount.
  • The method of conquering the pawn is based on zugzwang.
  • If you are going to lose your pawn anyway, the only way to a draw is when you step on the keysquare right after the opponent captures your pawn.
  • Take always the widest arc possible without losing tempos to the keysquare, to prevent the enemy king from winning a tempo by shouldering you away.
Level n. The concrete position.

Of course it will take time to formulate what there is between level 3 and level n-1. But I'm sure it will reveal new facts. I'm thinking of the following: Can the statement When there are two blocked pawns of opposite color on the board, one of them is going to fall be generalized to two pawns at the same file, no matter the distance? And how about the pawns being on adjacent files, or quite separate files?

The framework I describe above is the positive framework of the attacker. The defender has ofcourse a negative form of the framework. Prevent the attacker from removing the blockader etc..

If I try to give Blue Devil's position a place in the framework it would be something like:
This is an endgame position where the main goal is to queen a pawn. In order to do that, the attacker has to remove the blockading pawn. Since black is going to achieve that, white must blockade the road to promotion with his king by means of opposition, which is a special case of zugzwang.

The coming time I will experiment with building a framework and investigate its ability to connect seemingly different patterns in order to ease retrieval from memory.


  1. I agree that trying to understand a position at multiple levels is important. However, there is an entry point to the tower of abstraction, the level at which I enter being the level at which I have the initial "aha!" and feel that I've finally seen the solution. Usually this is not at the lowest level (moves), nor the highest level, but a kind of mid-level narrative using chunks like 'tribochet'. I call this the 'entrance level.' This entrace-level understanding is key, especially for remembering that particular problem.

    However, once I've entered the tower, I need to walk around in it. Go downstairs to the basement level, looking at various alternative moves, going upstairs in the heirarchy thinking about the general positional features that made the tactic possible (in the case of tactical problems). As you suggest, the important features of the position and solution form a kind of heirarchically organized structure, and the more floors I visit (once I am allowed in at the entrance level) the better.

    It is quite funny, actually, as your levels stuff is almost exactly like the new diagram of the "heirarchy of plans" that I recently put into my thought process document (at the top of the heirarchy of plans is to mate, then below that is the plan of increasing advantage/decreasing disadvantages and doing the opposite for the opponent. Below that are things that are actually useful like piece activity, material. Then more specific: activate the bishop is under the 'piece activity' plan. And once you get to this low level, you are at the location in the heirarchy of plans that immediately suggests sets of candidate moves because they talk about specific properties of specific pieces (groups of pieces) or squares (groups of squares).

  2. It really is funny: here is cut and paste from my thought process document in progress:
    This example illustrates some important features of the logical structure of plans. There exists a hierarchy of plans. At the top is the most general plan in the hierarchy: to win the game by mating your opponent. As we move down this hierarchy of plans, we end up at very specific plans, highly dependent on the features of the position, that often mention specific pieces and squares. These specific plans transparently dovetail with candidate moves.

    [Note there are examples and a crazy heirarchical figure drawn to show this in the document...]

  3. I think there are subtle differences between what you are saying and the way I'm thinking, but there is a remarkable convergence as well.