Copyright (c) 2013 John L. Jerz

Experimental Results
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Experimental Results

Software has been written to implement the ideas behind the proposed heuristic. The process is slow, but my goals are as follows:
 
1. Construct prototype evaluation function in C++ programming language.
 
2. Transplant evaluation function into micro-max (freely available, minimalist chess program).
 
3. Refine evaluation function.
 
4. Add more features to micro-max.
 
There were some surprises when writing the code:
 
1. There were a lot of redundant entries in the mobility tables.
2. There is a lot of data.
3. Keeping track of everything is difficult.
4. There are many opportunities for optimizing, but that will be done later.
5. The choice of a data structure is turning out to be critical.
 
- 1713 source lines of code (sloc) so far (method: counting semi-colons)
- C++ programming language
- minGW compiler (minimalist GNU for Windows)
 
Here is some of the output generated by the source code, as of January 5, 2008:
 

Test Position 1
qf2.jpg
Black to move after 13.Qf2

1st, 2nd & 3rd order mobility/ influence tables, automatically generated for Bishop on c1:
 
 5: root c1 w bishop UpperLeft (c1-b2: w pawn) move[b2-b3 0-0] BLOCKED-F-MOVEABLE-DIRECTIONAL_ISSUE
 6: root c1 w bishop UpperRight (c1-d2: w pawn) move[d2-d3 0-0] BLOCKED-F-MOVEABLE-DIRECTIONAL_ISSUE
 
 18: root c1 w bishop UpperLeft (c1-b2: w pawn) move[b2-b3 0-0] EMPTYFORMOBILITY
 19: root c1 w bishop UpperLeft (c1-a3: empty) move[b2-b3 0-0] EMPTYFORMOBILITY
 20: root c1 w bishop UpperRight (c1-d2: w pawn) move[d2-d3 0-0] EMPTYFORMOBILITY
 21: root c1 w bishop UpperRight (c1-e3: empty) move[d2-d3 0-0] EMPTYFORMOBILITY
 22: root c1 w bishop UpperRight (c1-f4: empty) move[d2-d3 0-0] EMPTYFORMOBILITY
 23: root c1 w bishop UpperRight (c1-g5: empty) move[d2-d3 0-0] EMPTYFORMOBILITY
 24: root c1 w bishop UpperRight (c1-h6: empty) move[d2-d3 0-0] EMPTYFORMOBILITY
 
 61: root c1 w bishop UpperRight (b2-c3: empty) move[b2-b3 0-0] EMPTYFORMOBILITY
 62: root c1 w bishop UpperRight (b2-d4: w pawn) move[b2-b3 0-0] BLOCKED-F-NOTMOVEABLE
 63: root c1 w bishop LowerRight (b2-c1: empty) move[b2-b3 0-0] EMPTYFORMOBILITY
 64: root c1 w bishop LowerLeft (b2-a1: w rook) move[b2-b3 0-0] BLOCKED-F-NOTMOVEABLE
 65: root c1 w bishop UpperRight (a3-b4: empty) move[b2-b3 0-0] EMPTYFORMOBILITY
 66: root c1 w bishop UpperRight (a3-c5: empty) move[b2-b3 0-0] EMPTYFORMOBILITY
 67: root c1 w bishop UpperRight (a3-d6: b bishop) move[b2-b3 0-0] BLOCKED-E-CAPTURABLE
 68: root c1 w bishop LowerRight (a3-c1: empty) move[b2-b3 0-0] EMPTYFORMOBILITY
 69: root c1 w bishop UpperLeft (d2-c3: empty) move[d2-d3 0-0] EMPTYFORMOBILITY
 70: root c1 w bishop UpperLeft (d2-b4: empty) move[d2-d3 0-0] EMPTYFORMOBILITY
 71: root c1 w bishop UpperLeft (d2-a5: empty) move[d2-d3 0-0] EMPTYFORMOBILITY
 72: root c1 w bishop LowerRight (d2-e1: w king) move[d2-d3 0-0] BLOCKED-F-MOVEABLE
 73: root c1 w bishop LowerLeft (d2-c1: empty) move[d2-d3 0-0] EMPTYFORMOBILITY
 74: root c1 w bishop UpperLeft (e3-d4: w pawn) move[d2-d3 0-0] BLOCKED-F-NOTMOVEABLE
 75: root c1 w bishop LowerRight (e3-f2: w queen) move[d2-d3 0-0] BLOCKED-F-MOVEABLE
 76: root c1 w bishop LowerLeft (e3-c1: empty) move[d2-d3 0-0] EMPTYFORMOBILITY
 77: root c1 w bishop UpperLeft (f4-e5: empty) move[d2-d3 0-0] EMPTYFORMOBILITY
 78: root c1 w bishop UpperLeft (f4-d6: b bishop) move[d2-d3 0-0] BLOCKED-E-CAPTURABLE
 79: root c1 w bishop LowerRight (f4-g3: w knight) move[d2-d3 0-0] BLOCKED-F-MOVEABLE
 80: root c1 w bishop LowerLeft (f4-c1: empty) move[d2-d3 0-0] EMPTYFORMOBILITY
 81: root c1 w bishop UpperLeft (g5-f6: empty) move[d2-d3 0-0] EMPTYFORMOBILITY
 82: root c1 w bishop UpperLeft (g5-e7: empty) move[d2-d3 0-0] EMPTYFORMOBILITY
 83: root c1 w bishop UpperLeft (g5-d8: empty) move[d2-d3 0-0] EMPTYFORMOBILITY
 84: root c1 w bishop LowerRight (g5-h4: b queen) move[d2-d3 0-0] BLOCKED-E-CAPTURABLE
 85: root c1 w bishop LowerLeft (g5-c1: empty) move[d2-d3 0-0] EMPTYFORMOBILITY
 86: root c1 w bishop UpperLeft (h6-g7: b pawn) move[d2-d3 0-0] BLOCKED-E-CAPTURABLE
 87: root c1 w bishop LowerLeft (h6-c1: empty) move[d2-d3 0-0] EMPTYFORMOBILITY
 
LEGEND:
 ▒  1 move influence
o  2 move influence
.  3 move influence
X piece is constrained from this square
 
┌───┬───┬───┬───┬───┬───┬───┬───┐     ┌───┬───┬───┬───┬───┬───┬───┬───┐
│   │ . │   │ . │ . │ . │ . │   │     │   │ X │   │ X │ X │ X │ X │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │ . │ . │ . │ . │ o │ . │ . │     │ X │ . │ X │ X │ X │ X │ X │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │ . │ . │ . │ . │ o │ . │ o │     │ . │ X │ . │ X │ X │ X │ X │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │ o │ . │ . │ . │ o │ o │ . │     │ . │ o │ X │ X │ X │ X │ X │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │ . │ o │ ▒ │ o │ o │ . │ o │     │ X │ X │ X │ ▒ │ o │ X │ X │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ o │ o │ o │ o │ ▒ │ ▒ │ ▒ │ . │     │ X │ X │ X │ X │ X │ X │ X │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │ o │ o │ ▒ │ ▒ │ Q │ ▒ │ o │     │ . │ o │ o │ ▒ │ ▒ │ Q │ ▒ │ o │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │ . │ o │ ▒ │ ▒ │ ▒ │ o │     │   │ . │ . │ o │ ▒ │ ▒ │ ▒ │ o │
└───┴───┴───┴───┴───┴───┴───┴───┘     └───┴───┴───┴───┴───┴───┴───┴───┘
Influence Diagram, w queen on f2 and corresponding Engagement Diagram.

┌───┬───┬───┬───┬───┬───┬───┬───┐     ┌───┬───┬───┬───┬───┬───┬───┬───┐
│   │   │ . │   │ . │   │ . │   │     │   │   │ X │   │ X │   │ X │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │   │ . │ o │ . │ o │ . │     │   │ X │   │ X │ X │ X │ X │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │   │ . │ o │ . │ o │ . │ o │     │ . │   │ X │ X │ X │ X │ X │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │ o │ . │   │ ▒ │ o │ ▒ │     │   │ X │ o │ X │   │ X │ X │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │   │ . │ o │ ▒ │ o │ . │ o │     │ X │   │ X │ X │ ▒ │ X │ X │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │ o │ . │ o │ . │ N │ . │     │   │ . │ o │ X │ X │ X │ N │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │   │ . │ o │ ▒ │ o │ . │ . │     │ . │   │ X │ o │ ▒ │ o │ X │ . │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │ o │ . │   │ ▒ │ o │ ▒ │     │   │ . │ o │ X │   │ ▒ │ o │ ▒ │
└───┴───┴───┴───┴───┴───┴───┴───┘     └───┴───┴───┴───┴───┴───┴───┴───┘
Influence Diagram, w knight on g3 and corresponding Engagement Diagram.
Note that the Queen mobility to squares f5, f6 and f7 can be accomplished in 2 moves if we play the capture move f3xe4 on move 1.
 
┌───┬───┬───┬───┬───┬───┬───┬───┐     ┌───┬───┬───┬───┬───┬───┬───┬───┐
│   │   │   │   │   │   │   │   │     │   │   │   │   │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │   │   │   │   │   │   │   │     │ X │   │   │   │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │   │   │   │   │   │   │   │     │ . │   │   │   │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │   │   │   │   │   │   │   │     │ . │   │   │   │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ o │   │   │   │   │   │   │   │     │ o │   │   │   │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ o │ . │ . │ . │ . │ . │   │   │     │ X │ X │ X │ X │ X │ X │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ ▒ │ . │   │   │   │   │   │   │     │ ▒ │ . │   │   │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ R │ ▒ │ o │   │   │   │   │   │     │ R │ ▒ │ o │   │   │   │   │   │
└───┴───┴───┴───┴───┴───┴───┴───┘     └───┴───┴───┴───┴───┴───┴───┴───┘
Influence Diagram, w rook on a1 and corresponding Engagement Diagram.
 
Note that the pawn on a2 can advance to a4 then capture the pawn on b5, opening the way for the rook to move to a6 and a7.
 
Look at all this useful information we can extract from the position of the pieces on the board!
 
The machine can use these influence maps as "probes" to determine how fully engaged each piece is in the game. Of course, there are other calculations we can do to gain insight into how well each piece is positioned, but there would have to be a corresponding performance increase in spending the time to make the additional calculation. The proposed heuristic has "bang for the buck" in terms of insight versus speed of calculation.
 
Once all the influence maps are generated for each piece, we can determine the limiting factors that are likely to restrict the movement of each piece.
 
We need to decide on a rule, and the simplest one is this: make a map of the lowest value piece (consider Bishops and Knights to be equivalent for now) that can attack each square, for each side. The idea here is that each lower value piece constrains the movement of the opponent's higher value pieces because it can threaten to capture it. For example, a white pawn that attacks a certain square will most likely prevent the opponent's pieces (bishops, knights, rooks, queens and the king) from moving there, but this is not so true for the opponent's pawns.
 
The other condition it makes sense to include in our constraint maps is a simple test for a "hung" piece (a piece moves to a square attacked by an enemy piece and is unsupported by a friendly piece).
 
Black constraint map
┌───┬───┬───┬───┬───┬───┬───┬───┐
│   │ r │ n │ r │ r │ n │ r │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ r │   │ n │ n │ n │ n │   │ r │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ p │   │ p │ p │ p │ p │ p │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │ n │ n │ n │ n │ q │ q │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ p │ n │ p │   │ q │ n │ n │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ n │   │ n │ p │ n │ p │ n │ n │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │   │   │   │   │   │ q │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │   │   │   │   │   │   │
└───┴───┴───┴───┴───┴───┴───┴───┘
 
White constraint map
┌───┬───┬───┬───┬───┬───┬───┬───┐
│   │   │   │   │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │   │   │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │   │   │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ N │ P │   │ P │ N │   │ N │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │ N │ Q │ P │   │ P │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ P │ P │ P │ N │ P │ P │ P │ P │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ R │ N │   │ N │ N │ K │ N │ R │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ R │   │ K │ Q │ N │ R │ N │
└───┴───┴───┴───┴───┴───┴───┴───┘
 
Now if we take the black constraint map and apply it to the White Bishop influence map (using the rules we mentioned above, including a simple test for a hung piece)  we get this engagement diagram:
 
┌───┬───┬───┬───┬───┬───┬───┬───┐     ┌───┬───┬───┬───┬───┬───┬───┬───┐
│   │   │   │ . │   │   │   │   │     │   │   │   │ X │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │   │   │ . │   │ . │   │     │   │   │   │   │ X │   │ X │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │   │ . │   │ . │   │ o │     │   │   │   │ X │   │ X │   │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │   │ . │   │ . │   │ o │   │     │ . │   │ . │   │ X │   │ X │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │   │ . │   │ o │   │ . │     │   │ X │   │ . │   │ X │   │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ o │   │ . │   │ o │   │ . │   │     │ o │   │ . │   │ o │   │ X │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ ▒ │   │ ▒ │   │ . │   │   │     │   │ ▒ │   │ ▒ │   │ . │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │   │ B │   │ . │   │   │   │     │ . │   │ B │   │ . │   │   │   │
└───┴───┴───┴───┴───┴───┴───┴───┘     └───┴───┴───┴───┴───┴───┴───┴───┘
Influence Diagram, w bishop on c1 and corresponding Engagement Diagram.
 
There are other places where the Bishop is not safe, but we decide to determine piece safety through search. The maps are useful in 1) steering our search efforts 2) deciding which lines of analysis we are no longer interested in and 3) evaluating the usefulness of the positions at the end points of our search tree
 
We can now use these maps to evaluate the level of engagement of this piece in the game.
 
Example Number 2

Hydra (computer)
NickelHydra.jpg
Arno Nickel

The above position was reached after 18.g4 was played in a 2004 correspondence game between correspondence GM Arno Nickel and the Hydra computer.
 
The black knight on h6 is trapped. Perhaps it is better just to say that it has been constrained to stay on the edge of the board by the white pawns and the location of the other pieces. The machine, calculating 200 million positions per second,  probably doesn't even realize this. The influence diagram and engagement diagram below strongly suggest that the black knight has problems with realistic attempts to move on the board.
 
┌───┬───┬───┬───┬───┬───┬───┬───┐     ┌───┬───┬───┬───┬───┬───┬───┬───┐
│   │   │ . │ . │ . │   │ ▒ │ . │     │   │   │ X │ . │ X │   │ ▒ │ . │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │   │ . │ o │ ▒ │ o │ . │     │   │ X │   │ X │ X │ ▒ │ X │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │ . │ o │ . │ o │ . │ n │     │   │   │ X │ X │ X │ X │ X │ n │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │   │ . │ o │ ▒ │ . │ . │     │   │ X │   │ X │ X │ X │ X │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │ . │ o │ . │   │ ▒ │ o │     │   │   │ X │ X │ X │   │ X │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │   │ . │ o │ . │ o │ . │     │   │ X │   │ X │ X │ X │ X │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │ . │   │ . │ o │ . │ o │     │   │   │ X │   │ X │ X │ X │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │   │ . │   │ . │   │ . │     │   │   │   │ X │   │ X │   │ X │
└───┴───┴───┴───┴───┴───┴───┴───┘     └───┴───┴───┴───┴───┴───┴───┴───┘
Influence Diagram, b knight on h6 and corresponding Engagement Diagram.
 
Example Number 3

Miralles
Miralles.jpg
Emms, black to move

John Emms, in his book Simple Chess, reached the above position as white with the idea of restricting the mobility of black's knight on b7. Exactly how bad is this piece? A traditional chess program may never figure this out. Let's see what the influence diagram and engagement diagram from the proposed heuristic shows us:
 
┌───┬───┬───┬───┬───┬───┬───┬───┐     ┌───┬───┬───┬───┬───┬───┬───┬───┐
│   │ . │   │ ▒ │   │ . │   │   │     │   │ X │   │ ▒ │   │ X │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │ n │ . │ . │ . │ o │ . │   │     │ X │ n │ X │ X │ X │ o │ X │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │ . │ o │ ▒ │ o │   │   │   │     │ X │ X │ X │ ▒ │ X │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ ▒ │   │ ▒ │   │ . │   │ . │   │     │ X │   │ X │   │ X │   │ X │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │ . │ o │ . │ . │ . │   │   │     │ X │ X │ X │ X │ X │ X │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │ o │   │ . │ . │   │   │   │     │ X │ X │   │ X │ X │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │   │ . │   │   │   │   │     │   │ X │   │ X │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │   │ . │   │   │   │   │   │     │ X │   │ X │   │   │   │   │   │
└───┴───┴───┴───┴───┴───┴───┴───┘     └───┴───┴───┴───┴───┴───┴───┴───┘
Influence Diagram, b knight on b7 and corresponding Engagement Diagram.
 
We can "see" that the movement of the piece has been constrained.
 
Example 4

Emms
bakeremms.jpg
Baker, black to move

In this position the black pawns are doing a great job restricting the mobility of the white Bishop, which is not fully engaged in the game. Just how bad is this piece?
┌───┬───┬───┬───┬───┬───┬───┬───┐     ┌───┬───┬───┬───┬───┬───┬───┬───┐
│ . │   │   │   │   │   │   │   │     │ X │   │   │   │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │   │ . │   │ . │   │   │     │   │ X │   │ X │   │ X │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │ . │   │ o │   │   │   │     │   │   │ X │   │ X │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │   │ o │   │ . │   │   │     │   │ X │   │ X │   │ X │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ o │   │ o │   │ . │   │ . │   │     │ X │   │ X │   │ X │   │ X │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ o │   │ ▒ │   │ . │   │ . │     │   │ X │   │ ▒ │   │ . │   │ X │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│ ▒ │   │ ▒ │   │ . │   │   │   │     │ ▒ │   │ ▒ │   │ . │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤     ├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ B │   │ o │   │   │   │   │     │   │ B │   │ o │   │   │   │   │
└───┴───┴───┴───┴───┴───┴───┴───┘     └───┴───┴───┴───┴───┴───┴───┴───┘
Influence Diagram, w bishop on b1 and corresponding Engagement Diagram.
 
The influence map shows that this piece has great mobility, if that is all you are counting. However, this piece has been effectively constrained by black. The engagement diagram tells us this. Neil McDonald in The Art of Planning in Chess says:
 
p.189"As so often, we need to remind ourselves that it isn't the fortune of the individual piece that matters - it is how it functions with the other pieces. For this reason it is often completely meaningless to count the number of squares, in the centre or otherwise, that a piece controls or attacks."
 
Conclusion
 
When we generate these maps, we obtain insight through an effect that is like an x-ray - we can tell at a glance from the influence diagram and engagement diagram of the proposed heuristic which pieces are "good" and "bad", and we can use these "constraints" to steer our search efforts, prune our search tree, and evaluate our endpoint positions.
 
We can, in addition, use concepts from positional chess theory in our evaluation function, and this can help us to predict, with a greater degree of accuracy, which positions are more promising than others.
 
 

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