Conspiracy Numbers

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Jonathan Schaeffer on Conspiracy Numbers. [1]

Conspiracy Numbers of the root or interior nodes of a search tree for some value V are defined as the least number of conspirators, that are leaves that must change their evaluation value to V in order to change the minimax value of the interior node or root [2]. Conspiracy Numbers and their possible application for Minimax search within a best-first search algorithm was first described by David McAllester [3].

Sample

Minimax Tree

A sample minimax tree T with some arbitrary values of the leaves [4]:

root                    ┌───────┐
max node                │  A=3  │
                        └───────┘
           ┌───────┐                 ┌───────┐
min nodes  │  B=2  │                 │  C=3  │
           └───────┘                 └───────┘
  ┌───────┐       ┌───────┐   ┌───────┐       ┌───────┐
  │  D=5  │       │  E=2  │   │  F=3  │       │  G=4  │
  └───────┘       └───────┘   └───────┘       └───────┘

Conspiracy Numbers

Conspiracy numbers for all possible values of the root A
v cn(A, v) conspirators
<= 1 2 (D or E) and (F or G)
2 1 (F or G)
3 0 none
4 1 (E or F)
5 1 E
>= 6 2 (D and E) or (F and G)
Conspiracy numbers for all possible values of node B
v cn(B, v) conspirators
<= 1 1 (D or E)
2 0 none
3,4,5 1 E
>= 6 2 (D and E)
Conspiracy numbers for all possible values of node C
v cn(C, v) conspirators
<= 2 1 (F or G)
3 0 none
4 1 F
>= 5 2 (F and G)

Recursive Definition

Following recursive definition in pseudo C is based on Van der Meulen's code [5]. V(J) represents the minimaxed value of node J. Opposed to McAllester's original definition which deals with pure game theoretic values, Van der Meulen's distinguished non terminal leaves with cn = 1 for values different of v from game theoretic terminal nodes to assign +oo, since it is impossible to change their value, independently been arrived at by Norbert Klingbeil and Jonathan Schaeffer [6]:

int cn(CNode J, int v) {
   int c;
   if ( V(J) == v ) {
      c = 0;
   } else if ( isTerminal(J) ) { 
      c = +oo; /* checkmate, stalemate, tablebase score, etc. */
   } else if ( isLeaf(J) ) {
      c = 1; 
   } else if (isMaxNode(J) && v < V(J) ) {
      c = 0;
      for (all childs J.j)
         if (v < V(J.j) ) c += cn(J.j, v); /* sum */
   } else if (isMinNode(J) && v > V(J) ) {
      c = 0;
      for (all childs J.j)
         if (v > V(J.j) ) c += cn(J.j, v); /* sum */
   } else {
      c = +oo;
      for (all childs J.j)
         c = min( cn(J.j, v), c);
   }
   return c;
}

Conspiracy Theory

Let δ be a number called the singular margin [7]. Conspiracy theory can be formulated using the following definition [8]:

Definition: Let T be a search tree with min-max value V[T]. The lower boand conspiracy number of T, denoted C<[T], is the number of leaf static values that must be changed to bring the root min-max value down to V[T]-δ. The upper boand conspiracy number of T, denoted C>[T], is the number of leaves that must be changed to bring the root value up to V[T]+δ. 

C<[T] expresses the confidence that the lower bound α will hold by further expansion of the search tree.

Search Algorithms

McAllester's aim was related to some drawbacks of alpha-beta, at the worst, the decision at the root is based on a single evaluation of one leaf. If that leaf has assigned an erroneous value, the decision may be disastrous [9]. The idea of Conspiracy Number Search (cn-search) and its variants is to continue until it is unlikely that the minimax value at the root will change.

Publications

[10]

1985 ...

1990 ...

1995 ...

2000 ...

2010 ...

External Links

Conspiracy Numbers

Conspiracy

References

  1. Photo from Advances in Computer Chess 5 by László Lindner, ICCA Journal, Vol. 10, No. 3, pp. 138
  2. Definition, Sample, and Pseudo code taken from Maarten van der Meulen (1990). Conspiracy-Number Search. ICCA Journal, Vol. 13, No. 1
  3. David McAllester (1988). Conspiracy Numbers for Min-Max Search. Artificial Intelligence, Vol. 35, No. 1, pp. 287-310. ISSN 0004-3702
  4. due to Jonathan Schaeffer (1989). Conspiracy Numbers. Advances in Computer Chess 5
  5. Maarten van der Meulen (1990). Conspiracy-Number Search. ICCA Journal, Vol. 13, No. 1
  6. Norbert Klingbeil, Jonathan Schaeffer (1988). Search Strategies for Conspiracy Numbers. Canadian Artificial Intelligence Conference, pp. 133-139
  7. The term singular margin comes from the singular extension algorithm (Anantharaman et al. 1990)
  8. David McAllester, Deniz Yuret (1993). Alpha-Beta Conspiracy Search. ps (draft)
  9. Ulf Lorenz, Valentin Rottmann (1996). Parallel Controlled Conspiracy-Number Search. Advances in Computer Chess 8
  10. ICGA Reference Database

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