Conspiracy Numbers

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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. Conspiracy Numbers and their possible application for Minimax search within a best-first search algorithm was first described by David McAllester.

=Sample=

Minimax Tree
A sample minimax tree T with some arbitrary values of the leaves : root                   ┌───────┐ max node               │  A=3  │ └───────┘          ┌───────┐                 ┌───────┐ min nodes  │  B=2  │                 │  C=3  │ └───────┘                └───────┘  ┌───────┐       ┌───────┐   ┌───────┐       ┌───────┐  │  D=5  │       │  E=2  │   │  F=3  │       │  G=4  │ └───────┘      └───────┘   └───────┘       └───────┘

Conspiracy Numbers
=Recursive Definition= Following recursive definition in pseudo C is based on Van der Meulen's code. 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 : 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. Conspiracy theory can be formulated using the following definition :

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 Bound|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. 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.


 * Alpha-Beta Conspiracy Search
 * Conspiracy Number Search
 * Proof-Number Search

=Publications=

1985 ...

 * David McAllester (1985). A New Procedure for Growing Minimax Trees. Technical Report, Artificial Intelligence Laboratory, MIT
 * David McAllester (1988). Conspiracy Numbers for Min-Max Search. Artificial Intelligence, Vol. 35, No. 1
 * Ingo Althöfer (1988). Root Evaluation Errors: How they Arise and Propagate. ICCA Journal, Vol. 11, Nos. 2/3
 * Maarten van der Meulen (1988). Parallel Conspiracy-Number Search. M.Sc. thesis, Faculty of Mathematics and Computer Science, Vrije Universteit, Amsterdam
 * Norbert Klingbeil (1988). Search Strategies for Conspiracy Numbers. M.Sc. thesis
 * Norbert Klingbeil, Jonathan Schaeffer (1988). Search Strategies for Conspiracy Numbers. Canadian Artificial Intelligence Conference, pp. 133-139
 * Jonathan Schaeffer (1989). Conspiracy Numbers. Advances in Computer Chess 5. » also published
 * Charles Elkan (1989). Conspiracy Numbers and Caching for Searching And/Or Trees and Theorem-Proving. IJCAI 1989, pdf

1990 ...

 * Maarten van der Meulen (1990). Conspiracy-Number Search. ICCA Journal, Vol. 13, No. 1
 * Norbert Klingbeil, Jonathan Schaeffer (1990). Empirical Results with Conspiracy Numbers. Computational Intelligence, Vol. 6, pp. 1-11, ps
 * Jonathan Schaeffer (1990). Conspiracy Numbers. Artificial Intelligence, Vol. 43, No. 1, pp. 67-84
 * Maarten van der Meulen, Victor Allis, Jaap van den Herik (1990). A Comment on `Conspiracy-Number Search. ICCA Journal, Vol. 13, No. 2
 * Victor Allis, Maarten van der Meulen, Jaap van den Herik (1991). Conspiracy-Number Search. Advances in Computer Chess 6
 * David McAllester, Deniz Yuret (1993). Alpha-Beta Conspiracy Search. ps (draft)
 * Lisa Lister, Jonathan Schaeffer (1994). An Analysis of the Conspiracy Numbers Algorithm. Computers & Mathematics with Applications Vol. 27, No. 1, Elsevier, pdf
 * Deniz Yuret (1994). The Principle of Pressure in Chess. TAINN 1994

1995 ...

 * Ulf Lorenz, Valentin Rottmann, Rainer Feldmann, Peter Mysliwietz (1995). Controlled Conspiracy-Number Search. ICCA Journal, Vol. 18, No. 3
 * Ulf Lorenz, Valentin Rottmann (1996). Parallel Controlled Conspiracy-Number Search. Advances in Computer Chess 8
 * Ulf Lorenz (1999). Controlled Conspiracy-2 Search. Technical Report, Paderborn University, ps
 * Robin Upton (1999). Dynamic Stochastic Control - A New Approach to Game Tree Searching. Ph.D. thesis, University of Warwick

2000 ...

 * Ulf Lorenz (2000). Controlled Conspiracy-2 Search. Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science (STACS)
 * David McAllester, Deniz Yuret (2002). Alpha-Beta Conspiracy Search. ICGA Journal, Vol. 25, No. 1
 * Ulf Lorenz (2002). Parallel Controlled Conspiracy Number Search. Euro-Par 2002, LNCS 2400, Springer

2010 ...

 * Mohd Nor Akmal Khalid, Umi Kalsom Yusof, Hiroyuki Iida, Taichi Ishitobi (2015). Critical Position Identification in Games and Its Application to Speculative Play. ICAART 2015
 * Mohd Nor Akmal Khalid, E. Mei Ang, Umi Kalsom Yusof, Hiroyuki Iida, Taichi Ishitobi (2015). Identifying Critical Positions Based on Conspiracy Numbers. Agents and Artificial Intelligence, ICAART 2015 - Revised Selected Papers
 * Jakub Pawlewicz, Ryan Hayward (2015). Sibling Conspiracy Number Search. SoCS 2015
 * Quang Vu, Taichi Ishitobi, Jean-Christophe Terrillon, Hiroyuki Iida (2016). Using Conspiracy Numbers for Improving Move Selection in Minimax Game-Tree Search. ICAART 2016, pdf
 * Zhang Song, Hiroyuki Iida (2018). Using single conspiracy number for long term position evaluation. CG 2018, ICGA Journal, Vol. 40, No. 3

=External Links=

Conspiracy Numbers

 * Conspiracy Numbers, Conspiracy Probailities & PCN* Search by Robin Upton
 * Reading: McAllister paper on "Consipracy Theory"? by Bruce Donald

Conspiracy

 * Conspiracy (disambiguation) from Wikipedia
 * Conspiracy theory from Wikipedia
 * List of conspiracy theories from Wikipedia
 * Conspiracy theory (disambiguation) from Wikipedia
 * Squire & Sherwood, Conspiracy - Conspiracy, YouTube Video

=References=

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