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 : code root                   ┌───────┐ max node               │   1   │ └───────┘                    /      3      \           ┌───────┐                 ┌───────┐ min nodes  │  1.1  │                 │  1.2  │ └───────┘                └───────┘          /    2    \               /    3    \  ┌───────┐       ┌───────┐   ┌───────┐       ┌───────┐  │ 1.1.1 │       │ 1.1.2 │   │ 1.2.1 │       │ 1.2.2 │  └───────┘       └───────┘   └───────┘       └───────┘      5               2           3               4 code

Conspiracy Numbers
The conspiracy numbers for all possible value of the root of tree T
 * ~ v ||~ cn(root, v) ||~ conspirators ||
 * = <= 1 ||~ 2 ||= (1.1.1 or 1.1.2) and (1.2.1 or 1.2.2) ||
 * = 2 ||~ 1 ||= (1.2.1 or 1.2.2) ||
 * = 3 ||~ 0 ||= none ||
 * = 4 ||~ 1 ||= (1.1.2 or 1.2.1) ||
 * = 5 ||~ 1 ||= 1.1.2 ||
 * = >= 6 ||~ 2 ||= (1.1.1 and 1.1.2) or (1.2.1 and 1.2.2) ||

The conspiracy numbers for all possible value of node 1.1 of tree T
 * ~ v ||~ cn(1.1, v) ||~ conspirators ||
 * = <= 1 ||~ 1 ||= (1.1.1 or 1.1.2) ||
 * = 2 ||~ 0 ||= none ||
 * = 3,4,5 ||~ 1 ||= 1.1.2 ||
 * = >= 6 ||~ 2 ||= (1.1.1 and 1.1.2) ||

The conspiracy numbers for all possible value of node 1.2 of tree T
 * ~ v ||~ cn(1.2, v) ||~ conspirators ||
 * = <= 2 ||~ 1 ||= (1.2.1 or 1.2.2) ||
 * = 3 ||~ 0 ||= none ||
 * = 4 ||~ 1 ||= 1.2.1 ||
 * = >= 5 ||~ 2 ||= (1.2.1 and 1.2.2) ||

=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 : code format="cpp" 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; } code

=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
 * Controlled Conspiracy Number Search
 * Parallel Controlled Conspiracy Number Search
 * Proof-Number Search

=Chess Programs=
 * Arachne
 * P.ConNerS
 * Ulysses

=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, pp. 287-310. ISSN 0004-3702
 * 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, University of Paderborn, [|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

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]

=External Links=

Conspiracy Numbers

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

Conspiracy
> feat.: [|Jack Gregg], [|Mark Whitecage], [|Steve McCall], Gunter Hampel, [|Sam Rivers], [|Marty Cook] > media type="youtube" key="CRqGVLEYbUc" > media type="youtube" key="af-8Of_uBxw"
 * [|Conspiracy (disambiguation) from Wikipedia]
 * [|Conspiracy theory from Wikipedia]
 * [|List of conspiracy theories from Wikipedia]
 * [|Conspiracy theory (disambiguation) from Wikipedia]
 * Jeanne Lee - Sundance, [|Conspiracy] (1975), [|YouTube] Video
 * Squire & [|Sherwood], [|Conspiracy] - [|Conspiracy], [|YouTube] Video

=References= =What links here?= include page="Conspiracy Numbers" component="backlinks" limit="60"
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