Difference between revisions of "Conspiracy Numbers"
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[[FILE:ACC5Schaeffer.jpg|border|right|thumb|[[Jonathan Schaeffer]] on ''Conspiracy Numbers.'' <ref>Photo from [[Advances in Computer Chess 5]] by [[László Lindner]], [[ICGA Journal#10_3|ICCA Journal, Vol. 10, No. 3]], pp. 138</ref> ]] | [[FILE:ACC5Schaeffer.jpg|border|right|thumb|[[Jonathan Schaeffer]] on ''Conspiracy Numbers.'' <ref>Photo from [[Advances in Computer Chess 5]] by [[László Lindner]], [[ICGA Journal#10_3|ICCA Journal, Vol. 10, No. 3]], pp. 138</ref> ]] | ||
− | '''Conspiracy Numbers''' of the [[Root|root]] or [[Interior Node|interior nodes]] of a [[Search Tree|search tree]] for some value ''' | + | '''Conspiracy Numbers''' of the [[Root|root]] or [[Interior Node|interior nodes]] of a [[Search Tree|search tree]] for some value '''V''' are defined as the least number of conspirators, that are [[Leaf Node|leaves]] that must change their evaluation value to '''V''' in order to change the minimax value of the interior node or root <ref>Definition, Sample, and Pseudo code taken from [[Maarten van der Meulen]] ('''1990'''). ''Conspiracy-Number Search''. [[ICGA Journal#13_1|ICCA Journal, Vol. 13, No. 1]]</ref>. Conspiracy Numbers and their possible application for [[Minimax]] search within a [[Best-First|best-first search]] algorithm was first described by [[David McAllester]] <ref>[[David McAllester]] ('''1988'''). ''Conspiracy Numbers for Min-Max Search''. [https://en.wikipedia.org/wiki/Artificial_Intelligence_(journal) Artificial Intelligence], Vol. 35, No. 1, pp. 287-310. ISSN 0004-3702</ref>. |
=Sample= | =Sample= | ||
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} | } | ||
</pre> | </pre> | ||
+ | =Conspiracy Theory= | ||
+ | Let δ be a number called the singular margin <ref>The term ''singular margin'' comes from the [[Singular Extensions|singular extension]] algorithm ([[Thomas Anantharaman|Anantharaman]] et al. 1990)</ref>. '''Conspiracy theory''' can be formulated using the following definition <ref>[[David McAllester]], [[Deniz Yuret]] ('''1993'''). ''Alpha-Beta Conspiracy Search''. [http://ttic.uchicago.edu/%7Edmcallester/abc.ps ps (draft)]</ref>: | ||
+ | |||
+ | '''Definition''': Let '''T''' be a search tree with min-max value '''<nowiki>V[T]</nowiki>'''. The [[Lower Bound|lower boand]] conspiracy number of '''T''', denoted '''C<sub><</sub><nowiki>[T]</nowiki>''', is the number of leaf static values that must be changed to bring the root min-max value down to '''<nowiki>V[T]</nowiki>-δ'''. The [[Upper Bound|upper boand]] conspiracy number of '''T''', denoted '''C<sub>></sub><nowiki>[T]</nowiki>''', is the number of leaves that must be changed to bring the root value up to '''<nowiki>V[T]</nowiki>+δ'''. | ||
+ | |||
+ | '''C<sub><</sub><nowiki>[T]</nowiki>''' expresses the confidence that the lower bound [[Alpha|α]] will hold by further expansion of the search tree. | ||
=Search Algorithms= | =Search Algorithms= | ||
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* [[Conspiracy Number Search]] | * [[Conspiracy Number Search]] | ||
* [[Proof-Number Search]] | * [[Proof-Number Search]] | ||
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− | |||
− | |||
− | |||
− | |||
=Publications= | =Publications= | ||
− | <ref>[ | + | <ref>[[ICGA Journal#RefDB|ICGA Reference Database]]</ref> |
==1985 ...== | ==1985 ...== | ||
* [[David McAllester]] ('''1985'''). ''A New Procedure for Growing Minimax Trees''. Technical Report, Artificial Intelligence Laboratory, MIT | * [[David McAllester]] ('''1985'''). ''A New Procedure for Growing Minimax Trees''. Technical Report, Artificial Intelligence Laboratory, MIT | ||
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* [[Norbert Klingbeil]], [[Jonathan Schaeffer]] ('''1990'''). ''Empirical Results with Conspiracy Numbers.'' Computational Intelligence, Vol. 6, pp. 1-11, [http://webdocs.cs.ualberta.ca/%7Ejonathan/Papers/Papers/compi.ps ps] | * [[Norbert Klingbeil]], [[Jonathan Schaeffer]] ('''1990'''). ''Empirical Results with Conspiracy Numbers.'' Computational Intelligence, Vol. 6, pp. 1-11, [http://webdocs.cs.ualberta.ca/%7Ejonathan/Papers/Papers/compi.ps ps] | ||
* <span id="AI"></span>[[Jonathan Schaeffer]] ('''1990'''). ''Conspiracy Numbers''. [https://en.wikipedia.org/wiki/Artificial_Intelligence_(journal) Artificial Intelligence], Vol. 43, No. 1, pp. 67-84 | * <span id="AI"></span>[[Jonathan Schaeffer]] ('''1990'''). ''Conspiracy Numbers''. [https://en.wikipedia.org/wiki/Artificial_Intelligence_(journal) 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 | + | * [[Maarten van der Meulen]], [[Victor Allis]], [[Jaap van den Herik]] ('''1990'''). ''A Comment on `Conspiracy-Number Search''. [[ICGA Journal#13_2|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]] | * [[Victor Allis]], [[Maarten van der Meulen]], [[Jaap van den Herik]] ('''1991'''). ''Conspiracy-Number Search.'' [[Advances in Computer Chess 6]] | ||
− | * [[David McAllester]], [[Deniz Yuret]] ('''1993'''). '' | + | * [[David McAllester]], [[Deniz Yuret]] ('''1993'''). ''Alpha-Beta Conspiracy Search''. [http://ttic.uchicago.edu/%7Edmcallester/abc.ps ps (draft)] |
* [[Lisa Lister]], [[Jonathan Schaeffer]] ('''1994'''). ''An Analysis of the Conspiracy Numbers Algorithm''. [https://en.wikipedia.org/wiki/Computers_and_Mathematics_with_Applications Computers & Mathematics with Applications] Vol. 27, No. 1, [https://en.wikipedia.org/wiki/Elsevier Elsevier], [http://webdocs.cs.ualberta.ca/%7Ejonathan/publications/ai_publications/icn.pdf pdf] | * [[Lisa Lister]], [[Jonathan Schaeffer]] ('''1994'''). ''An Analysis of the Conspiracy Numbers Algorithm''. [https://en.wikipedia.org/wiki/Computers_and_Mathematics_with_Applications Computers & Mathematics with Applications] Vol. 27, No. 1, [https://en.wikipedia.org/wiki/Elsevier Elsevier], [http://webdocs.cs.ualberta.ca/%7Ejonathan/publications/ai_publications/icn.pdf pdf] | ||
* [[Deniz Yuret]] ('''1994'''). ''[https://scholar.google.com/citations?view_op=view_citation&hl=en&user=EJurXJ4AAAAJ&cstart=40&citation_for_view=EJurXJ4AAAAJ:TQgYirikUcIC The Principle of Pressure in Chess]''. TAINN 1994 | * [[Deniz Yuret]] ('''1994'''). ''[https://scholar.google.com/citations?view_op=view_citation&hl=en&user=EJurXJ4AAAAJ&cstart=40&citation_for_view=EJurXJ4AAAAJ:TQgYirikUcIC The Principle of Pressure in Chess]''. TAINN 1994 | ||
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==2000 ...== | ==2000 ...== | ||
* [[Ulf Lorenz]] ('''2000'''). ''Controlled Conspiracy-2 Search''. Proceedings of the 17th Annual Symposium on Theoretical Aspects of Computer Science (STACS) | * [[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'''). '' | + | * [[David McAllester]], [[Deniz Yuret]] ('''2002'''). ''Alpha-Beta Conspiracy Search''. [[ICGA Journal#25_1|ICGA Journal, Vol. 25, No. 1]] |
* [[Ulf Lorenz]] ('''2002'''). ''[https://link.springer.com/chapter/10.1007/3-540-45706-2_57 Parallel Controlled Conspiracy Number Search]''. [https://dblp1.uni-trier.de/db/conf/europar/europar2002.html Euro-Par 2002], [https://en.wikipedia.org/wiki/Lecture_Notes_in_Computer_Science LNCS] 2400, [https://en.wikipedia.org/wiki/Springer_Science%2BBusiness_Media Springer] | * [[Ulf Lorenz]] ('''2002'''). ''[https://link.springer.com/chapter/10.1007/3-540-45706-2_57 Parallel Controlled Conspiracy Number Search]''. [https://dblp1.uni-trier.de/db/conf/europar/europar2002.html Euro-Par 2002], [https://en.wikipedia.org/wiki/Lecture_Notes_in_Computer_Science LNCS] 2400, [https://en.wikipedia.org/wiki/Springer_Science%2BBusiness_Media Springer] | ||
==2010 ...== | ==2010 ...== | ||
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* [[Mohd Nor Akmal Khalid]], [[E. Mei Ang]], [[Umi Kalsom Yusof]], [[Hiroyuki Iida]], [[Taichi Ishitobi]] ('''2015'''). ''[http://link.springer.com/chapter/10.1007%2F978-3-319-27947-3_6 Identifying Critical Positions Based on Conspiracy Numbers]''. [http://link.springer.com/book/10.1007/978-3-319-27947-3 Agents and Artificial Intelligence], [http://dblp.uni-trier.de/db/conf/icaart/icaart2015s.html#KhalidAYII15 ICAART 2015 - Revised Selected Papers] | * [[Mohd Nor Akmal Khalid]], [[E. Mei Ang]], [[Umi Kalsom Yusof]], [[Hiroyuki Iida]], [[Taichi Ishitobi]] ('''2015'''). ''[http://link.springer.com/chapter/10.1007%2F978-3-319-27947-3_6 Identifying Critical Positions Based on Conspiracy Numbers]''. [http://link.springer.com/book/10.1007/978-3-319-27947-3 Agents and Artificial Intelligence], [http://dblp.uni-trier.de/db/conf/icaart/icaart2015s.html#KhalidAYII15 ICAART 2015 - Revised Selected Papers] | ||
* [[Jakub Pawlewicz]], [[Ryan Hayward]] ('''2015'''). ''[https://www.aaai.org/ocs/index.php/SOCS/SOCS15/paper/view/11040 Sibling Conspiracy Number Search]''. [https://en.wikipedia.org/wiki/Symposium_on_Combinatorial_Search SoCS 2015] | * [[Jakub Pawlewicz]], [[Ryan Hayward]] ('''2015'''). ''[https://www.aaai.org/ocs/index.php/SOCS/SOCS15/paper/view/11040 Sibling Conspiracy Number Search]''. [https://en.wikipedia.org/wiki/Symposium_on_Combinatorial_Search SoCS 2015] | ||
+ | * [[Jakub Pawlewicz]], [[Ryan Hayward]] ('''2016'''). ''[https://www.sciencedirect.com/science/article/pii/S0304397516302729 Conspiracy number search with relative sibling scores]''. [https://en.wikipedia.org/wiki/Theoretical_Computer_Science_(journal) Theoretical Computer Science], Vol. 644 | ||
* [[Quang Vu]], [[Taichi Ishitobi]], [[Jean-Christophe Terrillon]], [[Hiroyuki Iida]] ('''2016'''). ''Using Conspiracy Numbers for Improving Move Selection in Minimax Game-Tree Search''. [http://www.icaart.org/?y=2016 ICAART 2016], [https://pdfs.semanticscholar.org/1bcf/bd2047bc1d74affda11bf2007cac442dd6f4.pdf pdf] | * [[Quang Vu]], [[Taichi Ishitobi]], [[Jean-Christophe Terrillon]], [[Hiroyuki Iida]] ('''2016'''). ''Using Conspiracy Numbers for Improving Move Selection in Minimax Game-Tree Search''. [http://www.icaart.org/?y=2016 ICAART 2016], [https://pdfs.semanticscholar.org/1bcf/bd2047bc1d74affda11bf2007cac442dd6f4.pdf pdf] | ||
* [[Zhang Song]], [[Hiroyuki Iida]] ('''2018'''). ''Using single conspiracy number for long term position evaluation''. [[CG 2018]], [[ICGA Journal#40_3|ICGA Journal, Vol. 40, No. 3]] | * [[Zhang Song]], [[Hiroyuki Iida]] ('''2018'''). ''Using single conspiracy number for long term position evaluation''. [[CG 2018]], [[ICGA Journal#40_3|ICGA Journal, Vol. 40, No. 3]] | ||
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* [https://en.wikipedia.org/wiki/List_of_conspiracy_theories List of conspiracy theories from Wikipedia] | * [https://en.wikipedia.org/wiki/List_of_conspiracy_theories List of conspiracy theories from Wikipedia] | ||
* [https://en.wikipedia.org/wiki/Conspiracy_theory_%28disambiguation%29 Conspiracy theory (disambiguation) from Wikipedia] | * [https://en.wikipedia.org/wiki/Conspiracy_theory_%28disambiguation%29 Conspiracy theory (disambiguation) from Wikipedia] | ||
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* [[:Category:Chris Squire|Squire]] & [https://en.wikipedia.org/wiki/Billy_Sherwood Sherwood], [https://en.wikipedia.org/wiki/Conspiracy_%28band%29 Conspiracy] - [https://en.wikipedia.org/wiki/Conspiracy_(band)#Conspiracy Conspiracy], [https://en.wikipedia.org/wiki/YouTube YouTube] Video | * [[:Category:Chris Squire|Squire]] & [https://en.wikipedia.org/wiki/Billy_Sherwood Sherwood], [https://en.wikipedia.org/wiki/Conspiracy_%28band%29 Conspiracy] - [https://en.wikipedia.org/wiki/Conspiracy_(band)#Conspiracy Conspiracy], [https://en.wikipedia.org/wiki/YouTube YouTube] Video | ||
: {{#evu:https://www.youtube.com/watch?v=af-8Of_uBxw|alignment=left|valignment=top}} | : {{#evu:https://www.youtube.com/watch?v=af-8Of_uBxw|alignment=left|valignment=top}} | ||
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'''[[Search|Up one Level]]''' | '''[[Search|Up one Level]]''' | ||
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[[Category:Chris Squire]] | [[Category:Chris Squire]] |
Latest revision as of 17:10, 16 November 2020
Home * Search * Conspiracy Numbers
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].
Contents
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
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
- Jakub Pawlewicz, Ryan Hayward (2016). Conspiracy number search with relative sibling scores. Theoretical Computer Science, Vol. 644
- 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
- ↑ Photo from Advances in Computer Chess 5 by László Lindner, ICCA Journal, Vol. 10, No. 3, pp. 138
- ↑ Definition, Sample, and Pseudo code taken from Maarten van der Meulen (1990). Conspiracy-Number Search. ICCA Journal, Vol. 13, No. 1
- ↑ David McAllester (1988). Conspiracy Numbers for Min-Max Search. Artificial Intelligence, Vol. 35, No. 1, pp. 287-310. ISSN 0004-3702
- ↑ due to Jonathan Schaeffer (1989). Conspiracy Numbers. Advances in Computer Chess 5
- ↑ Maarten van der Meulen (1990). Conspiracy-Number Search. ICCA Journal, Vol. 13, No. 1
- ↑ Norbert Klingbeil, Jonathan Schaeffer (1988). Search Strategies for Conspiracy Numbers. Canadian Artificial Intelligence Conference, pp. 133-139
- ↑ The term singular margin comes from the singular extension algorithm (Anantharaman et al. 1990)
- ↑ David McAllester, Deniz Yuret (1993). Alpha-Beta Conspiracy Search. ps (draft)
- ↑ Ulf Lorenz, Valentin Rottmann (1996). Parallel Controlled Conspiracy-Number Search. Advances in Computer Chess 8
- ↑ ICGA Reference Database