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'''[[Main Page|Home]] * [[Chess]] * Trajectory'''
[[FILE:StilmanTrajectory.JPG|border|right|thumb|Shortest and admissible trajectories <ref>from [[Boris Stilman]] ('''1994'''). ''A Linguistic Geometry of the Chess Model''. [[Advances in Computer Chess 7]], [http://www.stilman-strategies.com/bstilman/boris_papers/Jour94_CHESS7.pdf pdf draft], Figure 3</ref> ]]
'''Trajectory''',
a move path of a chess piece, a [[Pawn Spans|pawn]], a [[King|king]], a [[Knight|knight]] or a [[Sliding Pieces|sliding piece]] from its [[Origin Square|origin square]] to a [[Target Square|target square]] for attack or defense purposes, admissible in n [[Ply|plies]], according to a plan. A trajectory of a sliding piece consists of so called alpha-squares, move target squares where the piece is not actually [[En prise|en prise]], and empty beta-squares where the piece slides over, and which may be blocked as a defense.
=Mathematical Projection=
The concept of trajectories with the goal of winning [[Material|material]] was formalized by [[Mikhail Botvinnik]] in the 60s. Introduced in 1966 at [https://en.wikipedia.org/wiki/Chess_Centres Moscow Central Chess Club] <ref>[http://en.chessbase.com/home/TabId/211/PostId/4007513 The last day of the “Botvinnik Memorial”] by [http://de.wikipedia.org/wiki/Anna_Denissowna_Burtassowa Anna Burtasova], [[ChessBase|ChessBase News]], September 07, 2011</ref> , with the skeptical [[Georgy Adelson-Velsky]] and others attending, he found [[Vladimir Butenko]] as supporter and collaborator, who first implemented the [[Vector Attacks|15x15 vector attacks]] board representation on a [[M-20]] computer, to [[Distance#15x15|determine trajectories]]. In Botvinnik's hierarchical Mathematical Projection ('''MP''') of chess as a complex system, as apparently implemented in [[Pioneer]], trajectories build the lowest level of the hierarchy. The concepts of zones as intermediate level of the MP consists of a network of main trajectories conform to attacking of defending plans determined elsewhere, negation trajectories, that is opponent's counter trajectories which may block or combat the primary trajectory in time, and own supporting counter-counter trajectories. The MP controls the growth of a [[Search Tree|search tree]] inside a [[Best-First|best-first search]], and [[Pruning|prunes]] all branches forward which could not reach a goal in time.
=Linguistic Geometry=
Based on the research along with Botvinnik on the project [[Pioneer]], [[Boris Stilman]] further formalized the mathematical projection as [[Linguistic Geometry]] with the '''Language of Trajectories''', Languages of '''Trajectory Networks''' including the Language of '''Zones''', and Languages of '''Searches''' including its sub-family, the Languages of '''Transitions''' <ref>[[Boris Stilman]] ('''1994'''). ''A Linguistic Geometry of the Chess Model''. [[Advances in Computer Chess 7]], [http://www.stilman-strategies.com/bstilman/boris_papers/Jour94_CHESS7.pdf pdf draft]</ref>.
=See also=
* [[All Shortest Paths]]
* [[Botvinnik-Markoff Extension]]
* [[Combination]]
* [[Connectivity]]
* [[Distance#15x15|Determining Trajectories]]
* [[Distance]]
* [[Fill Algorithms]]
* [[Interception]]
* [[Intersection Squares]]
* [[Moves]]
* [[Pioneer]]
* [[Mobility#ProgressiveMobility|Progressive Mobility]]
* [[Réti Endgame Study]]
=Publications=
* [[Mikhail Botvinnik]] ('''1968'''). ''Algoritm igry v shakhmaty'' (The algorithm of chess)
* [[Mikhail Botvinnik]] ('''1970'''). ''Computers, Chess and Long-Range Planning''. Springer-Verlag, New York <ref>[[Paul Rushton]], [[Tony Marsland]] ('''1973'''). ''Current Chess Programs: A Summary of their Potential and Limitations''. INFOR Journal of the Canadian Information Processing Society Vol. 11, No. 1, [http://webdocs.cs.ualberta.ca/~tony/OldPapers/Rushton-Marsland-Feb73.pdf pdf]</ref> <ref>[http://www.johnljerz.com/superduper/tlxdownloadsiteMAIN/id128.html Computers, Chess and Long-range Planning by Botvinnik] by [[John L. Jerz]]</ref>
* [[Mikhail Botvinnik]] ('''1975'''). ''O Kiberneticheskoi Celi Igri''. (On the Cybernetic Goal of Games), Soviet Radio, Moscow
* [[Mikhail Botvinnik]] ('''1979'''). ''O Reshenii Netochnih Prebornih Zadach''. (On Solving Inexact Search Problems), Soviet Radio, Moscow
* [[Mikhail Botvinnik]], [[Boris Stilman]], [[Alexander Yudin]], [[Alexander Reznitskiy]], [[Michael Tsfasman]] ('''1980'''). ''Thinking of Man and Computer'', Proc. of the Second International Meeting on Artificial Intelligence, pp. 1-9, Repino, Leningrad, Russia.
* [[Mikhail Botvinnik]] ('''1982'''). ''Decision Making and Computers.'' [[Advances in Computer Chess 3]]
* [[Mikhail Botvinnik]] ('''1982'''). ''Meine neuen Ideen zur Schachprogrammierung''. Springer-Verlag, Berlin. [http://www.amazon.de/Meine-neuen-Schachprogrammierung-Informationstechnik-Datenverarbeitung/dp/3540110941 amazon.de] (German)
* [[Mikhail Botvinnik]] ('''1984'''). ''Computers in Chess: Solving Inexact Search Problems''. Springer-Verlag, New York.
* [[Rudolf Huber]] ('''1990'''). ''Selektive visuelle Aufmersamkeit: Untersuchungen zum Erlernen von Fokustrajektorien durch neuronale Netze''. Diplom thesis, Department of Computer Science, [[Technical University of Munich]] (German)
* [[Jürgen Schmidhuber]], [[Rudolf Huber]] ('''1991'''). ''Learning to Generate Artificial Fovea Trajectories for Target Detection''. [http://www.informatik.uni-trier.de/~ley/db/journals/ijns/ijns2.html#SchmidhuberH91 International Journal of Neural Systems, Vol. 2, No. 1-2], [ftp://ftp.idsia.ch/pub/juergen/attention.pdf pdf] <ref>[https://en.wikipedia.org/wiki/Fovea_centralis Fovea centralis from Wikipedia]</ref>
* [[Mikhail Botvinnik]], [[Evgeniĭ Dmitrievich Cherevik]], [[Vasily Vladimirov]], [[Vitaly Vygodsky]] ('''1994'''). ''[https://getinfo.de/app/Solving-Shannon-s-Problem-Ways-and-Means/id/BLCP%3ACN011979464 Solving Shannon's Problem: Ways and Means]''. [[Advances in Computer Chess 7]]
* [[Boris Stilman]] ('''1994'''). ''A Linguistic Geometry of the Chess Model''. [[Advances in Computer Chess 7]], [http://www.stilman-strategies.com/bstilman/boris_papers/Jour94_CHESS7.pdf pdf draft]
* [[Boris Stilman]] ('''1995'''). ''Linguistic geometry: a new paradigm for intelligent systems''. Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS '95), [http://www.computer.org/plugins/dl/pdf/proceedings/hicss/1995/6940/00/69400062.pdf pdf]
* [[Boris Stilman]] ('''1995'''). ''Deep Search in Linguistic Geometry''. Symposium on LINGUISTIC GEOMETRY AND SEMANTIC CONTROL, Proc. of the First World Congress on Intelligent Manufacturing: Processes and Systems, pp. 868-879, Mayaguez, Puerto Rico, [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.115.6220 CiteSeerX]
* [http://www.stilman-strategies.com/content/bios.html Vladimir Yakhnis], [[Boris Stilman]] ('''1995'''). ''Foundations of Linguistic Geometry: Complex Systems and Winning Strategies''. Symposium on LINGUISTIC GEOMETRY AND SEMANTIC CONTROL, Proc. of the First World Congress on Intelligent Manufacturing: Processes and Systems, pp. 843-854,Mayaguez, Puerto Rico
* [[Richard Sutton]], [[Andrew Barto]] ('''1998'''). ''[http://www.incompleteideas.net/sutton/book/the-book.html Reinforcement Learning: An Introduction]''. [https://en.wikipedia.org/wiki/MIT_Press MIT Press], Cambridge, Mass. ISBN 0-2621-9398-1. [http://webdocs.cs.ualberta.ca/~sutton/book/ebook/node100.html 9.6 Trajectory Sampling]
* [[Boris Stilman]] ('''2000'''). ''[http://atimopheyev.narod.ru/LG01pdf_in_HTML/LG01_eng.HTML Linguistic Geometry - From Search to Construction]''. (Operations Research/Computer Science Interfaces Series). Springer US, ISBN: 978-0-7923-7738-2, [http://www.amazon.com/exec/obidos/ASIN/0792377389/acmorg-20 amazon.com], [http://books.google.com/books?id=yBEcAksRfpwC&pg=PR11&lpg=PR11&dq=Alexander+A.+Yudin,+Botwinnik&source=bl&ots=Ulf83YReTS&sig=UvLUvNeGTVFczs5D0evzFtquLhc&hl=de&ei=GIySS9aHOtCi_Ab9vYz5DA&sa=X&oi=book_result&ct=result&resnum=2&ved=0CA4Q6AEwAQ#v=onepage&q=&f=false google books]
* [[Diego Rasskin-Gutman]] ('''2009, 2012'''). ''[http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=11847 Chess metaphors: artificial intelligence and the human mind]''. translated by [http://mitpress.mit.edu/catalog/author/default.asp?aid=36831 Deborah Klosky], [https://en.wikipedia.org/wiki/MIT_Press MIT Press]
* [[Marko Maliković]], [[Mirko Čubrilo]] ('''2010'''). ''Solving Shortest Proof Games by Generating Trajectories using Coq Proof Management System''. Proceedings of 21st Central European Conference on Information and Intelligent Systems, [https://en.wikipedia.org/wiki/Vara%C5%BEdin Varaždin], [https://en.wikipedia.org/wiki/Croatia Croatia] <ref>[https://en.wikipedia.org/wiki/Coq Coq from Wikipedia]</ref>
=Forum Posts=
* [http://www.talkchess.com/forum/viewtopic.php?t=21403 Search extensions at promising trajectories] by [[Reinhard Scharnagl]], [[CCC]], May 28, 2008
=External Links=
* [https://en.wikipedia.org/wiki/Trajectory Trajectory from Wikipedia]
* [https://en.wikipedia.org/wiki/Trajectory_%28disambiguation%29 Trajectory (disambiguation)]
=References=
<references />
'''[[Chess|Up one Level]]'''
[[FILE:StilmanTrajectory.JPG|border|right|thumb|Shortest and admissible trajectories <ref>from [[Boris Stilman]] ('''1994'''). ''A Linguistic Geometry of the Chess Model''. [[Advances in Computer Chess 7]], [http://www.stilman-strategies.com/bstilman/boris_papers/Jour94_CHESS7.pdf pdf draft], Figure 3</ref> ]]
'''Trajectory''',
a move path of a chess piece, a [[Pawn Spans|pawn]], a [[King|king]], a [[Knight|knight]] or a [[Sliding Pieces|sliding piece]] from its [[Origin Square|origin square]] to a [[Target Square|target square]] for attack or defense purposes, admissible in n [[Ply|plies]], according to a plan. A trajectory of a sliding piece consists of so called alpha-squares, move target squares where the piece is not actually [[En prise|en prise]], and empty beta-squares where the piece slides over, and which may be blocked as a defense.
=Mathematical Projection=
The concept of trajectories with the goal of winning [[Material|material]] was formalized by [[Mikhail Botvinnik]] in the 60s. Introduced in 1966 at [https://en.wikipedia.org/wiki/Chess_Centres Moscow Central Chess Club] <ref>[http://en.chessbase.com/home/TabId/211/PostId/4007513 The last day of the “Botvinnik Memorial”] by [http://de.wikipedia.org/wiki/Anna_Denissowna_Burtassowa Anna Burtasova], [[ChessBase|ChessBase News]], September 07, 2011</ref> , with the skeptical [[Georgy Adelson-Velsky]] and others attending, he found [[Vladimir Butenko]] as supporter and collaborator, who first implemented the [[Vector Attacks|15x15 vector attacks]] board representation on a [[M-20]] computer, to [[Distance#15x15|determine trajectories]]. In Botvinnik's hierarchical Mathematical Projection ('''MP''') of chess as a complex system, as apparently implemented in [[Pioneer]], trajectories build the lowest level of the hierarchy. The concepts of zones as intermediate level of the MP consists of a network of main trajectories conform to attacking of defending plans determined elsewhere, negation trajectories, that is opponent's counter trajectories which may block or combat the primary trajectory in time, and own supporting counter-counter trajectories. The MP controls the growth of a [[Search Tree|search tree]] inside a [[Best-First|best-first search]], and [[Pruning|prunes]] all branches forward which could not reach a goal in time.
=Linguistic Geometry=
Based on the research along with Botvinnik on the project [[Pioneer]], [[Boris Stilman]] further formalized the mathematical projection as [[Linguistic Geometry]] with the '''Language of Trajectories''', Languages of '''Trajectory Networks''' including the Language of '''Zones''', and Languages of '''Searches''' including its sub-family, the Languages of '''Transitions''' <ref>[[Boris Stilman]] ('''1994'''). ''A Linguistic Geometry of the Chess Model''. [[Advances in Computer Chess 7]], [http://www.stilman-strategies.com/bstilman/boris_papers/Jour94_CHESS7.pdf pdf draft]</ref>.
=See also=
* [[All Shortest Paths]]
* [[Botvinnik-Markoff Extension]]
* [[Combination]]
* [[Connectivity]]
* [[Distance#15x15|Determining Trajectories]]
* [[Distance]]
* [[Fill Algorithms]]
* [[Interception]]
* [[Intersection Squares]]
* [[Moves]]
* [[Pioneer]]
* [[Mobility#ProgressiveMobility|Progressive Mobility]]
* [[Réti Endgame Study]]
=Publications=
* [[Mikhail Botvinnik]] ('''1968'''). ''Algoritm igry v shakhmaty'' (The algorithm of chess)
* [[Mikhail Botvinnik]] ('''1970'''). ''Computers, Chess and Long-Range Planning''. Springer-Verlag, New York <ref>[[Paul Rushton]], [[Tony Marsland]] ('''1973'''). ''Current Chess Programs: A Summary of their Potential and Limitations''. INFOR Journal of the Canadian Information Processing Society Vol. 11, No. 1, [http://webdocs.cs.ualberta.ca/~tony/OldPapers/Rushton-Marsland-Feb73.pdf pdf]</ref> <ref>[http://www.johnljerz.com/superduper/tlxdownloadsiteMAIN/id128.html Computers, Chess and Long-range Planning by Botvinnik] by [[John L. Jerz]]</ref>
* [[Mikhail Botvinnik]] ('''1975'''). ''O Kiberneticheskoi Celi Igri''. (On the Cybernetic Goal of Games), Soviet Radio, Moscow
* [[Mikhail Botvinnik]] ('''1979'''). ''O Reshenii Netochnih Prebornih Zadach''. (On Solving Inexact Search Problems), Soviet Radio, Moscow
* [[Mikhail Botvinnik]], [[Boris Stilman]], [[Alexander Yudin]], [[Alexander Reznitskiy]], [[Michael Tsfasman]] ('''1980'''). ''Thinking of Man and Computer'', Proc. of the Second International Meeting on Artificial Intelligence, pp. 1-9, Repino, Leningrad, Russia.
* [[Mikhail Botvinnik]] ('''1982'''). ''Decision Making and Computers.'' [[Advances in Computer Chess 3]]
* [[Mikhail Botvinnik]] ('''1982'''). ''Meine neuen Ideen zur Schachprogrammierung''. Springer-Verlag, Berlin. [http://www.amazon.de/Meine-neuen-Schachprogrammierung-Informationstechnik-Datenverarbeitung/dp/3540110941 amazon.de] (German)
* [[Mikhail Botvinnik]] ('''1984'''). ''Computers in Chess: Solving Inexact Search Problems''. Springer-Verlag, New York.
* [[Rudolf Huber]] ('''1990'''). ''Selektive visuelle Aufmersamkeit: Untersuchungen zum Erlernen von Fokustrajektorien durch neuronale Netze''. Diplom thesis, Department of Computer Science, [[Technical University of Munich]] (German)
* [[Jürgen Schmidhuber]], [[Rudolf Huber]] ('''1991'''). ''Learning to Generate Artificial Fovea Trajectories for Target Detection''. [http://www.informatik.uni-trier.de/~ley/db/journals/ijns/ijns2.html#SchmidhuberH91 International Journal of Neural Systems, Vol. 2, No. 1-2], [ftp://ftp.idsia.ch/pub/juergen/attention.pdf pdf] <ref>[https://en.wikipedia.org/wiki/Fovea_centralis Fovea centralis from Wikipedia]</ref>
* [[Mikhail Botvinnik]], [[Evgeniĭ Dmitrievich Cherevik]], [[Vasily Vladimirov]], [[Vitaly Vygodsky]] ('''1994'''). ''[https://getinfo.de/app/Solving-Shannon-s-Problem-Ways-and-Means/id/BLCP%3ACN011979464 Solving Shannon's Problem: Ways and Means]''. [[Advances in Computer Chess 7]]
* [[Boris Stilman]] ('''1994'''). ''A Linguistic Geometry of the Chess Model''. [[Advances in Computer Chess 7]], [http://www.stilman-strategies.com/bstilman/boris_papers/Jour94_CHESS7.pdf pdf draft]
* [[Boris Stilman]] ('''1995'''). ''Linguistic geometry: a new paradigm for intelligent systems''. Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS '95), [http://www.computer.org/plugins/dl/pdf/proceedings/hicss/1995/6940/00/69400062.pdf pdf]
* [[Boris Stilman]] ('''1995'''). ''Deep Search in Linguistic Geometry''. Symposium on LINGUISTIC GEOMETRY AND SEMANTIC CONTROL, Proc. of the First World Congress on Intelligent Manufacturing: Processes and Systems, pp. 868-879, Mayaguez, Puerto Rico, [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.115.6220 CiteSeerX]
* [http://www.stilman-strategies.com/content/bios.html Vladimir Yakhnis], [[Boris Stilman]] ('''1995'''). ''Foundations of Linguistic Geometry: Complex Systems and Winning Strategies''. Symposium on LINGUISTIC GEOMETRY AND SEMANTIC CONTROL, Proc. of the First World Congress on Intelligent Manufacturing: Processes and Systems, pp. 843-854,Mayaguez, Puerto Rico
* [[Richard Sutton]], [[Andrew Barto]] ('''1998'''). ''[http://www.incompleteideas.net/sutton/book/the-book.html Reinforcement Learning: An Introduction]''. [https://en.wikipedia.org/wiki/MIT_Press MIT Press], Cambridge, Mass. ISBN 0-2621-9398-1. [http://webdocs.cs.ualberta.ca/~sutton/book/ebook/node100.html 9.6 Trajectory Sampling]
* [[Boris Stilman]] ('''2000'''). ''[http://atimopheyev.narod.ru/LG01pdf_in_HTML/LG01_eng.HTML Linguistic Geometry - From Search to Construction]''. (Operations Research/Computer Science Interfaces Series). Springer US, ISBN: 978-0-7923-7738-2, [http://www.amazon.com/exec/obidos/ASIN/0792377389/acmorg-20 amazon.com], [http://books.google.com/books?id=yBEcAksRfpwC&pg=PR11&lpg=PR11&dq=Alexander+A.+Yudin,+Botwinnik&source=bl&ots=Ulf83YReTS&sig=UvLUvNeGTVFczs5D0evzFtquLhc&hl=de&ei=GIySS9aHOtCi_Ab9vYz5DA&sa=X&oi=book_result&ct=result&resnum=2&ved=0CA4Q6AEwAQ#v=onepage&q=&f=false google books]
* [[Diego Rasskin-Gutman]] ('''2009, 2012'''). ''[http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=11847 Chess metaphors: artificial intelligence and the human mind]''. translated by [http://mitpress.mit.edu/catalog/author/default.asp?aid=36831 Deborah Klosky], [https://en.wikipedia.org/wiki/MIT_Press MIT Press]
* [[Marko Maliković]], [[Mirko Čubrilo]] ('''2010'''). ''Solving Shortest Proof Games by Generating Trajectories using Coq Proof Management System''. Proceedings of 21st Central European Conference on Information and Intelligent Systems, [https://en.wikipedia.org/wiki/Vara%C5%BEdin Varaždin], [https://en.wikipedia.org/wiki/Croatia Croatia] <ref>[https://en.wikipedia.org/wiki/Coq Coq from Wikipedia]</ref>
=Forum Posts=
* [http://www.talkchess.com/forum/viewtopic.php?t=21403 Search extensions at promising trajectories] by [[Reinhard Scharnagl]], [[CCC]], May 28, 2008
=External Links=
* [https://en.wikipedia.org/wiki/Trajectory Trajectory from Wikipedia]
* [https://en.wikipedia.org/wiki/Trajectory_%28disambiguation%29 Trajectory (disambiguation)]
=References=
<references />
'''[[Chess|Up one Level]]'''
