Trajectory,
a move path of a chess piece, a pawn, a king, a knight or a sliding piece from its origin square to a target square for attack or defense purposes, admissible in n 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, and empty beta-squares where the piece slides over, and which may be blocked as a defense.

The concept of trajectories with the goal of winning material was formalized by Mikhail Botvinnik in the 60s. Introduced in 1966 at Moscow Central Chess Club^{[2]} , with the skeptical Georgy Adelson-Velsky and others attending, he found Vladimir Butenko as supporter and collaborator, who first implemented the 15x15 vector attacks board representation on a M-20 computer, to 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 inside a best-first search, and 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^{[3]}.

Mikhail Botvinnik (1982). Meine neuen Ideen zur Schachprogrammierung. Springer-Verlag, Berlin. 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)

Boris Stilman (1995). Linguistic geometry: a new paradigm for intelligent systems. Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS '95), 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, CiteSeerX

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

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, Varaždin, Croatia^{[7]}

^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, pdf

Home * Chess * TrajectoryTrajectory,a move path of a chess piece, a pawn, a king, a knight or a sliding piece from its origin square to a target square for attack or defense purposes, admissible in n 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, and empty beta-squares where the piece slides over, and which may be blocked as a defense.

^{[1]}## Table of Contents

## Mathematical Projection

The concept of trajectories with the goal of winning material was formalized by Mikhail Botvinnik in the 60s. Introduced in 1966 at Moscow Central Chess Club^{[2]}, with the skeptical Georgy Adelson-Velsky and others attending, he found Vladimir Butenko as supporter and collaborator, who first implemented the 15x15 vector attacks board representation on a M-20 computer, to 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 inside a best-first search, and 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 theLanguage of Trajectories, Languages ofTrajectory Networksincluding the Language ofZones, and Languages ofSearchesincluding its sub-family, the Languages ofTransitions^{[3]}.## See also

## Publications

1968).Algoritm igry v shakhmaty(The algorithm of chess)1970).Computers, Chess and Long-Range Planning. Springer-Verlag, New York^{[4]}^{[5]}1975).O Kiberneticheskoi Celi Igri. (On the Cybernetic Goal of Games), Soviet Radio, Moscow1979).O Reshenii Netochnih Prebornih Zadach. (On Solving Inexact Search Problems), Soviet Radio, Moscow1980).Thinking of Man and Computer, Proc. of the Second International Meeting on Artificial Intelligence, pp. 1-9, Repino, Leningrad, Russia.1982).Decision Making and Computers.Advances in Computer Chess 31982).Meine neuen Ideen zur Schachprogrammierung. Springer-Verlag, Berlin. amazon.de (German)1984).Computers in Chess: Solving Inexact Search Problems. Springer-Verlag, New York.1990).Selektive visuelle Aufmersamkeit: Untersuchungen zum Erlernen von Fokustrajektorien durch neuronale Netze. Diplom thesis, Department of Computer Science, Technical University of Munich (German)1991).Learning to Generate Artificial Fovea Trajectories for Target Detection. International Journal of Neural Systems, Vol. 2, No. 1-2, pdf^{[6]}1994).Solving Shannon's Problem: Ways and Means. Advances in Computer Chess 71994).A Linguistic Geometry of the Chess Model. Advances in Computer Chess 7, pdf draft1995).Linguistic geometry: a new paradigm for intelligent systems. Proceedings of the 28th Hawaii International Conference on System Sciences (HICSS '95), pdf1995).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, CiteSeerX1995).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 Rico1998).Reinforcement Learning: An Introduction. MIT Press, Cambridge, Mass. ISBN 0-2621-9398-1. 9.6 Trajectory Sampling2000).Linguistic Geometry - From Search to Construction. (Operations Research/Computer Science Interfaces Series). Springer US, ISBN: 978-0-7923-7738-2, amazon.com, google books2009, 2012).Chess metaphors: artificial intelligence and the human mind. translated by Deborah Klosky, MIT Press2010).Solving Shortest Proof Games by Generating Trajectories using Coq Proof Management System. Proceedings of 21st Central European Conference on Information and Intelligent Systems, Varaždin, Croatia^{[7]}## Forum Posts

## External Links

## References

1994).A Linguistic Geometry of the Chess Model. Advances in Computer Chess 7, pdf draft, Figure 31994).A Linguistic Geometry of the Chess Model. Advances in Computer Chess 7, pdf draft1973).Current Chess Programs: A Summary of their Potential and Limitations. INFOR Journal of the Canadian Information Processing Society Vol. 11, No. 1, pdf## What links here?

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