Linguistic Geometry: From Search to Construction

402 Pages·2000·29.536 MB·English

by  Boris Stilman (auth.)

Linguistic Geometry
From Search to Construction
OPERATIONS RESEARCH/COMPUTER SCIENCE
INTERFACES SERIES
SeriesEditors
ProfessorRameshSharda Prof. Dr. StefanVoB
OklahomaState University Technische UniversitiitBraunschweig
Otherpublished titlesin theseries:
Jones,ChristopherV.
VisualizationandOptimization
Barr, RichardS./Helgason, RichardV./Kennington,JefferyL.
InterfacesinComputerScienceandOperationsResearch:Advancesin
Metaheuristics, Optimization, andStochasticModelingTechnologies
Ellacott,StephenW.IMason,JohnC.IAnderson, lainJ.
MathematicsofNeuralNetworks: Models, Algorithms& Applications
Woodruff, DavidL.
AdvancesinComputationalandStochasticOptimization, Logic
Programming, andHeuristicSearch
Klein, Robert
SchedulingofResource-ConstrainedProjects
Bierwirth,Christian
AdaptiveSearchandtheManagementofLogisticsSystems
Laguna,Manuel/Gonzalez-Velarde,JoseLuis
ComputingToolsforModeling, OptimizationandSimulation
************.*•••••••
Covercreated byMichaelStilman
StanfordUniversity
Linguistic Geometry
From Search to Construction
Boris Stilman
University of Colorado at Denver
&
Stilman Advanced Strategies, LLC
~.
"
Springer Science+Business Media, LLC
ISBN 978-1-4613-6999-8 ISBN 978-1-4615-4439-5 (eBook)
DOI 10.1007/978-1-4615-4439-5
Library of Congress Cataloging-in-Publication
Stilman, Boris.
Linguistie geometry : from seareh to eonstruetion / Boris Stilman.
p. em. --(Operations researeh/eomputer seienee interfaees ; ORCS 13)
Includes bibliographieal referenees and index.
1. Computer simulation. 2. Mathematieal models. 3. Intelligent agents (Computer
software) 1. Title. II. Series.
QA76.9.C65 S76 2000
003'.3--de21
99-056036
Copyright © 2000 by Springer Science+Business Media New York
Originally published by Kluwer Academic Publishers, New York in 2000
Softcover reprint ofthe hardcover Ist edition 2000
AII rights reserved. No part of this publieation may be reproduced, stored in a
retrieval system or transmitted in any form or by any means, meehanieal, photo
eopying, reeording, or otherwise, without the prior written permission of the
publisher, Springer Seienee+Business Media, LLC.
Printed an acid-free paper.
TABLE OF CONTENTS
1 Introduction
1.1 Problems 2
1.2 Current Approaches 4
1.3 LG Approach ataGlance 9
1.4 LG Approach: Deeper Account 13
1.5 LG Strategiesand GameTheory 19
1.6 LG: Three Stages of Development 22
1.7 Stage One: Project PIONEER 24
1.8 StageTwo: MathematicalTools 28
1.9 StageThree: Modern History 32
2 Hierarchy of Formal Languages 39
2.1 HierarchyOutline 39
2.2 Classof Problems 40
2.3 Various Problemsas Complex Systems 45
2.4 Set of Paths: Language ofTrajectories 50
2.5 Boardand State Distances 52
2.6 Networks of Paths: Languages of Networks 54
2.7 Representation of Movement: Translations 66
2.8 Search by Construction: Languages of Searches 71
2.9 Historical Remarks 75
3 Robot Combat for 20 District 77
3.1 Problem Statement 77
3.2 LG Search 79
3.3 Discussion 88
3.4 Historical Remarks 89
VI Linguistic Geometry:
4 Expanding to 3D Space 91
4.1 3D/4A Robot Combat 91
4.2 LG Search within Insufficient Horizon 94
4.3 LG Searchwithin Horizon 5 98
4.4 From 20to 3D: Running Time Change 105
4.5 Historical Remarks 106
5 Deeper Search, More Agents 107
5.1 2D/8AAirCombat 107
5.2 DeepSearch forAirCombat 112
5.3 3D/8A Space Combat 128
5.4 A New Level of Sophistication: Preliminary Conclusions 131
5.5 Historical Remarks 134
6 Concurrency, nxn District 137
6.1 SerialModeRelaxation: Partial Concurrency 137
6.2 LG Search forProblem with Partial Concurrency 139
6.3 Second Problem with Partial Concurrency 143
6.4 Second Problem: LG Search 145
6.5 Problem with Total Concurrencyand nxn District 150
6.6 Problem with Total Concurrency: LG Search 153
6.7 Impactof Concurrency 160
6.8 Historical Remarks 162
7 Scheduling: Artificial Conflict 165
7.1 Problem Statement 165
7.2 Conversion intoTwo-PlayerGame 168
7.3 Scheduling: Search and Solution 172
7.4 Formal Representation 174
7.5 Evaluation and Implementation 177
7.6 Applicability of LG 178
8 Generating Techniques 181
8.1 ChomskyGrammars 181
8.2 Controlled Grammars: Introduction 183
8.3 Towerof Hanoi: 3Discs 184
8.4 Towerof Hanoi: nDiscs 186
8.5 Controlled Grammars: Formal Definition 188
8.6 Historical Remarks 191
9 Language of Trajectories 193
9.1 ShortestTrajectories: Generating Grammar 193
9.2 Generation of Shortest 20Trajectories 196
9.3 Generation of Shortest 3DTrajectories 200
TableofContents vii
9.4 Shortest Trajectories: Correctness and Completeness 204
9.5 AdmissibleTrajectories: Generating Grammar 207
9.6 Avoiding Obstacles: Generation of AdmissibleTrajectories 210
9.7 Admissible Trajectories: Correctness and Completeness 215
9.8 Trajectories forthe GameofChess 219
9.9 Trajectories for Scheduling 224
9.10 Trajectories for Agents with Variable Speed 226
9.11 Language ofTrajectories: Efficient Implementation 232
9.12 Historical Remarks 233
10 Language of Zones 235
10.1 GrammarofZones 235
10.2 Generation of Zones 238
10.3 Geometry of Zones 250
10.4 Zones forthe Game ofChess 253
10.5 Zones for Scheduling 254
10.6 Historical Remarks 256
11 Translations 259
11.1 Approaching a Solution of the Problem of Change 259
11.2 Theorem aboutTranslations 265
11.3 Search with Translations 278
11.4 Translationsand Freeze 284
11.5 Historical Remarks 286
12 Languages of Searches 287
12.1 General Searches 287
12.2 Reduced Searches 290
12.3 GrammarofTranslations 294
12.4 QualityofTrajectoriesandZones 297
12.5 Tree Inspection Procedure 300
12.6 HistoricalRemarks 303
13 From Search to Construction 305
13.1 Problem Statement 305
13.2 Algorithm Outline 306
13.3 Preliminary Definitions 307
13.4 Terminal Sets Expansion 308
13.5 Structure of Expanded Terminal Sets 311
13.6 State Space Chart 319
13.7 Outline of Potential Strategies 323
13.8 Construction of Strategy atthe Start State 330
13.9 Strategiesfor Reti-like Problems 335
13.10 Discussion 337
13.11 Historical Remarks 339
Vlll Linguistic Geometry:
14 Computational Complexity 341
14.1 Running Time:
Conventional Algorithms for Reti-like problems 341
14.2 Running Time: Grammarof ShortestTrajectories 347
14.3 Running Time: GrammarofZones 350
14.4 Running Time:
Unfolding Bundles ofTrajectories and Cloning Zones 353
14.5 Computational Complexity: Reti-like Problems 357
14.6 Historical Remarks 359
Future Challenges 361
References 363
Index 377
PREFACE
This book is neither about linguistics nor geometry. It is about search
problems. Linguistic Geometry (LG) is an approach to construction of
mathematical models for knowledge representation and reasoning about
large-scale multiagent systems. Anumberof such systems (formally defined
here as Complex Systems), including air/space combat, robotic
manufacturing, software re-engineering, Internet cyberwar, etc. can be
modeled as abstract board games. These are multi-player games whose
moves can be represented by means of moving abstract pieces over
locations on an abstract board. The dimensions of the board (2D, nD, and
even non-linear space), its shape and size, the mobility of pieces, the turn of
moves (including concurrent moves) - all can betailored to model avarietyof
multiagent systems. Thus abstract board games are understood here as a
class of Complex Systems. The purpose of LG is to provide strategies to
guide the participants of a game to reach their goals. Traditionally, finding
such strategies required searches in giant game trees. Such searches are
often beyond capabilities of modern and even conceivable future
computers.
LG dramatically reduces the size of the search trees, thus making the
problems computationally tractable. LG provides a formalization and
abstraction of search heuristics of advanced experts. Essentially, these
heuristics replace search by construction of strategies. The formalized expert
strategies yield efficient algorithms for problem settings whose dimensions
may be significantly greater than the ones for which the experts developed
their strategies. Moreover, these formal strategies allowed to solve problems
from different problem domains far beyond the areas envisioned by the
experts. It is really fascinating that for certain classes of problems these
expertstrategiesyield provably optimal solutions. Toformalize the heuristics,
LG employs the theoryofformal languages (i.e., formal linguistics), aswell as
certain geometric structures over the abstract board. Since both, the
linguistics and the geometry, were involved, this approach was named
Linguistic Geometry.
x Preface
This is the first book on the subject. It is not my intention to present a
complete theory of LG, because such a theory is yet to be completed.
However, the book includes foundations of such theory.
LG already has a long and enlightening history. Chess, a Drosophila of
Artificial Intelligence, contributed in a major way to the origin of LG. It
continues to be an everlasting source of ideas and a driving force behind its
development. Having started with computer experiments in 1972, continued
with theoretical insights in the 80s, and formally established in 1991, LG has
moved in many directions. Some ofthem are based on rigorous foundations,
others - on sophisticated experiments. The purpose of this book is to
introduce the reader to a multiplicity of approaches, ideas, and experiments.
Every chapter is written as an essay on one topic in LG, theoretical or
experimental, though all chapters are linked together and support each
other.
Chapter 1 includes introduction to the subject and a brief research
background. Chapter 2 includes basic definitions and a survey of the LG
formal tools, a hierarchy of formal languages. Chapters 3 through 7 cover a
number of experiments with LG tools. They demonstrate how the LG
algorithms solve problems of gradually increasing complexity. Besides
interesting actual results, adetailed description of experiments is intended to
develop an intuitive understanding of LG algorithms and their underlying
structure. Armed with this intuition, a reader will be able to digest a formal
description of LG tools in Chapters 8-12. Chapter 13 gives anew, much
deeper account in the foundations of LG. By redeveloping the experiment
from Chapter 3, it points a new direction in LG: solving search problems by
construction of strategies (without any tree-based search). In Chapter 14 we
discuss some issues of computational complexity. The book suggests that
LG tools allow us to identify a wide subclass of tractable (polynomial)
problems among those that are usually considered as intractable
(exponential). The LG algorithms providetheirsolutions.
This book can be used as a textbook for a graduate or senior
undergraduate course on LG for one or two semesters or a supplement to a
course on AI. Some of this material was taught in the courses Knowledge
Representation for Intelligent Systemsand Complex Intelligent Systems at
the University of Colorado at Denver. Also, many topics included in the book
were presented as tutorials and shortcourses on LG around the world.
I hope that the reader will share my excitement with the subject and join
the army of researchers and practitioners that develop the theory and
applications of LG. The book is not going to be an untouchable sacred
sourceof information on LG for decadestocome. My intention is to move the
frontier, elicit interest, ignite research and development, to bring this book
out of date, and write a new book that will further revolutionize our
understanding of the sUbject.
Boris Stilman,
Denver, Colorado, USA