Media theory: interdisciplinary applied mathematics PDF
Preview Media theory: interdisciplinary applied mathematics
Media Theory First edition David Eppstein · Jean-Claude Falmagne Sergei Ovchinnikov Media Theory Interdisciplinary Applied Mathematics First edition 123 DavidEppstein SergeiOvchinnikov UniversityofCalifornia,Irvine SanFranciscoStateUniversity DepartmentofComputerScience DeptartmentofMathematics Irvine,92697-3425 HollowayAvenue1600 USA SanFrancisco,94132 [email protected] USA [email protected] Jean-ClaudeFalmagne UniversityofCalifornia,Irvine SchoolofSocialSciences DepartmentofCognitiveSciences SocialSciencePlazaA3171 Irvine,92697-5100 USA [email protected] ISBN978-3-540-71696-9 e-ISBN978-3-540-71697-6 DOI10.1007/978-3-540-71697-6 LibraryofCongressControlNumber:2007936368 ©2008Springer-VerlagBerlinHeidelberg Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. CoverDesign:KünkelLopka,Heidelberg Printedonacid-freepaper 987654321 springer.com Preface The focus of this book is a mathematical structure modeling a physical or biological system that can be in any of a number of ‘states.’ Each state is characterized by a set of binary features, and differs from some other neigh- bor state or states by just one of those features. In some situations, what distinguishes a state S from a neighbor state T is that S has a particular fea- turethatT doesnothave.Afamiliarexampleisapartialsolutionofajigsaw puzzle, with adjoining pieces. Such a state can be transformed into another state, that is, another partial solution or the final solution, just by adding a single adjoining piece. This is the first example discussed in Chapter 1. In othersituations,thedifferencebetweenastateS andaneighborstateT may resideintheirlocationinaspace,asinoursecondexample,inwhichinwhich S and T are regions located on different sides of some common border. We formalize the mathematical structure as a semigroup of ‘messages’ transforming states into other states. Each of these messages is produced by the concatenation of elementary transformations called ‘tokens (of informa- tion).’ The structure is specified by two constraining axioms. One states that any state can be produced from any other state by an appropriate kind of message. The other axiom guarantees that such a production of states from other states satisfies a consistency requirement. What motivates our interest in this semigroup is, first, that it provides an algebraic formulation for mathematical systems researched elsewhere and earlier by other means. A prominent example is the ‘isometric subgraph of a hypercube’ (see Djokovi´c, 1973, for an early reference), that is, a subraph in which the distance between vertices is identical to that in the parent hy- percube. But there are many other cases. We shall outline some of them in our first chapter, reserving in depth treatment for later parts of this book. Until recently, however, no common algebraic axiomatization of these out- wardly different concepts had been proposed. Our purpose is to give here the first comprehensive treatment of such a structure, which we refer to as a ‘medium.’ A second, equally importantly reason for studying media, is that they offerahighlyconvenientrepresentationforavastclassofempiricalsituations ranging from cognitive structures in education to the study of opinion polls in political sciences and including, conceivably, genetics, to name just a few pointers.Theyprovideanappropriatemedium1 wherethetemporalevolution of a system can take place. Indeed, it turns out that, for some applications, the set of states of a medium can be profitably cast as the set of states of a random walk. Moreover, under simple hypotheses concerning the stochastic processinvolved,theasymptoticprobabilitiesofthestatesareeasytocompute and simple to write. Accordingly, some space is devoted to the development 1 There lies the origin of the term. VI Preface of a random walk on the set of states of a medium, and to the description of a substantial application to the analysis of an opinion poll. Inthismonograph,westudymediafromvariousangles:algebraicinChap- ters 2, 3, and 4; combinatoric in Chapters 5 and 6; geometric in Chapters 7 to 9; algorithmic in Chapters 10 and 11. Chapters 12 and 13 are devoted to random walks on media and to applications. Throughthebook,eachchapterisorganizedintosectionscontainingpara- graphs, which often bear titles such as Definition, Example, or Theorem. Forsimplicityofreferenceandtofacilitateasearchthroughthebook,asingle numerical system is used. For instance: 2.4.12 Lemma. 2.4.13 Definition. arethetitlesofthetwelfthandthethirteenthparagraphsofChapter2,Section 2.4. We refer to the above lemma as “Lemma 2.4.12.” Defined technical terms are typed in slanted font just once, at the place where they are defined, which is typically within a “Definition” paragraph. Technical terms used before their definition are put between single quotes (at the first mention). The text of theorems and other results are also set in slanted font. A short history of the results leading to the concept of a medium and ultimately to this monograph can be found in Section 1.9. In the course of our work, we benefitted from exchanges with many re- searchers,whosereactionstoourideasinfluencedourwriting,sometimessub- stantially. We want to thank, in particular, Josiah Carlson, Dan Cavagnaro, Victor Chepoi, Eric Cosyn, Chris Doble, Aleks Dukhovny, Peter Fishburn, Bernie Grofman, Yung-Fong Hsu, Geoff Iverson, Duncan Luce, Louis Narens, Michel Regenwetter, Fred Roberts, Pat Suppes, Nicolas Thi´ery, and Hasan Uzun. A special mention must be made of Jean-Paul Doignon, whose joint work withFalmagneprovidedmuchofthefoundationalideasbehindtheconceptof a medium. For a long time, we thought that Jean-Paul would be a co-author. However, other commitments prevented him to be one of us. No doubt, had he been a co-author, our book would have been a better one. Last but not least, Diana, Dina, and Galina deserve much credit for var- iously letting us be—the relevant one, that is—whenever it seemed that the call of the media was too strong, or for gently drawing us away from them, forourowngoodsake,whentherewasanopening.Tothosethree,wearethe most grateful. David Eppstein Jean-Claude Falmagne Sergei Ovchinnikov August 11, 2007 Contents 1 Examples and Preliminaries ............................... 1 1.1 A Jigsaw Puzzle......................................... 1 1.2 A Geometrical Example .................................. 4 1.3 The Set of Linear Orders ................................. 6 1.4 The Set of Partial Orders................................. 7 1.5 An Isometric Subgraph of Zn ............................. 8 1.6 Learning Spaces......................................... 10 1.7 A Genetic Mutations Scheme ............................. 11 1.8 Notation and Conventions ................................ 12 1.9 Historical Note and References ............................ 17 Problems............................................... 19 2 Basic Concepts ............................................ 23 2.1 Token Systems .......................................... 23 2.2 Axioms for a Medium.................................... 24 2.3 Preparatory Results ..................................... 27 2.4 Content Families ........................................ 29 2.5 The Effective Set and the Producing Set of a State .......... 30 2.6 Orderly and Regular Returns ............................. 31 2.7 Embeddings, Isomorphisms and Submedia .................. 34 2.8 Oriented Media ......................................... 36 2.9 The Root of an Oriented Medium ......................... 38 2.10 An Infinite Example ..................................... 39 2.11 Projections ............................................. 40 Problems............................................... 45 3 Media and Well-graded Families ........................... 49 3.1 Wellgradedness.......................................... 49 3.2 The Grading Collection .................................. 52 VIII Contents 3.3 Wellgradedness and Media................................ 54 3.4 Cluster Partitions and Media ............................. 57 3.5 An Application to Clustered Linear Orders ................. 62 3.6 A General Procedure .................................... 68 Problems............................................... 68 4 Closed Media and ∪-Closed Families....................... 73 4.1 Closed Media ........................................... 73 4.2 Learning Spaces and Closed Media ........................ 78 4.3 Complete Media......................................... 80 4.4 Summarizing a Closed Medium............................ 83 4.5 ∪-Closed Families and their Bases ......................... 86 4.6 Projection of a Closed Medium............................ 94 Problems............................................... 98 5 Well-Graded Families of Relations .........................101 5.1 Preparatory Material ....................................102 5.2 Wellgradedness and the Fringes ...........................103 5.3 Partial Orders ..........................................106 5.4 Biorders and Interval Orders..............................107 5.5 Semiorders .............................................110 5.6 Almost Connected Orders ................................114 Problems...............................................119 6 Mediatic Graphs...........................................123 6.1 The Graph of a Medium..................................123 6.2 Media Inducing Graphs ..................................125 6.3 Paired Isomorphisms of Media and Graphs .................130 6.4 From Mediatic Graphs to Media...........................132 Problems...............................................136 7 Media and Partial Cubes ..................................139 7.1 Partial Cubes and Mediatic Graphs........................139 7.2 Characterizing Partial Cubes .............................142 7.3 Semicubes of Media......................................149 7.4 Projections of Partial Cubes ..............................151 7.5 Uniqueness of Media Representations ......................154 7.6 The Isometric Dimension of a Partial Cube .................158 Problems...............................................159 Contents IX 8 Media and Integer Lattices ................................161 8.1 Integer Lattices .........................................161 8.2 Defining Lattice Dimension ...............................162 8.3 Lattice Dimension of Finite Partial Cubes ..................167 8.4 Lattice Dimension of Infinite Partial Cubes .................171 8.5 Oriented Media .........................................172 Problems...............................................174 9 Hyperplane arrangements and their media.................177 9.1 Hyperplane Arrangements and Their Media.................177 9.2 The Lattice Dimension of an Arrangement..................184 9.3 Labeled Interval Orders ..................................186 9.4 Weak Orders and Cubical Complexes ......................188 Problems...............................................196 10 Algorithms ................................................199 10.1 Comparison of Size Parameters............................199 10.2 Input Representation ....................................202 10.3 Finding Concise Messages ................................211 10.4 Recognizing Media and Partial Cubes......................217 10.5 Recognizing Closed Media ................................218 10.6 Black Box Media ........................................222 Problems...............................................227 11 Visualization of Media.....................................229 11.1 Lattice Dimension .......................................230 11.2 Drawing High-Dimensional Lattice Graphs..................231 11.3 Region Graphs of Line Arrangements ......................234 11.4 Pseudoline Arrangements.................................238 11.5 Finding Zonotopal Tilings ................................246 11.6 Learning Spaces.........................................252 Problems...............................................260 12 Random Walks on Media ..................................263 12.1 On Regular Markov Chains ...............................265 12.2 Discrete and Continuous Stochastic Processes ...............271 12.3 Continuous Random Walks on a Medium...................273 12.4 Asymptotic Probabilities .................................279 12.5 Random Walks and Hyperplane Arrangements ..............280 Problems...............................................282 X Contents 13 Applications ...............................................285 13.1 Building a Learning Space................................285 13.2 The Entailment Relation .................................291 13.3 Assessing Knowledge in a Learning Space...................293 13.4 The Stochastic Analysis of Opinion Polls ...................297 13.5 Concluding Remarks.....................................302 Problems...............................................303 Appendix: A Catalog of Small Mediatic Graphs ...............305 Glossary.......................................................309 Bibliography...................................................311 Index..........................................................321
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