Media theory: interdisciplinary applied mathematics

330 Pages·2008·1.731 MB·English

by  David Eppstein, Jean-Claude Falmagne, Sergei Ovchinnikov (auth.)

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
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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