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Introduction to Lattice Theory with Computer Science Applications


Introduction to Lattice Theory with Computer Science Applications


1. Aufl.

von: Vijay K. Garg

76,99 €

Verlag: Wiley
Format: PDF
Veröffentl.: 02.03.2016
ISBN/EAN: 9781119069737
Sprache: englisch
Anzahl Seiten: 272

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Beschreibungen

<b>A computational perspective on partial order and lattice theory, focusing on algorithms and their applications</b> <p>This book provides a uniform treatment of the theory and applications of lattice theory. The applications covered include tracking dependency in distributed systems, combinatorics, detecting global predicates in distributed systems, set families, and integer partitions. The book presents algorithmic proofs of theorems whenever possible. These proofs are written in the calculational style advocated by Dijkstra, with arguments explicitly spelled out step by step. The author’s intent is for readers to learn not only the proofs, but the heuristics that guide said proofs.</p> <p><i>Introduction to Lattice Theory with Computer Science Applications</i>:</p> <ul> <li>Examines; posets, Dilworth’s theorem, merging algorithms, lattices, lattice completion, morphisms, modular and distributive lattices, slicing, interval orders, tractable posets, lattice enumeration algorithms, and dimension theory</li> <li>Provides end of chapter exercises to help readers retain newfound knowledge on each subject</li> <li>Includes supplementary material at www.ece.utexas.edu/~garg</li> </ul> <p><i>Introduction to Lattice Theory with Computer Science Applications</i> is written for students of computer science, as well as practicing mathematicians.</p>
<p><b>List of Figures xiii</b></p> <p><b>Nomenclature xv</b></p> <p><b>Preface xvii</b></p> <p><b>1 Introduction 1</b></p> <p>1.1 Introduction 1</p> <p>1.2 Relations 2</p> <p>1.3 Partial Orders 3</p> <p>1.4 Join and Meet Operations 5</p> <p>1.5 Operations on Posets 7</p> <p>1.6 Ideals and Filters 8</p> <p>1.7 Special Elements in Posets 9</p> <p>1.8 Irreducible Elements 10</p> <p>1.9 Dissector Elements 11</p> <p>1.10 Applications: Distributed Computations 11</p> <p>1.11 Applications: Combinatorics 12</p> <p>1.12 Notation and Proof Format 13</p> <p>1.13 Problems 15</p> <p>1.14 Bibliographic Remarks 15</p> <p><b>2 Representing Posets 17</b></p> <p>2.1 Introduction 17</p> <p>2.2 Labeling Elements of The Poset 17</p> <p>2.3 Adjacency List Representation 18</p> <p>2.4 Vector Clock Representation 20</p> <p>2.5 Matrix Representation 22</p> <p>2.6 Dimension-Based Representation 22</p> <p>2.7 Algorithms to Compute Irreducibles 23</p> <p>2.8 Infinite Posets 24</p> <p>2.9 Problems 26</p> <p>2.10 Bibliographic Remarks 27</p> <p><b>3 Dilworth’s Theorem 29</b></p> <p>3.1 Introduction 29</p> <p>3.2 Dilworth’s Theorem 29</p> <p>3.3 Appreciation of Dilworth’s Theorem 30</p> <p>3.4 Dual of Dilworth’s Theorem 32</p> <p>3.5 Generalizations of Dilworth’s Theorem 32</p> <p>3.6 Algorithmic Perspective of Dilworth’s Theorem 32</p> <p>3.7 Application: Hall’s Marriage Theorem 33</p> <p>3.8 Application: Bipartite Matching 34</p> <p>3.9 Online Decomposition of Posets 35</p> <p>3.10 A Lower Bound on Online Chain Partition 37</p> <p>3.11 Problems 38</p> <p>3.12 Bibliographic Remarks 39</p> <p><b>4 Merging Algorithms 41</b></p> <p>4.1 Introduction 41</p> <p>4.2 Algorithm to Merge Chains in Vector Clock Representation 41</p> <p>4.3 An Upper Bound for Detecting an Antichain of Size <i>K</i> 47</p> <p>4.4 A Lower Bound for Detecting an Antichain of Size <i>K</i> 48</p> <p>4.5 An Incremental Algorithm for Optimal Chain Decomposition 50</p> <p>4.6 Problems 50</p> <p>4.7 Bibliographic Remarks 51</p> <p><b>5 Lattices 53</b></p> <p>5.1 Introduction 53</p> <p>5.2 Sublattices 54</p> <p>5.3 Lattices as Algebraic Structures 55</p> <p>5.4 Bounding The Size of The Cover Relation of a Lattice 56</p> <p>5.5 Join-Irreducible Elements Revisited 57</p> <p>5.6 Problems 59</p> <p>5.7 Bibliographic Remarks 60</p> <p><b>6 Lattice Completion 61</b></p> <p>6.1 Introduction 61</p> <p>6.2 Complete Lattices 61</p> <p>6.3 Closure Operators 62</p> <p>6.4 Topped ∩-Structures 63</p> <p>6.5 Dedekind–Macneille Completion 64</p> <p>6.6 Structure of Dedekind--Macneille Completion of a Poset 67</p> <p>6.7 An Incremental Algorithm for Lattice Completion 69</p> <p>6.8 Breadth First Search Enumeration of Normal Cuts 71</p> <p>6.9 Depth First Search Enumeration of Normal Cuts 73</p> <p>6.10 Application: Finding the Meet and Join of Events 75</p> <p>6.11 Application: Detecting Global Predicates in Distributed Systems 76</p> <p>6.12 Application: Data Mining 77</p> <p>6.13 Problems 78</p> <p>6.14 Bibliographic Remarks 78</p> <p><b>7 Morphisms 79</b></p> <p>7.1 Introduction 79</p> <p>7.2 Lattice Homomorphism 79</p> <p>7.3 Lattice Isomorphism 80</p> <p>7.4 Lattice Congruences 82</p> <p>7.5 Quotient Lattice 83</p> <p>7.6 Lattice Homomorphism and Congruence 83</p> <p>7.7 Properties of Lattice Congruence Blocks 84</p> <p>7.8 Application: Model Checking on Reduced Lattices 85</p> <p>7.9 Problems 89</p> <p>7.10 Bibliographic Remarks 90</p> <p><b>8 Modular Lattices 91</b></p> <p>8.1 Introduction 91</p> <p>8.2 Modular Lattice 91</p> <p>8.3 Characterization of Modular Lattices 92</p> <p>8.4 Problems 98</p> <p>8.5 Bibliographic Remarks 98</p> <p><b>9 Distributive Lattices 99</b></p> <p>9.1 Introduction 99</p> <p>9.2 Forbidden Sublattices 99</p> <p>9.3 Join-Prime Elements 100</p> <p>9.4 Birkhoff’s Representation Theorem 101</p> <p>9.5 Finitary Distributive Lattices 104</p> <p>9.6 Problems 104</p> <p>9.7 Bibliographic Remarks 105</p> <p><b>10 Slicing 107</b></p> <p>10.1 Introduction 107</p> <p>10.2 Representing Finite Distributive Lattices 107</p> <p>10.3 Predicates on Ideals 110</p> <p>10.4 Application: Slicing Distributed Computations 116</p> <p>10.5 Problems 117</p> <p>10.6 Bibliographic Remarks 118</p> <p><b>11 Applications of Slicing to Combinatorics 119</b></p> <p>11.1 Introduction 119</p> <p>11.2 Counting Ideals 120</p> <p>11.3 Boolean Algebra and Set Families 121</p> <p>11.4 Set Families of Size <i>k</i> 122</p> <p>11.5 Integer Partitions 123</p> <p>11.6 Permutations 127</p> <p>11.7 Problems 129</p> <p>11.8 Bibliographic Remarks 129</p> <p><b>12 Interval Orders 131</b></p> <p>12.1 Introduction 131</p> <p>12.2 Weak Order 131</p> <p>12.3 Semiorder 133</p> <p>12.4 Interval Order 134</p> <p>12.5 Problems 136</p> <p>12.6 Bibliographic Remarks 137</p> <p><b>13 Tractable Posets 139</b></p> <p>13.1 Introduction 139</p> <p>13.2 Series–Parallel Posets 139</p> <p>13.3 Two-Dimensional Posets 142</p> <p>13.4 Counting Ideals of a Two-Dimensional Poset 145</p> <p>13.5 Problems 146</p> <p>13.6 Bibliographic Remarks 147</p> <p><b>14 Enumeration Algorithms 149</b></p> <p>14.1 Introduction 149</p> <p>14.2 BFS Traversal 150</p> <p>14.3 DFS Traversal 154</p> <p>14.4 LEX Traversal 154</p> <p>14.5 Uniflow Partition of Posets 160</p> <p>14.6 Enumerating Tuples of Product Spaces 163</p> <p>14.7 Enumerating All Subsets 163</p> <p>14.8 Enumerating All Subsets of Size <i>k</i> 165</p> <p>14.9 Enumerating Young’s Lattice 166</p> <p>14.10 Enumerating Permutations 167</p> <p>14.11 Lexical Enumeration of All Order Ideals of a Given Rank 168</p> <p>14.12 Problems 172</p> <p>14.13 Bibliographic Remarks 173</p> <p><b>15 Lattice of Maximal Antichains 159</b></p> <p>15.1 Introduction 159</p> <p>15.2 Maximal Antichain Lattice 161</p> <p>15.3 An Incremental Algorithm Based on Union Closure 163</p> <p>15.4 An Incremental Algorithm Based on BFS 165</p> <p>15.5 Traversal of the Lattice of Maximal Antichains 166</p> <p>15.6 Application: Detecting Antichain-Consistent Predicates 168</p> <p>15.7 Construction and Enumeration of Width Antichain Lattice 169</p> <p>15.8 Lexical Enumeration of Closed Sets 171</p> <p>15.9 Construction of Lattices Based on Union Closure 174</p> <p>15.10 Problems 174</p> <p>15.11 Bibliographic Remarks 175</p> <p><b>16 Dimension Theory 177</b></p> <p>16.1 Introduction 177</p> <p>16.2 Chain Realizers 178</p> <p>16.3 Standard Examples of Dimension Theory 179</p> <p>16.4 Relationship Between the Dimension and the Width of a Poset 180</p> <p>16.5 Removal Theorems for Dimension 181</p> <p>16.6 Critical Pairs in the Poset 182</p> <p>16.7 String Realizers 184</p> <p>16.8 Rectangle Realizers 193</p> <p>16.9 Order Decomposition Method and Its Applications 194</p> <p>16.10 Problems 196</p> <p>16.11 Bibliographic Remarks 197</p> <p><b>17 Fixed Point Theory 215</b></p> <p>17.1 Complete Partial Orders 215</p> <p>17.2 Knaster–Tarski Theorem 216</p> <p>17.3 Application: Defining Recursion Using Fixed Points 218</p> <p>17.4 Problems 226</p> <p>17.5 Bibliographic Remarks 227</p> <p><b>Bibliography 229</b></p> <p><b>Index 235</b></p>
<p>"This nice book on lattices and their applications in computer science is written from the perspective of a computer scientist rather than a mathematician...Given its emphasis on algorithms and their complexity, it seems to be mainly intended for students of computer science and engineering. The author's approach is based on the premise that a student needs to learn the heuristics that guide the proofs, besides the proofs themselves, and to learn ways to extend and analyze theorems...One of the most important and valuable features of the book is its focus on applications of lattice theory. The author intends to treat applications on par with the theory." Altogether a "lovely book". (<i>Mathematical Reviews/MathSciNet</i> April 2017)
<b>Vijay K. Garg</b>, PhD, is a Cullen Trust Endowed professor at the University of Texas at Austin. His research focuses on applications of lattice theory to distributed computing. He has worked in the areas of distributed systems and discrete event systems for the past thirty years. Dr. Garg is the author of <i>Elements of Distributed Computing</i> (Wiley, 2002), <i>Concurrent and Distributed Computing in Java</i> (Wiley, 2004) and <i>Modeling and Control of Logical Discrete Event Systems</i> (co-authored with Ratnesh Kumar).
<b>A computational perspective on partial order and lattice theory, focusing on algorithms and their applications</b> <p>This book provides a uniform treatment of the theory and applications of lattice theory. The applications covered include tracking dependency in distributed systems, combinatorics, detecting global predicates in distributed systems, set families, and integer partitions. The book presents algorithmic proofs of theorems whenever possible. These proofs are written in the calculational style advocated by Dijkstra, with arguments explicitly spelled out step by step. The author’s intent is for readers to learn not only the proofs, but also the heuristics that guide these proofs.</p> <p><i>Introduction to Lattice Theory with Computer Science Applications:</i></p> <ul> <li>Examines posets, Dilworth's theorem, merging algorithms, lattices, lattice completion, morphisms, modular and distributive lattices, slicing, interval orders, tractable posets, lattice enumeration algorithms, and dimension theory</li> <li>Provides end of chapter exercises to help readers retain newfound knowledge on each subject</li> <li>Includes supplementary material at www.ece.utexas.edu/~garg</li> </ul> <p><i>Introduction to Lattice Theory with Computer Science Applications</i> is written for students of computer science, as well as for practicing mathematicians.</p>

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