Preface ix
About the Authors xiii
1 Foundation 1
1.1 Categories of Ideas: Entity, Species, Genus 1
1.2 Values 2
1.3 Objects 4
1.4 rocedures6
1.5 Regular Types 6
1.6 Regular Procedures 8
1.7 Concepts 10
1.8 Conclusions14
2 TransformationsandTheirOrbits1 5
2.1 Transformations 15
2.2 Orbits 18
2.3 Collision Point 21
2.4 Measuring Orbit Sizes 27
2.5 Actions 28
2.6 Conclusions 29
3 Associative Operations 31
3.1 Associativity 31
3.2 Computing Powers 33
3.3 Program Transformations 35
3.4 Special-Case Procedures 39
3.5 Parameterizing Algorithms 42
3.6 Linear Recurrences 43
3.7 Accumulation Procedures 46
3.8 Conclusions 47
4 Linear Orderings 49
4.1 Classification of Relations 49
4.2 Total and Weak Orderings 51
4.3 Order Selection 52
4.4 Natural Total Ordering 61
4.5 Clusters of Derived Procedures 62
4.6 Extending Order-Selection Procedures 63
4.7 Conclusions 63
5 Ordered Algebraic Structures 65
5.1 Basic Algebraic Structures 65
5.2 Ordered Algebraic Structures 70
5.3 Remainder 71
5.4 Greatest Common Divisor 76
5.5 Generalizinggcd 79
5.6 Steingcd 81
5.7 Quotient 81
5.8 Quotient and Remainder for Negative Quantities 83
5.9 Concepts and Their Models 85
5.10 Computer Integer Types 87
5.11 Conclusions 88
6 Iterators 89
6.1 Readability 89
6.2 Iterators 90
6.3 Ranges 92
6.4 Readable Ranges 95
6.5IncreasingRanges 103
6.6 Forward Iterators 106
6.7 Indexed Iterators 110
6.8 Bidirectional Iterators 111
6.9 Random-Access Iterators 113
6.1 0Conclusions 114
7 Coordinate Structures 115
7.1 ifurcate Coordinates 115
7.2 Bidirectional Bifurcate Coordinates 119
7.3 Coordinate Structures 124
7.4 Isomorphism, Equivalence, and Ordering 124
7.5 Conclusions 131
8 Coordinates with Mutable Successors 133
8.1 Linked Iterators 133
8.2 Link Rearrangements 134
8.3 Applications of Link Rearrangements 140
8.4 Linked Bifurcate Coordinates 143
8.5 Conclusions148
9 Copying 149
9.1 Writability 149
9.2 Position-Based Copying 151
9.3 Predicate-Based Copying 157
9.4 Swapping Ranges 164
9.5 Conclusions 168
10 Rearrangements 169
10.1 Permutations 169
10.2 Rearrangements 172
10.3 Reverse Algorithms 174
10.4 Rotate Algorithms 178
10.5 Algorithm Selection 186
10.6 Conclusions 189
11 Partitionand Merging 191
11.1 Partition 191
11.2 Balanced Reduction 198
11.3 Merging 202
11.4 Conclusions 208
12 Composite Objects 209
12.1 Simple Composite Objects 209
12.2 Dynamic Sequences 216
12.3 Underlying Type 222
12.4 Conclusions 225
Afterword 227
Appendix A Mathematical Notation 231
Appendix B Programming Language 233
B.1 Language Definition 233
B.2 Macros and Trait Structures 240
Bibliography 243
Index 247
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