From Mathematics to Generic Programming

In this substantive yet accessible book, pioneering software designer Alexander Stepanov and his colleague Daniel Rose illuminate the principles of generic programming and the mathematical concept of abstraction on which it is based, helping you write code that is both simpler and more powerful. If you’re a reasonably proficient programmer who can think logically, you have all the background you’ll need. Stepanov and Rose introduce the relevant abstract algebra and number theory with exceptional clarity. They carefully explain the problems mathematicians first needed to solve, and then show how these mathematical solutions translate to generic programming and the creation of more effective and elegant code. To demonstrate the crucial role these mathematical principles play in many modern applications, the authors show how to use these results and generalized algorithms to implement a real-world public-key cryptosystem. As you read this book, you’ll master the thought processes necessary for effective programming and learn how to generalize narrowly conceived algorithms to widen their usefulness without losing efficiency. You’ll also gain deep insight into the value of mathematics to programming–insight that will prove invaluable no matter what programming languages and paradigms you use. You will learn about How to generalize a four thousand-year-old algorithm, demonstrating indispensable lessons about clarity and efficiency Ancient paradoxes, beautiful theorems, and the productive tension between continuous and discrete A simple algorithm for finding greatest common divisor (GCD) and modern abstractions that build on it Powerful mathematical approaches to abstraction How abstract algebra provides the idea at the heart of generic programming Axioms, proofs, theories, and models: using mathematical techniques to organize knowledge about your algorithms and data structures Surprising subtleties of simple programming tasks and what you can learn from them How practical implementations can exploit theoretical knowledge Alexander A. Stepanov has been programming since 1972–first in the Soviet Union and, since emigrating in 1977, in the United States. He has programmed operating systems, programming tools, compilers, and libraries. His work on the foundations of programming has been supported by GE, Polytechnic University, Bell Labs, HP, SGI, Adobe, and, since 2009, A9.com, Amazon’s search subsidiary. In 1995, he received the Dr. Dobb’s Journal Excellence in Programming Award for the design of the C++ Standard Template Library. Daniel E. Rose is a research scientist who has held management positions at Apple, AltaVista, Xigo, Yahoo, and A9.com. His research focuses on all aspects of search, ranging from low-level algorithms for index compression to human—computer interaction. Rose led the Apple team that created desktop search for the Mac. He holds a Ph.D. in cognitive science and computer science from the University of California, San Diego, and a B.A. in philosophy from Harvard. In this substantive yet accessible book, pioneering software designer Alexander Stepanov and his colleague Daniel Rose illuminate the principles of generic programming and the mathematical concept of abstraction on which it is based, helping you write code that is both simpler and more powerful. If you’re a reasonably proficient programmer who can think logically, you have all the background you’ll need. Stepanov and Rose introduce the relevant abstract algebra and number theory with exceptional clarity. They carefully explain the problems mathematicians first needed to solve, and then show how these mathematical solutions translate to generic programming and the creation of more effective and elegant code. To demonstrate the crucial role these mathematical principles play in many modern applications, the authors show how to use these results and generalized algorithms to implement a real-world public-key cryptosystem. As you read this book, you’ll master the thought processes necessary for effective programming and learn how to generalize narrowly conceived algorithms to widen their usefulness without losing efficiency. You’ll also gain deep insight into the value of mathematics to programming–insight that will prove invaluable no matter what programming languages and paradigms you use. You will learn about * How to generalize a four thousand-year-old algorithm, demonstrating indispensable lessons about clarity and efficiency * Ancient paradoxes, beautiful theorems, and the productive tension between continuous and discrete * A simple algorithm for finding greatest common divisor (GCD) and modern abstractions that build on it * Powerful mathematical approaches to abstraction * How abstract algebra provides the idea at the heart of generic programming * Axioms, proofs, theories, and models: using mathematical techniques to organize knowledge about your algorithms and data structures * Surprising subtleties of simple programming tasks and what you can learn from them * How practical implementations can exploit theoretical knowledge **Alexander A. Stepanov** has been programming since 1972–first in the Soviet Union and, since emigrating in 1977, in the United States. He has programmed operating systems, programming tools, compilers, and libraries. His work on the foundations of programming has been supported by GE, Polytechnic University, Bell Labs, HP, SGI, Adobe, and, since 2009, A9.com, Amazon’s search subsidiary. In 1995, he received the __Dr. Dobb’s Journal__ Excellence in Programming Award for the design of the C++ Standard Template Library. **Daniel E. Rose** is a research scientist who has held management positions at Apple, AltaVista, Xigo, Yahoo, and A9.com. His research focuses on all aspects of search, ranging from low-level algorithms for index compression to human—computer interaction. Rose led the Apple team that created desktop search for the Mac. He holds a Ph.D. in cognitive science and computer science from the University of California, San Diego, and a B.A. in philosophy from Harvard. In Searching for Algorithms, Alexander Stepanov (creator of C++ STL) and Daniel Rose introduce math that can make any serious programmer more effective - and they do so in an engaging and accessible fashion, revealing how this math was first discovered, how programmers recognized its value, and the many surprising ways they have applied it. The perfect complement to Stepanov's classic Elements of Programming, Searching for Algorithms journeys through three key algorithms: multiplication; division with remainder; and adding 1. Those algorithms may sound pretty basic - even elementary school basic. But the authors show how they have played a profound role in the development of mathematics - and how, at a much deeper level, they are still essential to the work of today's programmers. In exploring these case studies, Stepanov and Rose show how to implement and read algorithms of all kinds, how to generalize them to the broadest possible set of applications, and how to define programming interfaces based on them This book is a great introduction to the core principles of generic programming for the experienced programmer. The authors work through examples showing how to analyse the requirements of an algorithm and make it as general as possible. The book includes several programming ""laws"" of particular interest to those building software components. The authors shsw how programmers can become more effective by learning about the idea of abstraction and the math it relies on. In an engaging and accessible fashion, they describe how these mathematical results were first discovered and are surprisingl Acknowledgments About the Authors Author's Note What This Book Is About The First Algorithm Ancient Greek Number Theory Euclid's Algorithm The Emergence of Modern Number Theory Abstraction in Mathematics Deriving a Generic Algorithm More Algebraic Structures Organizing Mathematical Knowledge Fundamental Programming Concepts Permutation Algorithms Extensions of GCD A Real-World Application Conclusions Further Reading Notation Common Proof Techniques C++ for Non-C++ Programmers Bibliography Index A detailed, mathematically-oriented examination of abstraction in software development leading to concepts and techniques in generic programming.