Computational complexity is one of the most beautiful fields of modern mathematics, and it is increasingly relevant to other sciences ranging from physics to biology. But this beauty is often buried underneath layers of unnecessary formalism, and exciting recent results like interactive proofs, phase transitions, and quantum computing are usually considered too advanced for the typical student. This book bridges these gaps by explaining the deep ideas of theoretical computer science in a clear and enjoyable fashion, making them accessible to non-computer scientists and to computer scientists who finally want to appreciate their field from a new point of view. The authors start with a lucid and playful explanation of the P vs. NP problem, explaining why it is so fundamental, and so hard to resolve. They then lead the reader through the complexity of mazes and games; optimization in theory and practice; randomized algorithms, interactive proofs, and pseudorandomness; Markov chains and phase transitions; and the outer reaches of quantum computing. At every turn, they use a minimum of formalism, providing explanations that are both deep and accessible. The book is intended for graduate and undergraduate students, scientists from other areas who have long wanted to understand this subject, and experts who want to fall in love with this field all over again.

Computational complexity is one of the most beautiful fields of modern mathematics, and it is increasingly relevant to other sciences ranging from physics to biology. But this beauty is often buried underneath layers of unnecessary formalism, and exciting recent results like interactive proofs, phase transitions, and quantum computing are usually considered too advanced for the typical student. This book bridges these gaps by explaining the deep ideas of theoretical computer science in a clear and enjoyable fashion, making them accessible to non-computer scientists and to computer scientists who finally want to appreciate their field from a new point of view. The authors start with a lucid and playful explanation of the P vs. NP problem, explaining why it is so fundamental, and so hard to resolve. They then lead the reader through the complexity of mazes and games; optimization in theory and practice; randomized algorithms, interactive proofs, and pseudorandomness; Markov chains and phase transitions; and the outer reaches of quantum computing. At every turn, they use a minimum of formalism, providing explanations that are both deep and accessible. The book is intended for graduate and undergraduate students, scientists from other areas who have long wanted to understand this subject, and experts who want to fall in love with this field all over again.

The boundary between physics and computer science has become a hotbed of interdisciplinary collaboration. In this book the authors introduce the reader to the fundamental concepts of computational complexity and give in-depth explorations of the major interfaces between computer science and physics.

Computing systems are ubiquitous in contemporary life. Even the brain is thought to be a computing system of sorts. But what does it mean to say that a given organ or system "computes"? What is it about laptops, smartphones, and nervous systems that they are deemed to compute - and why does itseldom occur to us to describe stomachs, hurricanes, rocks, or chairs that way? These questions are key to laying the conceptual foundations of computational sciences, including computer science and engineering, and the cognitive and neural sciences.Oron Shagrir here provides an extended argument for the semantic view of computation, which states that semantic properties are involved in the nature of computing systems. The first part of the book provides general background. Although different in scope, these chapters have a common theme-namely,that the linkage between the mathematical theory of computability and the notion of physical computation is weak. The second part of the book reviews existing non-semantic accounts of physical computation. Shagrir analyze three influential accounts in greater depth and argues that none of theseaccounts is satisfactory, but each of them highlights certain key features of physical computation that he eventually adopts in his own semantic account of physical computation - a view that rests on a phenomenon known as simultaneous implementation (or "indeterminacy of computation"). Shagrircompletes the characterization of his account of computation and highlights the distinctive feature of computational explanations.

An introduction to computational complexity theory, its connections and interactions with mathematics, and its central role in the natural and social sciences, technology, and philosophy Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field, with strong links to most mathematical areas and to a growing number of scientific endeavors. Avi Wigderson takes a sweeping survey of complexity theory, emphasizing the field’s insights and challenges. He explains the ideas and motivations leading to key models, notions, and results. In particular, he looks at algorithms and complexity, computations and proofs, randomness and interaction, quantum and arithmetic computation, and cryptography and learning, all as parts of a cohesive whole with numerous cross-influences. Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. He ends with a comprehensive look at the theory of computation, its methodology and aspirations, and the unique and fundamental ways in which it has shaped and will further shape science, technology, and society. For further reading, an extensive bibliography is provided for all topics covered. Mathematics and Computation is useful for undergraduate and graduate students in mathematics, computer science, and related fields, as well as researchers and teachers in these fields. Many parts require little background, and serve as an invitation to newcomers seeking an introduction to the theory of computation. Comprehensive coverage of computational complexity theory, and beyond High-level, intuitive exposition, which brings conceptual clarity to this central and dynamic scientific discipline Historical accounts of the evolution and motivations of central concepts and models A broad view of the theory of computation's influence on science, technology, and society Extensive bibliography

The identity of computing has been fiercely debated throughout its short history. Why is it still so hard to define computing as an academic discipline? Is computing a scientific, mathematical, or engineering discipline? By describing the mathematical, engineering, and scientific traditions of computing, The Science of Computing: Shaping a Discipline presents a rich picture of computing from the viewpoints of the field’s champions. The book helps readers understand the debates about computing as a discipline. It explains the context of computing’s central debates and portrays a broad perspective of the discipline. The book first looks at computing as a formal, theoretical discipline that is in many ways similar to mathematics, yet different in crucial ways. It traces a number of discussions about the theoretical nature of computing from the field’s intellectual origins in mathematical logic to modern views of the role of theory in computing. The book then explores the debates about computing as an engineering discipline, from the central technical innovations to the birth of the modern technical paradigm of computing to computing’s arrival as a new technical profession to software engineering gradually becoming an academic discipline. It presents arguments for and against the view of computing as engineering within the context of software production and analyzes the clash between the theoretical and practical mindsets. The book concludes with the view of computing as a science in its own right—not just as a tool for other sciences. It covers the early identity debates of computing, various views of computing as a science, and some famous characterizations of the discipline. It also addresses the experimental computer science debate, the view of computing as a natural science, and the algorithmization of sciences.

Computation permeates our world, but a satisfactory philosophical theory of what it is has been lacking. Gualtiero Piccinini presents a mechanistic account of what makes a physical system a computing system. He argues that computation does not entail representation or information-processing, although information-processing entails computation.