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This is a book about computation, something which is ubiquitous in the modern world. More precisely, it examines computability theory and computational complexity theory. Computability theory is the part of mathematics and computer science which seeks to clarify what we mean by computation or algorithm. When is there a computational solution possible to some question? How can we show that none is possible? How computationally hard is the question we are concerned with? Arguably, this area lead to the development of digital computers. (Computational) complexity theory is an intellectual heir of computability theory. Complexity theory is concerned with understanding what resources are needed for computation, where typically we would measure the resources in terms of time and space. Can we perform some task in a feasible number of steps? Can we perform some algorithm with only a limited memory? Does randomness help? Are there standard approaches to overcoming computational difficulty?
"The textbook achieves a balance between theoretical concepts and numerous examples and exercises, scattered efficiently throughout the chapters. Solutions and hints to the exercises are also included. The comprehensive set of references also increases the usefulness of the book. Due to its structure, it is suitable for a wide range of audiences, from students to researchers, and it represents a fundamental stepping stone towards expanding knowledge in the field." (Irina Ioana Mohorianu, zbMATH 1555.68005, 2025)
Rodney Downey is an Emeritus Professor at Victoria University of Wellington, NZ. He is the co-author of the Springer books, Fundamentals of Parameterized Complexity, and Algorithmic Randomness and Complexity. He has won many prizes for his work, including (twice) the Shoenfield Prize for writing, as well as the Rutherford Medal, New Zealand's premier science award.
Introduction.- Some Naive Set Theory.- Regular Languages and Finite Automata.- General Models of Computation.- Deeper Computability.- Computational Complexity.- NP- and PSPACE-Completeness.- Some Structural Complexity.- Parameterized Complexity.- Average Case, Smoothed Analysis, and Generic Case.- Complexity.- References.
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