More precisely, it examines computability theory and computational complexity theory. (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.

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.

The ideas and techniques comprised in the mathematical framework for understanding computation should form part of the standard background of a graduate in mathematics or computer science, as the issues of computability and complexity permeate modern science.This textbook/reference offers a straightforward and thorough grounding in the theory of computability and computational complexity. Among topics covered are basic naive set theory, regular languages and automata, models of computation, partial recursive functions, undecidability proofs, classical computability theory including the arithmetical hierarchy and the priority method, the basics of computational complexity and hierarchy theorems.Topics and features: ·       Explores Conway’s undecidability proof of the ``3x+1’’ problem using reductions from Register Machines and “Fractran” ·       Offers an accessible account of the undecidability of the exponential version of Hilbert’s 10th problem due to Jones and Matijacevič ·       Provides basic material on computable structure, such as computable linear orderings ·       Addresses parameterized complexity theory, including applications to algorithmic lower bounds and kernelization lower bounds ·       Delivers a short account of generic-case complexity and of smoothed analysis ·       Includes bonus material on structural complexity theory and priority arguments in computability theory This comprehensive textbook will be ideal for advanced undergraduates or beginning graduates, preparing them well for more advanced studies or applications in science.  Additionally, it could serve such needs for mathematicians or for scientists working in computational areas, such as biology. 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.

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