Computability and Complexity

Beschreibung

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 10^th 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.

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?

Autor*in

Rod Downey

Themen in »Computability and Complexity«

Finite automata NP-completeness Parameterized complexity Reducability Recursion Structural complexity

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“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)

“Computability and Complexity isn't just another book about computation; it is the authoritative contribution of an old hand, one of the living masters of the subject - one of the founders of parametrised complexity theory.  Reassuringly it also shows throughout its invaluable roots in classroom experience.  A distinctive contribution is its treatment of the Collatz conjecture and its connections with Conway's FRACTRAN programming language, whose generality explains why the Collatz conjecture is so intractable.  In principle this is a book for beginners (at least initially), but its 360-odd pages will be the reader's companion into what is definitely fourth-year material and beyond.” (Dr. Thomas Forster, Queen’s College, University of Cambridge)

“Computability Theory and Complexity Theory share much in common:  They both adopt a hierarchical view of problems, and there is considerable overlap in the communities working in these fields. This book appears to be unique in achieving a genuine balance between the two subjects, while also revealing the deep and organic connections between them. Professor Rod Downey is a leading researcher in both areas. Owing to his expertise, the author is able to cover an impressively wide range of topics, from foundational definitions to the current research frontier. As for the exposition, the writing is often remarkably accessible. My suggestion is to use it both as a textbook (focusing on the foundational material) and as a source for mini-projects (particularly the starred sections and more challenging exercises). I would recommend it to advanced undergraduate students and beginning graduate students in both mathematics and computer science.” (Prof. Yang Sue, National University of Singapore)

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