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Title: Computability: An Introduction to Recursive Function Theory
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Manufacturer: Cambridge University Press
List Price: $43.00
Our Price: $34.40
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| Customer Reviews: |
| Computability: An Introduction to Recursive Function Theory by Cambridge University Press better than the high priced spred | | Elements of the Theory of Computation (2nd Edition) I just got through reading other book and I realized the only reason I understood it at all was that I had read this book.When I first read this book I thought some of the parts elementary, but comparing it to the above text I realized it was doing the job of teaching by giving good illustrations of the processes involved. | | Computability: An Introduction to Recursive Function Theory by Cambridge University Press We used syrup on salad and danced away the night | Beware the intended audience--undergraduates and infants. Just kidding about the infants, but the exercises are very easy for anyone with decent mathematical/logical sophistication. The results are easy to follow and some motivation is given, but a lot of the proofs are not particularly rigorous, and not because they invoke Church's thesis. A number of proofs of important theorems are excluded, but references are given to texts which include them. A number of important results are excluded, e.g. those of Mostowski and Kleene (the "arithmetical hierarchy").
Cutland does cover the s-m-n theorem, universal functions, the Kleene normal form theorem, degrees of unsolvability, reducibility of decision problems, and equivalent formulations of computability. Of course the usual stuff is in there too--e.g., effective operations on functions, recursion, minimalization, etc. There is a rather nonstandard introduction to computational complexity that is greatly lacking, but at least something is there.
Unlimited Register Machine (URM) computability is the central notion of computability employed in the text. Other than that, there really isn't any notation or concepts that would prevent one from reading chapters independently from the rest of the text. You just need to know how URMs work. (Obviously there is some nonstandard notation but it's listed at the end in a notation section.)
The most annoying feature of the text was the lack of rigor. As such it is best suited as an introductory text for undergraduates. | | Computability: An Introduction to Recursive Function Theory by Cambridge University Press Excellent book for the right audience; often incomplete or informal | This is a well-written book, and gives a satisfying account of the field of recursion theory. It covers basic aspects of recursion theory, Godel numbering, the structure of recursive and recursively enumerable sets, and even a brief (and quite sketchy) foray into complexity results at the end. It is, however, worth deciding whether you are in the target audience before making a purchase.
If you are trying to make a first transition over into theory topics from, say, a career of practical software development tasks, then this is the wrong book. Try Sipser's Introduction to the Theory of Computation instead. Sipser is more willing to spend time on demonstrating the intuitive picture, and relies less on formal mathematical arguments. This book can come later to fill in some of the mathematical properties.
On the opposite end of the spectrum, this is a passable but mediocre reference book for recursion theory. It omits major topics, such as the arithmetic hierarchy. It deviates considerably from other traditional treatments. These decisions will get annoying if you plan to read bits and pieces rather than learn in sequence according to the author's presentation. A better reference is Hartley Rogers' Theory of Recursive Functions and Effective Computability.
Buy this book if you are in the middle. It's a great book if you've seen some decidability results, but not a formal mathematical treatment; and if you intend to follow the book and learn what it decides rather than look up specific topics. In that situation, it's hard to see how you could do better. | | Computability: An Introduction to Recursive Function Theory by Cambridge University Press The best introduction to computability theory | | Cutland's book "Computability: An Introduction to Recursive Function Theory" is without doubt the best introduction to recursion theory available on the market. Elementary Recursion Theory is a logician's expression for theoretical computer science, with an emphasis on negative results, i.e. what computers cannot do. Cutland could have exploited Church's Thesis in the manner of H. Rogers, and this could have perhaps make the book readable for a larger public from the human science. Instead Cutland defines the computable functions in a rather standard way through the use of the Register Machine. Then all the basic chapter of recursion theory are introduced, including a gentle explanation of Kleene second recursion theorem, on recursive operators, on formal arithmetic and Godel's incompleteness theorem, Post creative and productive sets. It contains a chapter on Blum complexity theory, with the gap theorem and Blum's speed-up theorem. The book is a must for those who want to penetrate this fundamental subject. | | Computability: An Introduction to Recursive Function Theory by Cambridge University Press Clearest mathematical introduction | This introduction for undergrads assumes no specifics other than general experience in college math. The writing is clear and exercises are interspersed and follow naturally from the explanations. Proofs are explanatory and easy to follow, though often rather informal.
However, it looses some of its best qualities about halfway through the book. The early chapters give excellent context and motivation, but by chapter 4 that is mostly gone. It seems that the push to cover more topics is what led to making the introduction of new topics more and more brief. The later chapters give little feeling for how it all fits together and why we should care. You can look up the same topics in Rogers or Odifreddi to get an idea of the interesting things that could have been said. | | Computability: An Introduction to Recursive Function Theory by Cambridge University Press Product Description | | What can computers do in principle? What are their inherent theoretical limitations? These are questions to which computer scientists must address themselves. The theoretical framework which enables such questions to be answered has been developed over the last fifty years from the idea of a computable function: intuitively a function whose values can be calculated in an effective or automatic way. This book is an introduction to computability theory (or recursion theory as it is traditionally known to mathematicians). Dr Cutland begins with a mathematical characterisation of computable functions using a simple idealised computer (a register machine); after some comparison with other characterisations, he develops the mathematical theory, including a full discussion of non-computability and undecidability, and the theory of recursive and recursively enumerable sets. The later chapters provide an introduction to more advanced topics such as Gildel's incompleteness theorem, degrees of unsolvability, the Recursion theorems and the theory of complexity of computation. Computability is thus a branch of mathematics which is of relevance also to computer scientists and philosophers. Mathematics students with no prior knowledge of the subject and computer science students who wish to supplement their practical expertise with some theoretical background will find this book of use and interest. |
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