Editorial Review

This book is a rigorous exposition of formal languages and models of computation, with an introduction to computational complexity. The authors present the theory in a concise and straightforward manner, with an eye out for the practical applications. Exercises at the end of each chapter, including some that have been solved, help readers confirm and enhance their understanding of the material. This book is appropriate for upper-level computer science undergraduates who are comfortable with mathematical arguments. ... Read more

Reviews (31)

Need some challenge? Come here!
I started to learn this course at the beginning of this semester and I just brought this book from Amazon in August.
first edition is a classic, the second one unremarkable
The first edition is one of the best book in its field. A classic. A reference for many advanced courses in computer theory. Excellent introductory text, but has several weaknesses
This was my textbook for an introductory course on Finite Automata and Languages - I enjoyed it a lot and I think that the chapters until the Turing Machines are covered very well, along with good examples. As one previous reviewer has already mentioned, the exercises can get very hard as compared to what's actually presented - this I found not too good.Read more

Subjects:  1. Computational complexity    2. Computer Science    3. Formal languages    4. Logic    5. Machine theory    6. Mathematics    7. Number Theory    8. Science/Mathematics    9. Artificial intelligence    10. Computers / Computer Science    11. General Theory of Computing    12. Programming languages   

Editorial Review

"God created the integers," wrote mathematician Leopold Kronecker, "All the rest is the work of Man." In this collection of landmark mathematical works, editor Stephen Hawking has assembled the greatest feats humans have ever accomplished using just numbers and their brains. Each of the 17 sections opens with a historical introduction of the featured author, and proceeds to a faithful translation of their most famous work. While most mathematicians will already have complete editions of Isaac Newton's Read more

Reviews (15)

Stephen's talk of stars and the Universe
During his career as a cosmologist Professor Stephen Hawking has proposed many ground-breaking theories. His theories on the origins and nature of the universe are quite complex, but Stephen believes it is important to communicate his ideas to the general public. He has written two books - A Brief History of Time, and The Universe in a Nutshell, to try to explain his ideas to everyone. Hawking is the leading scientist on Quantum Mechanics -- which he describes the behaviour of the smallest particles in the Universe. Unlike everyday objects which tend to obey predictable laws, tiny objects like atoms can behave in very unpredictable ways. Hawking's theories include -- Steady State vs the Big Bang. When Hawking was a student, most cosmologists believed in the steady state theory. The steady state theory says the universe had no beginning and will exist in the same state forever. In contrast, the Big Bang theory says that everything in our Universe (and therefore space, matter and time) began in an explosion about 15 billion years ago. This is the theory that most cosmologists agree on today. The theory of relativity was one of the most important ideas of the twentieth century. Albert Einstein realised that space and time were not independent concepts - how you measure time depends on how you are moving through space, and vice versa. He suggested that gravitational fields could bend what he called space-time. Relativity predicts that time slows down near very massive objects like Black Holes. Black Holes are large stars that have undergone total gravitational collapse and whose gravity is so great that nothing, not even light, can escape from them. The heart of a Black Hole is infinitely dense, and is an example of what mathematicians call a singularity. Singularities are points where space and time are warped so much that the laws of physics break down. Singularities occur at the heart of black holes. The Big Bang is also a singularity; before the Big Bang, time does not mathematically exist.
"The writing fingers having writ ...", In the Shoes of the "God"? - -From the Pen of Dr. Hawking .. ?!
It is no wonder that this great work has attracted so many diverse type of reviews from some of the very competent reviewers.The omissions of many good things bring to the mind the work of (Sir) Roger Penrose; really this great project pales in comparison.
Need an editor, couldn't agree more
I really want to give this book at least 4 stars as I love the idea of encapsulating those great mathematical breakthroughs in one book and giving each a proper account instead of over simplistic summaries like many other math readings do.
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Subjects:  1. History & Philosophy    2. Logic    3. Mathematics    4. Science/Mathematics    5. History of mathematics    6. Popular science   

Editorial Review

"Intended as an upper-level undergraduate or introductory graduate text in computer science theory," this book lucidly covers the key concepts and theorems of the theory of computation. The presentation is remarkably clear; for example, the "proof idea," which offers the reader an intuitive feel for how the proof was constructed, accompanies many of the theorems and a proof. Read more

Reviews (45)

Terrible
I do not understand why this book is rated so highly.
Most appropriate for CS students
As a teacher of the subject, I have had the chance to evaluate numerous books on the theory of computation.Of all the available texts, I think this one is the most appropriate for CS students.In the past I taught out of Dexter Kozen's book, which is incredibly elegant, but had some resistance from the students.Thinking it over I decided that Kozen's text, although beautiful, may be better suited to students pursuing a degree in pure math.Sipser's book, on the other hand, is more gentle.I find that Sipser demands far less mathematical maturity from his readers, and thus allows the difficulty to be shifted from excessive formalism to the inherent challenges present in the material.In addition, following Sipser's treatment, I was able to cover finite state machines and pushdown automata in far less time, thus allowing me to concentrate on computability and beyond.The book really shines in its treatment of computability theory, eloquently directing attention to some of the most beautiful aspects.
Excellent accessible textbook on the theory of computation
The theory of computation is the branch of computer science that deals with whether and how efficiently problems can be solved on a computer. The field is divided into two major branches: computability theory and complexity theory, but both branches deal with formal models of computation, and both of these subjects are dealt with in this book. This is an important subject because no matter what leaps forward computers make, something that is proved undecidable and not computable will always be so, thus the theory behind this subject is very important.
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Subjects:  1. Advanced    2. Computational complexity    3. Computers - General Information    4. Discrete Mathematics    5. General    6. Logic    7. Machine Learning    8. Machine theory    9. Mathematics    10. Science/Mathematics    11. Systems Analysis    12. Computers / Information Theory    13. Mathematical theory of computation   

Reviews (4)

Great intro to Real Analysis
Lay's book is my second favorite book on introductory level real analysis--please see my review of Elementary Analysis by Kenneth Ross.Ross somehow covers a lot more material, and quite a bit more clearly than Lay, BUT, the first two chapters of Lay's book covering axiomatic set theory are wonderfully written.I particularly love the clearly written coverage of the paradoxes contained in the Fraenkel-Zermalo Axioms, in Particular the Banach-Tarski paradox which states that a solid sphere of radius A < B can be cut an pasted into a sphere of radius B. Then, from chapter 3 onwards, Lay and Ross pretty much follow each other, Lay being more wordy than Ross, Ross being far more clear than Lay.
good to go
The book arrived in good condition and I have not had any problems with it.

Definitely a good first text
I bought this book because I have been looking for a Introductory analysis text that isn't too advanced, but yet doesn't gloss over the essential stuff, and I found it in Lay's book.For the self-studier, this book is excellent!I have several books on analysis: Shilov, Kolomogorov, Rosenlicht, Ross,etc...For the beginner, this book is superior to all of them.A plethora of examples.Also, a good range of problems:from straight forward problems requiring only the use of a definition to more advanced problems requiring a little thought.If you already have had some Analysis, then this book is probably not for you.But, if you are a student who wants to learn Analysis on your own, then this book would be hard to beat.After this book, one should be able to tackle "Papa Rudin".For according to Rudin, all that is needed to study his "Real and Complex Analysis" tome, is the first seven chapters of his "Principles of Mathematical Analysis".This book covers all that Rudin covers with the exception of Riemann-Stieltjes integration.On the whole, this is a great start!If proof-based math is new to you, then you will appreciate the first chapter on proofs.Would have given five stars, but I would have liked to seen Riemann-Stieltjes integration.That's really only nit picking, though. ... Read more

Subjects:  1. Functional Analysis    2. Logic    3. Mathematical And Symbolic Logic    4. Mathematical analysis    5. Mathematics    6. Proof theory    7. Science/Mathematics    8. Set Theory    9. Mathematical logic    10. Mathematics / Advanced    11. Mathematics and Science    12. Real analysis   

Reviews (23)

Rigorous but accesible..an engineer's intro to proofs
I write this review on the context of having done all the math required for mechanical engineer but never havin to do proofs... as B.Rusell once said do and the by faith you will believe... trying to escape that state of mind I got this book some years ago, Im glad for the author follows a structure approach similar to learning a programming language, once you master some elemenatry techniques you can stack up to create refined algorithms.

5 Stars for the book, 1 star for an unnecessary 2nd edition
I have the first edition of this excellent book, and I compared it to the recently published second edition. Quite frankly, I can't see why they bothered. It's not like the methods of mathematical proofs change that much over time. We're not talking about some 1400 page shovelware book on being a MCSE/CCNA/etc. where the qualifications change every six months. There are only three basic changes to this book in its second edition:
overlay logic on any other type of maths
The math notation in this book is that typically of texts used by maths majors. Most of those people should already have a grasp of proving statements. But some do not, and the book is directed to them.
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Subjects:  1. Combinatorics    2. General    3. Logic    4. Logic, Symbolic and mathematic    5. Logic, Symbolic and mathematical    6. Mathematics    7. Science    8. Science/Mathematics    9. Mathematical logic    10. Mathematics / Combinatorics   

Reviews (17)

Not ideal for self study
A primary group of people this book is aimed at is those preparing for Math contests such as the Olympiads. Many such people study the subject on their own. An important step in such preparation is solving a lot of problems. While it is important to try to solve the problems on one's own, it is equally important to be able to verify that one's solutions are correct. Unfortunately, this book does not provide solutions to the problems. Hence, it is not very helpful for those who are studying on their own. There are many other books in the market which are better from that point of view: For example, many books by Titu Andreescu, "Problem Solving Strategies" book by Arthur Engel are all good books that provide solutions as well. The "Art of Problem Solving" (Vol. 1 & Vol. 2) by Rusczyk et al. are also very good and have separate solution manuals available for purchase.

Outstanding, but you will need other sources
The Art and Craft of Problem Solving is an excellent book that covers the essentials of Algebra, Combinatorics, Number Theory, and even Calculus from a problem-solving point of view. However, there are very few solved problems. I strongly recommend this book, but also suggest that the student study other books as well, books with many more solved examples, because this book alone does not provide enough experience in putting the concepts into practice.

a marvelous introduction to problem solving
this is one of the rare books which truly
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Subjects:  1. Logic    2. Mathematics    3. Problem solving    4. Science/Mathematics    5. Mathematics / Logic   

Reviews (7)

Excellent, Accessible Book
This an excellent book.Basically, the whole point of it is to mathematically define what a computer is and prove that it works.The author does this by defining and manipulating mathematical alphabets and languages without resorting to any kind of advanced math.Starting from nothing, the whole thing leads up to Turing Machines.More specifically, according to the Preface, the goals of the book are:
Excellent
I must say this is one of the best books I have ever read.The auther is humorous and insightful.He manages to take very abstract concepts and explain them in clear concrete terms and metaphors.

Discursive presentation. Helpful for novices.
The book has one important attribute: it's clear, undoubtedly. Having a minimum of prerequisites, I think there's no way to not understand what Prof. Cohen says through its pages. It makes the job of learning this part of theory easier than any other text.Read more

Subjects:  1. Computer Books: General    2. Computer Science    3. Computers    4. Electronic digital computers    5. General    6. Logic    7. Mathematics    8. Theory Of Computing    9. Computers / General    10. General Theory of Computing   

Reviews (30)

Content left as an exercise
I had the "pleasure" of being exposed to this nightmare of a book in a bachelor level course. I am told that it is normal to use this book on masters-degree level, so maybe it's because I wasn't "prepared" enough for this book that my take on it is so negative. The book does have it's moments, where things are understandable, but thats mainly if the stuff is easy. There is a lack of explenations throughout the entire book. It seems the author(s) view of "explenation" is the words "it is easy to see..." or "left as an exercise"....The proof of my feelings toward this book, can be seen on the teeth-marks that decorate the books cover. Im not writing this, because I need to vent steam...I passed the course on the first try, and it is now behind me, but I advice anyone faced with this book, to seek alternatives..this book is not a teaching book, it is a telling book.
A good textbook
I taught a couple of classes from the first edition of this textbook, and my students did fairly well.On the whole, they were able to understand the material and solve the homework problems.I certainly wouldn't mind teaching a class on this subject from the second edition as well, which I feel is a mild improvement over the first one.
First and foremost, a math book
I enjoyed this book because I enjoy formal mathematics.This is not an applications book, but a formal study of the mathematics that underly algorithmic design and analysis.I'm no math wizard, and I found this book readable (but I had to take it very slowly).The course for which I bought the book only covered chapters 1 - 4 and glossed over the final 3 chapters, but I intend to read the rest over the summer between semesters because it's so well and thoroughly written.This book is *dense*.I had to re-read everything three times before I absorbed it all, but ultimately I've understood everything I've read.The hardest parts to understand were the formal "proofs by induction" on the lengths of strings and sets - and, as any math student knows, you can gloss over the proofs on the first reading.A lot of the formal definitions (finite automata, pushdown automata, Turing machines, context-free grammars, etc.) baffled me on the first reading, but after reviewing the examples and working through a few problems, I could go back, re-read the formal definition and understand it.
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Subjects:  1. Computational complexity    2. Computer Books: General    3. Computer Mathematics    4. Computer Science    5. Computers    6. Computers - General Information    7. Discrete Mathematics    8. Formal languages    9. Logic    10. Logic, Symbolic and mathematic    11. Machine theory    12. Computers / Programming / Algorithms    13. Mathematical theory of computation    14. Probability & statistics   

Reviews (3)

Perhaps the best written written elementary book of logic
I bought the book just because my teacher of elementary philosophy in the university respected Tarski as a master of formal logic. It took me 26 years to get this book in my hands. What makes Tarski unique is, that he was a great logician and a great teacher, too.
TIMELESS CORE HOLDING IN ANY LOGIC LIBRARY
This timeless classic by one of the five greatest logicians of all time should be owned by anyone who cares about logic - especially at this illogically low price.The Greek philosopher Aristotle (384-322 BCE), the English mathematician George Boole (1815-1864), the German mathematician Gottlob Frege (1848-1925), the Austrian-American mathematician Kurt G�del and the Polish mathematician Alfred Tarski (1901-1983) are considered to be the five greatest logicians of history.Today it is difficult to appreciate the astounding permanence of what is accomplished in the works of Aristotle, Boole, and Frege without seeing their ideas surviving in the work of a modern master.Of the two modern master logicians Tarski is by far the most suitable for this purpose since he was by far the one most interested in the articulation of the conceptual basis of logic, he was by far the one most interested in history and philosophy of logic, and he was the only one to write an introductory book attempting to explain his perspective in accessible terms. This book, together with Aristotle's Prior Analytics and Boole's Laws of Thought, should form the core of any logic library. All three are still in print and available in inexpensive paperback editions.Hackett publishes an excellent up-to-date translation of Prior Analytics by Robin Smith and Prometheus recently reprinted Laws of Thought with an introduction by John Corcoran.- Frango Nabrasa.

I will always keep it as a reference
This is one of the classic introductory mathematics books. When I was learning logic, I relied on it heavily, although it was not the text for the course. Over my years as a teacher, I have consulted it often and when I was working on a recent book on logic, there were very few days when I did not open it in search of an idea or clarification. Read more

Subjects:  1. Arithmetic    2. Foundations    3. General    4. Logic    5. Mathematics    6. Philosophy    7. Science/Mathematics    8. Mathematical logic    9. Science / General   

Reviews (22)

A book that teaches Math and thinking at once
Polya struck gold with this book! "How To Solve It" contains a simple, 5 step method for solving problems that's applicable in multitudes of disciplines. While the emphasis of this book is on story problems; Polya's method for problem solving is useful in areas such as computer programming, automotive troubleshooting, electronics repair, heating and cooling services, research writing, and much more.
Excellent and useful book
A really helpful book that goes way beyond math. In our hurry to get through algebra we almost completely sidestep problem solving in teaching. It's true that we have `story problems' but they're usually just applications that restate the chapter's contents. That's why so many people get to a class like physical chem in college and flounder: they've never really been taught how to solve a problem only to regurgitate what's been read. This book by Polya fills in that gap.
A delightful and satisfying classic
Are you like a dog with a bone when you're working on a brain teaser? After pages of scribbles, do you get a big grin on your face when you turn to the answers and say: "I'm right!" Then this book is for you.
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Subjects:  1. Advanced    2. Applied    3. Logic    4. Mathematics    5. Science/Mathematics    6. Mathematics / Logic   

Reviews (8)

Numero Uno
The Scandanavian scholarship of Torkel Franzen has brought light where there should already be light in lieu of Godel's theorem itself.I recall some odd-ball who tried to use Godel's theorem as an argument against the Apostle Paul.Dr. Franzen clarifies the meaning and purpose of Godel's theorem so that people won't use it to defend their pet ideology.It is a theorem that has been much abused.Indeed, I would argue that Kenneth Arrow's use of the theorem has caused many to take liberties they shouldn't.Kenneth Arrow, of course, knew what he was doing.However, when philosophers or professional people or even atheistic mathematicians with no background in mathematical logic think that their capacity to understand the general import of the theorem try and venture beyond it and apply it to domains incongruous with it, Dr. Franzen's labor in this short but important book is greatly needed.If you want an intellectual grasp of Godel's theorem, then this book is for you.In a sense, people cannot be blamed for their foolish conduct since hindsight is 20/20.Torkel Franzen put that hindsight in stone.

Valuable Counterbalance to Widespread Misconceptions and Nonsense in Print and on the Internet
Torkel Franzen has created an immensely valuable, deeply fascinating examination of misunderstandings, misconceptions, and outright abuse of Godel's theorems found in print as well as on the Internet. He does so in a cogent, non-confrontational style that makes enjoyable reading. Godel's Theorem - An Incomplete Guide to Its Use and Abuse warrants five stars.
Franzen's guide incomplete but consistently good nonetheless
For certain numbering systems, one must content oneself with either being consistent but incomplete or complete but inconsistent.
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Subjects:  1. Gèodel's theorem    2. History & Philosophy    3. Incompleteness theorems    4. Logic    5. Mathematics    6. Science/Mathematics    7. Mathematical logic   

Reviews (4)

Mathematical Proofs
This book is concise, clearly written and easy to understand. However, I would have liked to have seen more examples of new concepts.

Good book for what it is
I am currently using this book for a class right now. I find this book easy to read and work with, but wish there was more examples explaining each kind of proof. Although there are quite a few examples to the proofs, I just need a few more to work with. I am going to purchase another proof writing book to supplement this one, as this is my first dealings with proofs. All in all though, this is a good book.

Mathematical Proofs:A Transition to Advanced Mathematics
I purchased this book as a preparation to take advanced courses in mathematics.It is well laid out and explains the material clearly.I especially liked the chapter on "proofs found in calculus" which explains delta epsilon proofs in a manner that is easy and understandable.Big thumbs up! ... Read more

Subjects:  1. Logic    2. Mathematical And Symbolic Logic    3. Mathematics    4. Philosophy Of Mathematics    5. Proof theory    6. Science/Mathematics    7. Set Theory    8. Mathematics / Advanced   

Editorial Review

Twenty years after it topped the bestseller charts, Douglas R. Hofstadter's Read more

Reviews (222)

AI's hardest problem...
Whenever I read a book in translation I often wonder if it's the translator I love or hate, or the author themselves.Literature in translation is essentially a complete re-writing of the book itself.Consider Lewis Carroll:How does one go about translating seemingly endless word play and words that are often made up?Now consider the fact that Alice in Wonderland had been translated into over 50 languages.If you could read all 50 languages, some would approximate your understanding of "Alice", and some would appear tangenital, or worse, dishonest to the original.
Transfiguration in Print
I read this book in 1980, and each chapter of it still stands out in my mind in glittering bas-relief against all I read before or have read since. What stands out the most, however, is that my thinking changed after reading this book. Everything since reading it has been inevitably processed through its filter, which has quite simply changed my life. Perhaps a first-reading today would not have an equal effect; in 1980 it was timely. Things we take for granted now were difficult to conceptualize then--the internet, for instance. But I'll leave it for contemporary readers to decide whether Gödel, Escher, Bach: An Eternal Golden Braid is as eternal as the title claims. For me, the answer is a resounding yes. I can say without doubt that my life would have been different if I had not read it at that time.
Clever (occasionally to the exclusion of accuracy)
This book is a great way to expose yourself to new ideas, but, in the words of a previous reviewer, it was written by a passionate dilettante.
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Subjects:  1. 1685-1750    2. Artificial Intelligence    3. Artificial Intelligence - General    4. Bach, Johann Sebastian,    5. General    6. Logic    7. Metamathematics    8. Philosophy    9. Speculative Philosophy    10. Symmetry    11. Cognition & cognitive psychology    12. Philosophy of science   

Reviews (14)

The ideal guide for making and breaking arguments
So many significant decisions are made in the workplace based on majority opinion or by the highest ranking person in the room.
Even the best of us sometimes disappoint, ironically, and otherwise
As you can read from the excellent reviews of the fourth Edition of Attacking Faulty Reasoning (which anyone contemplating buying this volume should), Damer's "positive theory of argument" represents a huge breakthrough in informal logic.To my knowledge, Damer was the first to put foward a coherent schema of the fallacies of informal, organizing them as violations of four comprehensive rubrics, which stand as criteria for effective argument: 1) relevance 2) acceptability 3) sufficiency 4) resiliency (actually a change which Damer might have contemplated making in this new edition) - he uses 'rebuttal' - which, I suppose would have to be 'rebuttal-ness' - to stay in adjectival mode (appropriate to the idea of finding suitable and exhaustive 'criteria' for argument).Still, the term 'resiliency' more adequately than 'rebuttal' describes the criterion he is delineating in his theory.
Invaluble for everyday life, not just the classroom...
Although I gathered that this book was written primarily as a text for logic and critical thinking courses, it helped me immensely in terms of learning how to make sense of argumentative quality in everyday life, not only the classroom.I did not read this book for a class as I have graduated college, but I found it to be a real help in determining the strengths and weaknesses in arguments and other forms of persuasive speech that we encounter daily.
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Subjects:  1. Fallacies (Logic)    2. General    3. Logic    4. Philosophy    5. Reasoning    6. Business ethics    7. Business negotiation    8. Mathematics / Logic   

Reviews (1)

Interesting but very uneven
The stated goal of the book is to have no prerequisites other than freshman math, but you really need at least undergrad logic and set theory. It is a thick book and covers a lot of interesting stuff, but there can be a frustrating lack of order and detail. He is an engaging writer, but sloppy in that he gives only part of the information you need to understand something, forgetting that he didn't give it or you don't know it. Sections are often either too sketchy or wordy and unfocused. Sometimes, though, he comes through with an elegant explanation of something. He is at his best when relating recursion theory to science, philosophy, and other branches of math. These discussions tend to be rather handwavy, but he gives references to the literature. The book as a whole is not very unified and it doesn't clearly indentify and relate the central ideas of recursion theory, so it wouldn't make a good introductory text. But it is a good reference for those with a moderate background. ... Read more

Subjects:  1. Information Theory    2. Logic    3. Mathematics    4. Recursion theory    5. Science/Mathematics    6. Computers / Information Theory    7. Mathematical logic   

Reviews (8)

Moderately difficult and very effective
This is the most clear book on intermediate level logic that is available.I have many of the logic books that are on its level, and this one is per