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Can you talk about how this new edition differs from the old edition?
Sure, no problem. I don't own the old edition, but I have heard enough bad comments that I feel I can comment on the new one.
I have a copy of the first edition of this book, and I've always thought it was one of the worst computer science texts I'd ever read.
This has profoundly changed, I'd say. The new book comes nowhere close to being "one of the worst". The topic of the book will not appeal to all people, but the coverage of the material is clear and easy to understand. Please note again that I have no formal background in mathematics (I originally have studied psychology and brain research, and statistics is pretty much the only part of higher math I feel somehow educated about). Despite this, I was able to cover a lot of ground in short time, due to the excellent introductory sections in the book.
I found it turgid and confusing. The arguments are all overformalized, with excessive notation that obscures what's really going on.
That has definitively changed. The authors also note in the introduction that they did change this for a number of reasons:
You say that the formal proofs are preceded by informal sketches, and I hope that's true, because they sure didn't do it in the first edition.
Oh yes, it's true. Just take a look at the book in your nearest library.
For example, I know from experience that many people are deeply confused by the first edition's explanation of the pumping lemma for regular languages. But I also know that there's nothing hard about the pumping lemma, because I've been able to teach high school students about it in half an hour. This isn't a boast, because I think anyone could do the same. But the first edition of this book didn't do it.
The pumping lemma took me about half an hour to cover, mostly because there is a missing assumption in the proof (the language must have no bound on the length of its members, meaning it is infinite, for the pumping lemma to work). I found the explanation in the book very clear and easy.
The first edition's treatment of NP-completeness is similarly turgid.
Sorry, I can't really comment on this, since I have not yet covered this part of the book. I am still with Turing machines and undecidability. Maybe next week :-) ...
You also said that "The style of the book is vey application-oriented." That would be a welcome change from the first edition, but I wish you had given an example, because I'm skeptical.
Take a look at sections 2.1 (using a finite automaton to validate a simple e-commerce protocol), 3.3 and 3.4 (UNIX Regexes and Algebraic Laws for Regular Expressions) and 5.3 (YACC and XML as applications of context free grammars). That pretty much explains what I mean. OK, maybe I should have said "The book is written with an eye towards possible applications of the theoretical material covered." and not "very application oriented". The latter may convey a wrong message.
Anyway, I'm glad you liked the new edition, and if what you say is true, I guess the authors have learned something since 1979, when the first edition appeared.
Indeed I think they have.
But I wish you had been able to compare the new edition with the first edition, and I'd be reluctant to buy the new version without taking a very close look over some of the material to make sure it really had been improved.
I'd do so, too, if I were you. It is hard to believe a book can change so much. OTOH, I *really* like this text. It opened up the world of a big part of "classic" formal computer science for me. That is no little thing. Maybe I am just too excited about the book, but I still think it is definitively worth a look on the next trip to the library or bookstore.