Complexity Books
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Beautiful and Reflective BookReview Date: 2008-04-08
The Tao of Switchbacks Review Date: 2008-03-21
book as vacation breakReview Date: 2008-03-11
- Kare, movingfrommetowe.com
A Wonderful Guide to LifeReview Date: 2008-02-11
I can say that the book was not at all what I thought it was going to be. BUT...I am very, very happy with my purchase. I call my experience with this book my happy, little accident. It came to me at a time where I really needed to have simple and clear messages to help me interpret some situations in my life. I would recommend this book to anyone who is looking for some insight into themselves and the world around them.
A remarkable 114-page compendiumReview Date: 2008-02-05

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Definitely a classic but not good for beginnersReview Date: 2007-10-24
comprehensive book for NP-completenessReview Date: 2007-09-21
Published in 1979 and still the bestReview Date: 2007-06-16
Arrived in time, good conditionReview Date: 2006-02-24
A Beautiful Book on a Beautiful SubjectReview Date: 2005-12-10

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Unexpectedly relevantReview Date: 2008-06-15
Becoming a Resilient OrganizationReview Date: 2008-01-07
Good luck!Review Date: 2007-12-28
The main point could be explained in a single sentence: We can get valuable lessons if we pay attention to organizations who work in high risk and unpredictable environments.
This is my own view and, actually, I tried to show this using aviation as a kind of learning field. That is why I hope the authors will be lucky. My own experience was unsuccessful and that itself shows that the authors are right.
When I started to get conclusions from aviation to business management, I found that the more interested people came precisely from aviation. I'm afraid the authors could suffer the same experience and people interested in their concepts could come from air carriers, nuclear-powered plants and some other examples they use.
The authors could be three or four steps in advance of the present situation in business management. They try to extract the right lessons from other fields. However, they would not be surprised if their intent "bounces back" and it is picked-up precisely from the fields that they try to show as examples, not from business management.
Recipe for a Learning Organization Review Date: 2007-11-06
1. Preoccupation with failure - treating any failure (often small ones) as a symptom that something is wrong with the system, they are continually updating their understanding.
2. Reluctance to simplify interpretations - ensuring a more complete and nuanced picture, simplifying less and seeing more.
3. Sensitivity to operations - paying attention to relationships at the front line, where the work gets done.
4. Commitment to resilience - maintaining a deep knowledge of the technology, the system, one's coworkers, and one's self as avenues for improvising and keeping the system functioning.
5. Deference to expertise - cultivating diversity to do more with complexities, they push decisions down to the people with the most expertise, not the most rank. They also move issues around/across the system, migrating problems to someone with the knowledge and capabilities to address them.
Together, these elements give the organization `mindfulness', and this organizational mindset allows it to handle the unexpected with more responsibility and thus a higher probability of success in the face of change. Although the HRO's analyzed (aircraft carriers, nuclear power plants, and others) operate in more dangerous environments than the average business, today's rapidity of change causes the unexpected to happen to every organization and it would seem that the five elements of mindfulness could benefit nearly every organization today.
Dennis DeWilde, author of "The Performance Connection"
Unexpectedly a good readReview Date: 2007-01-18

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Groundbreaking Part II...Review Date: 2002-05-15
This is truly paradigm-shifting, moreso than anything else you are likely to read about in science. The Sante Fe crowd such as Stuart Kauffman obviously did not even grasp what Rosen was talking about when they met back in 1994 and that is even more tragic. So much time has been wasted with such money-wasters like the genome mapping fiasco when it could have been going into exploring new axioms for science.
For you see, this is what Rosen so eloquently points out in his work: the present axioms of science are much too limiting to explain anything we really would like to know about the universe. It is very interesting to see that Rosen grasped the implications of what also caught Einstein and Schrodinger's attention: the problem of inertial and gravitational mass. Rosen also points out the myriad of other areas where science has been busy putting band-aid after band-aid on the present set of theories to try to make them predict real phenomena.
For this is the problem with the present-day paradigms: they are only useful for predicting the N+1 state for some dead (and therefore uninteresting) mechanistic universe. The evidence has been staring us in the face for quite a while and I am not sure why Rosen should have been the first to analyze where the problems lie; it is even more surprising why his work appears to be so little known.
I also like the fact that this book is much more polished than his previous work. The index is mostly complete and there is also a list of references. I didn't note very many editorial erros and the language is quite friendly. This is a very high-quality science book and I suspect the first editions will be going for large prices in about 20 years when the "establishment" finally figures out where they went wrong.
Buy this and read it. And read it again. Then wonder why we are rushing pell-mell to "engineer" the world when we don't understand it at all.
Profound.....Utterly ProfoundReview Date: 2002-11-15
Rosen was a very insightful and technically capable theoretical biologist. His work - first as a student of physicist and theoretical biologist Nicholas Rashevsky, and later as professor emeritus at Dalhousie - is unquestionably of the level of importance of Einstein's Special/General Theory of Relativity, or Godel's Incompleteness Theorems. This is a grand claim to make, but once you read Rosen's work, you will see for yourself.
These are not the easiest books to read, despite Rosen's excellent writing skills. The difficulty is two-fold. First and foremost, the new concepts and paradigms presented are of such breadth and profundity that it can take several readings to begin to fully grasp them adequately. Secondly, Rosen is mathematically (and otherwise) quite astute. The reader will encounter to some degree: category theory, topology, catastrophe theory (Rosen dedicates a chapter on genericity in _Essays_ to Rene Thom), differential equations, dynamical systems, Godel, Church-Turing, as well as philosophical topics of epistemology, ontology, and foundations of biology, mathematics and physics.
This should not, however, deter even the non-professional. Particularly in _Life Itself_, Rosen progresses carefully and patiently, even including a short intro to Category Theory. One can gloss over some of the math and still garner most of the insights from the text alone. _Essays_ utilizes a wider range of math skills, since that book covers a broader range of topics, but it is still quite accessible to the careful and astute reader.
In _Life Itself_, Rosen was investigating the question posed by Erwin Shrodinger originally in his 1943 lecture "What is Life?". Rosen's search led him to peel back in careful detail the foundations of Newtonian mechanics and reveal the underlying tacit assumptions of a state/phase-based physics and the repercussions for science in general, and biology in particular.
By setting aside state/phase-based physics, Rosen then proceeded to layout the groundwork for an atemporal relational biology based on functional organization and to methodically investigate the theoretical limits of mechanistic systems, including along the way: simulation, Turing machines, and the epistemology and ontology of such systems. The distinction eventually becomes clear that any such algorithmic mechanisms cannot embody the kinds of impredicative complexity that are characteristic of an organism. Because the syntax of Newtonian physics can express no such closed loops of entailment, "life" cannot even be described in that model of physics, much less modeled in any complete way. Thus it is that biological organisms are not a mere subset of current physics, but are representative of complexities that require physics to be enlarged.
In _Essays on Life Itself_, Rosen uses his considerable abilities across a broad spectrum of topics to continue the ideas from _Life Itself_. It is difficult to describe how topics as diverse as the assumptions of Pythagoras, the Turing test, universal unfoldings, morphogenesis, mind-brain problem, and more can be in the same book. Mostly, they all in one way or another accomplish one task: to look beyond the limits of how a problem is currently being viewed, and to see it from a larger perspective. Often, these perspectives take Rosen into terrain others would avoid, since they sometimes lead into the non-algorithmic / noncomputable, or the breakdown of the presumed subject-object division, or other kinds of "messy" scenarios.
Often they lead into "complex systems", where Rosen uses the word "complex" to define a certain class of systems - those systems have symptoms of being: impredicative, non-algorithmic, context-dependent, semantic, nonformalizable. This classification is not a desire for obfuscation or ineffability, but is as rigorous as the nonformalizability of Number Theory or the unsolvability in closed form of the n-body problem. It is a complexity akin to the size of a transfinite number: it is not simply a matter of merely being hugely complicated, it is rather an entirely different order of system structure.
However, guided by Rosen, one does not feel uneasy following his path. Rather one feels enriched both in knowledge and in paradigm. Distinguishing the broader generic case from the degenerate or special is a characteristic theme in Rosen. The unfamiliar terrain he argues to is thus not some void, but a grander scale that subsumes the orthodox view.
In that grander view, it may become more clear that some problems are based on incorrect assumptions, while some are more difficult or complex than in the more limited original view. However, it is apparent that Rosen is uninterested in making problems appear simpler by ignoring those difficulties - he is interested in where the science leads. It is an immensely richer, complex view of the physical world that one comes away with. As such, it presents some difficult challanges, but it also opens up vast opportunities - opportunities not visible in the neat and tidy fantasy model of science that generally prevails where it is assumed that with enough effort everything can be reduced or calculated.
Rosen writes deliberately and with precision, and is both a critical and a profound thinker. I hope that he one day receives the recognition and admiration he rightfully deserves.
Powerful critique, but ...Review Date: 2004-04-01
More than once he mentiones Goedel Theorem, as well as various paradoxes, encountered by science over the centuries, emphasizing the fact, that they all are directly related to the impossibility to draw definite border between an observer and her object (not just in quantum physics).
Although the book was very interesting for me, i felt that some essays essentially repeated the material, already covered in other parts of the same book. Also, this "new science", which Rosen thought is needed to deal with open systems, is never really described in any way, so we are left with critique only.
I am not sure i fully agree with Rosen's view of the Turing Test, which he only sees as a simulation approach to the mind (intelligence) problem. My understanding is that Turing Test should be rather understood in the "observer/object" context, meaning that the participant makes a judgement, being, at the same time, fully incorporated into the system.
In one of the essays Rosen says: "If somebody wants to call this 'vitalism' - then ... so be it." With no constructive theory in site it's a bit like this, to my understanding.
A wonderful collection of essaysReview Date: 2001-05-26
My main contribution to what has already been said in other reviews is to note that this work might be best viewed as a complement to Rosen's earlier work, Life Itself. Or, said differently, it might be best to read Life Itself first (if you haven't already). These are very complex topics that are explained from the standpoint of biology and mathematics and those without a previous foundation in Rosen's work--as I was when I bought this book--may find they have to establish that foundation first.
Answers: Why is the whole is more than the sum of its partsReview Date: 2002-03-13
Bob was an eloquent speaker and reading this set of essays is almost as good as hearing him in person. The essays were written to be published in a number of places, usually as invited talks, yet they may as well have been set down to be a book from the start. There is a thread of continuity that makes this the case. In addition, even though I had read many of the essays as they appeared earlier, their juxtaposition in this volume proves that "the whole is more than the sum of its parts"!
His stated purpose of this collection is to, in a sense, "flesh out" arguments in Life Itself (LI) that had to be short or even omitted for what might be called "logistic" reasons. In my opinion the essays do that at least. In LI he began with a caveat with which I am totally sympathetic. He warned the reader that he was weaving a very intricate cloth with a single linear thread and therefore much was being laid upon the reader's shoulders. My own experience is that it took numerous readings to begin to see how the weave was manifest. Once there, things fell into place more and more quickly, yet still a lot more was required because the design is so highly interconnected and rich in levels of meaning. I hope this book of essays will spare others that struggle. It will never be my place to evaluate that possibility since I can never go back.
The first part deals with the relationship of biology and physics within science, which can sound like an innocent enough topic until one understands that it is a revolutionary view.
Underlying it all is the common notion that physics is the source of all scientific laws and that chemistry and biology somehow must utilize physics to be scientific. Rosen rejects this notion and thereby opens a Pandora's Box. He uses the now more than fifty year old essay by Schrödinger, What is Life? as a springboard to the revealing argument about biology's more generic character in comparison to physics. As he does this he develops his notion of complexity as a description of this more generic view promoted by biology in contrast to the kind of "simple systems" which are the subject matter of physics. None of this should sound new to anyone who has read his earlier work, especially Life Itself, except for the new connections and new depths to which the arguments are taken. The result is a more solid whole than ever before
His introduction to this part of the book is worth having here to get a flavor for where he is going: "I claim that Gödelian noncomputability results are a symptom, arising within mathematics itself, indicating that we are trying to solve problems in too limited a universe of discourse." This is a nice capsule version of Rosen's message. If nothing else comes from his writings, this alone should change everyone who understands the message.
The book develops this theme along with the idea that science has limited itself unnecessarily. It created a surrogate world and then insisted that any observations about the real world not compatible with this model were "unscientific". The consequences are many and he explores them systematically. Whether you agree or disagree, an honest reading will require you to re-examine your beliefs.

The only one of its kind....Review Date: 2001-09-23
Biggest return for the biggest investmentReview Date: 2005-05-07
But the payback!! I've gotten more return on investment from this book than from any other book I've ever read. If you dilligently read and master this book, you will be able to analyze and solve problems your collegues just can't.
The basic idea behind Kolmogorov complexity is straighforward: a good measure of the complexity of an object is the length of the shortest computer program which will construct that object. From this basic idea an amazing variety of insights and powerful techniques have been developed, and this book is quite comprehensive in cataloging and explaining them.
For computer scientists and working programmers, probably the most useful result of Kolmogorov complexity would be the "Incompressibility Method", which is a powerful technique for the analysis of the runtime of algorithms. Typically, it is relatively easy to figure out what the best case or the worst case runtime of an algorithm is. Until now, it was hard to calculate the average runtime of an algorithm, because it usually involved a tricky counting problem, to enumerate all possible runs of the the algorithm and summing over them. The incompressibility method eliminates the need for doing these complicated enumerations, by letting you perform the analysis on a single run of the algorithm which is guarunteed to be representative of the average runtime of the algorithm. If you program for a living like I do, this will give you an edge, because if you can accurately predict that the worst-case runtimes almost never happen, you can usually simplify and streamline your programs by optimizing it for the average case. If your competitors are wasting time optimizing for a worst case which almost never happens--at the expense of _not_ optimizing for the average case, you win bigtime.
For philosophers of science and AI/knowledge representation folks, the most useful results of Kolmogorov complexity are probably the contributions of Kolmogorov complexity to Baysianism. To be a Baysian is to follow a two step process: (STEP 1) for every possible sentence, assign to it a number between 0 and 1 which represents how certain you are that that sentence is true. This initial assignment should be a probability distribution over all possible sentences. It should be a "good" probability distrubution, but of course it won't be perfect, since you don't know everything. (STEP 2) when confronted with new evidence, e.g. an observation, update your current "good" degrees of belief by using Bayes' law, to yield a new "better" set of degrees of belief.
The Baysians always had a good story for Step 2--just use Bayes law. But until now, they were mostly hand-waving on Step 1--what would constitude a "good" initial probability distribution? There were many proposals (e.g. maximum entropy) but all proposals had benefits and drawbacks. What Kolmogorov complexity provides is the so-called "universal" distribution, which is guarunteed to be a "good" initial distirbution. This book devotes much time to explaining and exploring this, and shows how previous techniques, like maximum entropy, minimum description length, etc all can be seen as computable approximations to the (unfortunately uncomputable) universal distribution. This really gives a nice framework for evalutating and formulating good prior distributions.
After remarking on how hard this book was to read, I should emphasize that this is not due to bad writing on the part of the authors! Indeed, after throwing the book across the room, I was always drawn back by Li & Vitanyi's most engaging writing style to pick the book back up, dust it off, and have another go at it. If it were not for their wonderul ability to expain a very complicated subject matter, I never would have gotten through it.
An unsung hero of this book is Peter Gacs, who wrote a set of lecture notes which really could be considered to be an Urtext for this book. If you tackle this book, I highly recommend that you also get ahold of these notes, because it is sometimes very useful, when trying to puzzle out a difficult argument, to get another description/explaination of it from a different point of view. These notes are available on the web, just google for "Lecture note on descriptional complexity and randomness" by Peter Gacs.
If you're up to the challange, then buy this book, dilligently read it, swear at it--then swear by it.
Comprehensive and ExcellentReview Date: 1999-07-30
A mustReview Date: 2003-10-29
The authors are careful to place the development of the theory in its historical context, give a face to the main players in the field and explore frictions with other lines of thought. But the main storyline is the mathematical world of Kolmogorov complexity. Neccessary background knowledge is provided, most proofs are given and the open problems are presented. Most chapters are more or less self sufficient, making it possible to skip those that are of less relevance to you. In the later chapters much thought is given to the different fields of application.
A third edition is in the making which will include recent advances. But since the authors make new discoveries available on the web, the present edition will continue for a long time to hold a prominent place in the book shelves of many computer scientist.
Excellent if you have the math...Review Date: 2002-08-13
The reviews below give more than enough information so I won't belabour the Kolmogorov complexity here. Suffice it to say you won't find the subject detailed more fully in any other reference work in existence today.
However, this book does need to be revised and updated. There has been a lot of development in the field and the sections overviewing Solomonoff's work, in particular, could be expanded. Also, I found it hard to believe that nothing about the 'philosophical' importance of the whole induction question - this is at the core of many very important questions and should not be treated trivially.
There should also be some overview of two other areas that, in combination with the theory outlined in this text, are starting to form the nexus of a "new kind of science" (definitely not Wolfram's pathetic attempt). I refer to some information regarding non-classical logical systems as well as anticipatory computing systems. Both will, I predict, become core areas in addition to extensions to Kolmogorov/Chaitin complexity in the future.
All textbooks should be as clear and concise as this example.

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Brilliant but incredibly obtuseReview Date: 2008-01-29
You're Smarter Than You Think You AreReview Date: 2006-03-30
I read this book in a Bantam mass market edition after sampling a piece of it in some science magazine (maybe Discover). Gregory Bateson was a renaissance man (which is one of the delights in reading him), the former husband of anthropologist Margaret Mead, and best known for the double bind theory of schitzophrenia, included as an essay in The Ecology of Mind. That theory may not sound well-known at all, but it's the basis of family counseling and why we talk about dysfunctional families (instead of just individuals). And we've all been in situations that are double binds, or as these no-win situations are known in everyday jargon: "damned if you do, damned if you don't".
Bateson wrote this book as metafiction, which is to say he talks about the book in the book, and he includes a handful or metalogues with his daughter, Catherine Bateson, herself now a writer for such magazines as Smithsonian, although he made them up. These metalogues reflect on ideas in the book and widen the feedback loop, as it were, to include the reader. They are relaxed and leisurely and not meant to be persuasive.
My experience reading this book was that it changed the way I saw everything. That sounds like an over-reaching claim or a self-help book gone wild, but the reason is, as Bateson points out, that many of our educations are simply based on gathering information, like Number Five in the film Short Circuit, with no help at all on how to think about it.
I certainly didn't understand everything in this book. But then, if you already understand and agree with everything in a book, why read it? What I did glean was a few tidbits from an engaging and knowledgeable prof who gave me not just more to think about but ways to think about it, and the happy realization that we're all smarter than we think we are.
The most important book on epistemology there isReview Date: 2005-08-18
This book is one of his most important. It is a testament of his view of science and coming from a person who helped revolutionize more scientific fields than the average person has even heard of it should be taken seriously. In its pages Bateson tells us what science is and how it should be properly exercised. Given the confusion and nihilism that have followed on the pseudoscientific revolutions of postmodernism and decostructivism (read Focault, Derida or Judith Butler for instance) such readings are necessary if at times disturbing. Not all ways of doing science are equal and many of them are based on logical confusion. Bateson is clear on that point. On page 24 he tells us "Some tools of thought are so blunt that they are almost useless". Self-evident to most people this maxim needs to be restated and taken seriously, especially within the social sciences that have only succeeded in making minor steps since the time of Aristotle. In this book we learn the why of this unfortunate situation. The question is if anybody wants to listen...
Still Bateson is not in any way preaching like some untouchable headmaster, unlike many other philosophers of his rank (read Jerry Fodor for instance). He is aware of the difficulties and obstacles involved and most of the time keeps his voice low. He also is not a techno-freak like many of the newest cognitive scientists, modern rationalists or evolutionary psychologists though he is one of their intellectual fathers. Instead he often talks of the need of a holistic approach, of looking out for the pattern which connects mind to nature and nature to the universe, and warns against the dangers of degrading the ecosystem and turning our backs to the fellow living creatures of this, still wonderful, planet.
If you only read one book on the history of science or on epistemology make this one your choice. You wont regret it. It is a cybernetically quided misile which will hit you on the head, and change you forever. To the better that is.
That reminds me of a story...Review Date: 2007-07-16
"Mind and Nature" is both Gregory Bateson's most accessible and most difficult book. It is a deeply personal exploration of what has come to be called cognitive science from a brilliant man and great scientist who pioneered a deep synthesis of anthropology, language and communications, and biology over the course of a remarkable life. Be advised that it is more of a progress report on a lifelong quest than a coherent whole. If you have an enduring interest in cognitive science and you haven't read Bateson, you don't know what you are missing.
Bateson's starting point is, "How is it possible for the same evolutionary forces that shaped our survival as a species failed to shape our minds?" The answer, of course, is that it is not. It ought to be self-evident that the phenomenon that we call the "mind" is shaped by natural selection. Bateson does not claim to understand all the implications of this empiricist stance, his focus instead is on how to start asking the right questions about the mind and cognition. For instance: What is learning? What is play? (Is it true that only mammals play? Why is that?) If you think about it, these are phenomenon central to the human experience and there is no one that discussed them more insightfully than Bateson does here (and in "Steps...".
I find myself returning to this book again and again over the years. Its effect on me has been profound. I am sure I will never understand more than a small part of what Bateson is trying to tell me here, but the feeble fraction that I do understand is remarkable. The wisdom that animates this book has shaped many of the foundational notions of my life. It is full of life lessons.
And that reminds me of a story about the time I incorporated one of Bateson's teaching parables from this book into a speech I had to give not too long ago....
Inspiration Beyond Imagination!Review Date: 2006-08-17

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Required reading material, definitely!!Review Date: 2004-09-10
Keep It Simple Whenever and Wherever PossibleReview Date: 2002-01-19
Required reading material, definitely!!Review Date: 2001-07-18
Clear and simple, the best adviceReview Date: 2000-10-28
It is as much a thesis for life as it is for the way you run your marketing and sales worlds. Having read the book twice, once for practical and once for pleasure the book is an excellent combination of examples, ripe for metaphor and theory.
As a head of strategy for a leading e services company this book was well worth my time and the time of all our senior managers. The four "R's" are the best way to simplify a horribly complicated world.
If I could give six stars I would
Innovative MarketingReview Date: 2001-01-23
Finally, marketing professionals who will intelligently embrace and (with ease) outline "repackaging" and "replenishing" strategies. These sections made the book worth the price for me.
The only drawback(s) with the book (though I gave it a full 5 stars), are the case studies. Too many of the online grocers are performing poorly in the marketplace to rely on them as standards, whether or not their marketing strategies are exciting and workable.

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O. BlochReview Date: 2007-09-06
instructiveReview Date: 2007-09-05
An excellent introduction for astrophysicists to a wide range of topics in nonlinear dynamics and chaos theoryReview Date: 2007-09-12
Regev's Chaos and Complexity in AstrophysicsReview Date: 2007-10-03
"Very readable, engaging and understandable. Regev offers insights into and explanations of complex concepts and methods by using frameworks familiar to students (particularly of astronomy) on the graduate level. I found that there are a number of typos and misprints. However, the author offers a useful, detailed and up-to-date errata that can be accessed on his site (http://physics.technion.ac.il/~regev/erratum.pdf)."
-Timothy Purcell
An excellent text on nonlinear dynamics in strophysics in particular, but on nonlinear dynamics in generalReview Date: 2007-09-05
However, I was deeply impressed by this book. It is very clearly written, very thorough yet tractable, and at times witty. Most importantly, some issues I consider subtle, tricky or often overlooked or addressed head-on here, meticulously , scientifically soundly and, astonishingly, in a manner I think is suitable for even first or second year graduate students.
In short, I STRONGLY recommend this book to anybody interested in nonlinear dynamics, in the heavens or on earth...
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Pure mathematical view of Computability and ComplexityReview Date: 2002-02-14
A special note goes to the chapter on Blum's complexity, which is about the only good place where I found it and from where I studied for my course on Complexity I.
For this reason the book requires quite more attention than others, but it really worths all the time one can spend reading it. Truly understanding Computability and Complexity as Professor Davis teaches them with this book is in my opinion a definitely high achievement, bringing the sensation that you grasp it totally, with no space for ambiguity or weakness.
Beautiful overviewReview Date: 2001-07-11
After a brief review of elementary mathematics and mathematical logic in chapter 1, the authors move right into the consideration of computable functions in chapter 2. They choose a particular abstract programming language in which to study the computability theory, which is built from variables, and programs that can be built from lists of instructions. Examples of programs are given, which have a Fortran flavor, with examples of computing partial functions. Unfortunately, a plethora of GOTO statements appear in the programs, and throughout the rest of the book, which is surprising given the publishing date. The use of these GOTO statements in the book is a major annoyance.
Then in chapter 3, the authors discuss primitive recursive functions, beginning with a treatment of composition, followed by the all-important concept of recursion. The class (PRC) of primitive recursive functions is introduced, and shown to be computable. The primitive recursive predicates are introduced, followed by a proof that the existential and universal quantifiers over an element of a PRC class are also PRC. This is followed by a discussion of minimalization and Godel numbers.
The next chapter is very interesting, wherein the famous halting problem is discussed and related to Church's thesis. The authors stress, most importantly, that an algorithm cannot be defined outside of the choice of a language, and therefore Church's thesis cannot be proved as a theorem. The authors also introduce recursively enumerable sets and show, via diagonalization, that non-recursively enumerable sets exist. They give an interesting example of a function that is computable but not primitive recursive.
The next chapter extends the results to strings of symbols instead of just numbers, and the authors introduce programming languages for doing string computations. One of these is the famous Post-Turing language, which they use to discuss the halting problem, with a variant used in the next chapter on Turing machines. The authors discuss the famous halting problem for Turing machines in this chapter. This is followed in chapter 7 by a discussion of productions and simulation of nondeterministic Turing machines. A very lucid treatment of Post's correspondence problem is given.
Things get somewhat more complicated in chapter 8, where the authors attempt to classify unsolvable problems. It contains one of the best discussions I have seen in the literature on oracles, and the authors give a very clear treatment of arithmetic hierarchies.
The second part of the book reads more like a book on compilers, as the authors delve into the area of grammars and automata. Regular languages, deterministic and non-deterministic finite automata are discussed, and Kleene's theorem, which states that regular languages and finite automata define the same languages, is proven. The context-free languages, so familiar from the study of compilers, are discussed also, along with a proof that a context-free grammar can be reduced to a Chomsky normal form grammar. Pushdown automata, needed for accepting context-free languages, are treated in detail. The authors give a good explanation here as to the additional facilities needed for a finite automaton to decide if a word belongs to a "bracket" language. Chomsky hierarchies are also discussed, and the authors motivate nicely the need for a linear bounded automaton to accept context sensitive languages.
Part three of the book is an overview of mathematical logic, and begins with a treatment of the propositional calculus. The satisfiability problem is discussed for this system, along with how to reduce formulas to normal form. The important compactness theorem is given a very detailed proof. Predicate calculus is then discussed, and Herbrand's theorem, which effectively reduces logical inference in predicate calculus to a problem of satisfiability of universal sentences, is proven. This theorem is fascinating and has important applications to automated theorem proving, as it ties together semantic and syntactical properties of a formal system. The Godel incompleteness theorem and the unsolvability of the satisfiability problem in predicate logic is proven.
In part 4, issues in computational complexity are addressed, the measure of complexity given in terms of the Blum axioms. This is a very abstract way of introducing complexity theory, as it introduces measures of complexity that more general than time and space complexity. The fascinating gap theorem, comparing program performance on two computing machines via complexity measures, is proven. This is followed by a detailed discussion of the speedup theorem, which essentially states that there is a wildly complicated recursive function such that for any program computing this function, there exists another program computing the function that works a lot faster for almost every input. The polynomial-time computability is discussed along with the famous P vs NP problem, with the discussion given in terms of Turing machines. Examples of NP-complete problems are given.
The last part of the book covers semantics, with operational and denotational semantics defined and compared. The emphasis in this part is on programming languages and constructions that one would actually find in practice, and so the preceding chapters on computable functions must be extended. The concept of an approximate ordering is introduced to allow for the instantaneous of a computation at some point before its completion. The denotational semantics of recursion equations and infinitary data structures are discussed, with the latter put it in to deal with the sophisticated systems that are constructed here. The discussion here is very involved, but the authors do a fair job of explaining the need for these types of data structures. The same is done for operational semantics, and the authors finally show that the computable numerical functions are actually partially computable. They then show the existence of computable irrational numbers.
CS Theory at it's bestReview Date: 2000-03-30
My favorite book on the theory of computationReview Date: 2000-05-10
This is a wonderful text about the theory of computation.Review Date: 1999-02-25


Analytical, informative and creative...Review Date: 2005-10-11
I hope Davies will continue to do what he does best-- analyze, synthesize and share his ideas.
Is there a cosmic blueprint?Review Date: 2006-07-15
What follows is a fast-paced and critical tour-de-force of the state of current and emerging scientific theories and prospects (promising and otherwise) for the future. There are many outstanding discussions, one centered on the mathematics of self-similar scaling -- the "Mandelbrot set" being a famous example. Davies believes that, in principle, science will one day explain, comprehensively, how the world works. Don't hold your breath, we're not exactly close to that day just yet. In some significant areas, notably the deepest theoretical understandings of biological and mind sciences, there seems to have been rather little progress at all. From popular treatments [like glossy spreads in National Geographic magazine, or Discovery Channel shows], one might be led to believe that great insights have been gained into how biological evolution proceeds and how life arises spontaneously from non-life. Davies surveys the competing claims and theories in these disciplines and exposes them as being starkly impotent to date. (There is a popular myth that only religious fundamentalists are skeptical of the neo-Darwinian story line -- but many of the most penetrating minds of modern theoretical science and mathematics, including Werner Heisenberg, Niels Bohr, John von Neumann, and Kurt Godel, among others, have found the Darwinian story to be non-compelling at best, and on some points glaringly wrong. As Davies points out, a world in which 'natural selection' was The Great Generative Engine, supporting only reproductive advantages, many life forms that we observe, like elephants [low birth rate, long gestation period, etc], could not have been 'selected' into existence. It does no good to protest that elephants should not and could not reproduce like bunnies -- in a truly Darwinian world there simply should not be elephants [or humans: striving to discern whether the universe might be headed toward a 'big crunch' or a 'heat death' can offer no reproductive advantage for beings given to contemplating such things!]. One thinks of many Darwin-confuters in the plant kingdom. A world built by Darwinian mechanisms would be populated only by masters of mitosis, and perhaps sex-maniacs that mated like bunnies, although sex itself, a comparatively inefficient means of reproduction [obviously when compared to mitotic reproduction], is another intractable problem for the Darwinian story.) Davies, like many noted physicists, is well studied and articulate in theoretical biology, and finds the state of neo-Darwinian evolutionary theory to be mostly a patched-up, just-so story that is easy fodder for skepticism. He does, however, believe that Stuart Kaufmann's ideas concerning holistic approaches to understanding complexity may be more fruitful. However I give a wrong impression if I seem to suggest that the book is largely about biological evolution, it is not. This is but one of several illuminating discussions, most of which understandably center on mathematical physics.
Davies, an epistemological optimist at any rate, expects that theoretical biology and mind science will one day succumb to our advancing knowledge, and that we will know, exhaustively, how the world works. But to know how the world works is not to know why the world works. Even if science should attain a reductionistic "theory of everything" and a stunning cadre of holistic theories explaining all features of 'emergence' and 'complexity', the big Why question(s) will remain, and any answers will remain as matters of faith, not strict science. Science speaks to "how," but why should this be so? Why should there be an explainable world and why should it contain world-explainers? One can deny teleological inference and many materialists insist [religiously] on doing so, but the denial is one of personal choice, not actual science. The world IS 'up to something,' and this fact IS fundamentally written into the new physics. The mysterious profundity of "why" always remains. In the day that Davies foresees, when physical science has achieved its final triumph, it will not have dispensed with God and it will not have written purpose out of the world.
Well, that's the book in a nutshell, but there's much more to it. I've read something like ten of Davies' books; most of them being either 'good' or 'very good' (with one notable exception) and this volume fits either category, except, perhaps, for that wild assertion in the new preface.
IT CANNOT BE LEGO!Review Date: 2000-07-31
The book strikes a perfect balance - not in the sense of compromise at all costs but in that, more useful and enlightening, of creative dialectical synthesis - between a steady faith in the capacity of science to investigate and eventually unveiling natural truths and a sort of rational and humane optimism which makes one feel that our universe is a formidable work-in-progress with a built-in, but not mechanistic inclination at producing new principles and meaning.
Mystery and freedom (and that flavour of "philosophic poetry" associated with them!) are preserved in the frame of a non-deterministic worldview, because no precise and mandatory evolutionary path seems to have been established at the "beginning", which Davies assumes, like the majority of today's cosmologists, to have been the Big Bang, the "moment" at which all - space, time, matter and energy - broke into being. Rather a potentiality for progress which in the course of billions of years has reached and will reach numerous transition points, from where the universe can branch out into a wide choice of meaningful possibilities. The only "constraint" being represented by a sort of tacit, maternal invitation to follow the route to self-consciousness as if the universe was sketched in such a way as to eventually reach this fundamental stage. For this reason Davies appears to believe, but never in a fixed, dogmatic sense, that just something like a loose cosmic blueprint, whence the book's title, lies hidden at the very core of creation, secretly fostering the growth of that substance we call, with still a bit of approximation, Intelligence.
I think that, like any truly important book, this one was written to revisit old questions and pose new ones rather than to provide standard ready-made answers, because, Davies seems to imply, no definite and irreversible answer is written on the giant cosmic page before us. That's to say: open arguments, for open minds!
During his exposition Davies touches many of the issues on which the debate and the contrasts between reductionism and holism are more vivid and intellectually productive. And he affirms that a new turn towards holism is gaining momentum among the scientific communities the world over as the lego-like philosophy of reductionism shows all its conceptual inadequacy to provide convincing prospects for the advancement of our understanding of complexity and significance.
Such rich notions as Jung's principle of synchronicity and Elsasser's "biotonic laws" are discussed, as well as that nice and inevitable philosophical animal known under the name of Schroedinger's cat, which as every well-read scientific layman already knows implies a plunge into the spectral world of quantum-mechanics. These and many other ideas and hypotheses are presented in a fascinating review of the more suggestive attempts to forge new visions of the universe and its destiny.
As a final, if trivial, consideration I may say that after reading this kind of books you come back to your everyday routine problems with a refreshed notion of their limited importance and consequently with the conviction that you should not permit them to take away a too large amount of your intellectual and spiritual energies.
The Cosmic BlueprintReview Date: 2003-05-28
Davies' book clearly explains the fundamental concepts and ties them all in - emergence, nonlinearity, the second law, self-organization, stochastic structures, complex and dynamic systems, darwinism and creativity - in all their cosmological and terrestial implications, with excellent philosophy to back it.
This is an absolute must-read for anyone interested in contemporary science. It still stands as a classic explication of an emerging new scientific paradigm which is now in full swing, and which Davies called and contextualized years ahead of his time.
If you read one science book in your life, this should be it.
The Cosmic BlueprintReview Date: 2003-05-28
Davies' book clearly explains the fundamental concepts and ties them all in - emergence, nonlinearity, the second law, self-organization, stochastic structures, complex and dynamic systems, darwinism and creativity - in all their cosmological and terrestial implications, with excellent philosophy to back it.
This is an absolute must-read for anyone interested in contemporary science. It still stands as a classic explication of an emerging new scientific paradigm which is now in full swing, and which Davies called and contextualized years ahead of his time.
If you read one science book in your life, this should be it.
Related Subjects: Conferences
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This is a beautiful book that quietly dances around your soul causing you to reflect on your own life. Using the metaphor of her own lifelong journey hiking in the mountains, the author takes the experiences of those hikes and connects them to the choices which fill life with joy and wonder. The book is illustrated with wonderful art that is worth the price of the book on its own. Eileen is a great communicator in person and on paper through the written word. This book is a gift from the mountain to your soul. You will want to keep it my your bedside and read more than once.
John Izzo, Author of The Five Secrets You Must Discover Before You Die