Computability, Complexity, and Languages, Second Edition: Fundamentals of Theoretical Computer Science (Computer Science and Scientific Computing)

Book Description

This introductory text covers the key areas of computer science, including recursive function theory, formal languages, and automata. It assumes a minimal background in formal mathematics. The book is divided into five parts: Computability, Grammars and Automata, Logic, Complexity, and Unsolvability.

* Computability theory is introduced in a manner that makes maximum use of previous programming experience, including a "universal" program that takes up less than a page.
* The number of exercises included has more than tripled.
* Automata theory, computational logic, and complexity theory are presented in a flexible manner, and can be covered in a variety of different arrangements.

Customer Reviews:

Pure mathematical view of Computability and Complexity.......2002-02-14

This is not a common book on Computability and Complexity as Hopcroft-Ullman, Sipser or Papadimitrou. You won't find here too many words describing topics: you'll find the power and elegance of a superlative mathematical approach from one the best authors of the century in the field. Conversely, you'll find here a detailed and elegant treatment of the whole history of computational models that starts at the Primitive Recursive Functions, something you won't find in the other books above mentioned.
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 overview.......2001-07-11

The authors of this book define theoretical computer science as the mathematical study of models of computation, and they do an excellent job of detailing the major results in the theory of computation as related to mathematical logic. Mathematicians, programmers, and philosophers will find the book an effective one in which to learn computability theory, and it serves well as a textbook for courses in the subject.

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.

My favorite book on the theory of computation.......2000-05-11

I first learned computability from this book and I loved every minute of it. It has lots of material and is superbly written. In fact, I think the chapters on logic are the most painless way to learn that subject. There are many other books around on this subject, but this is the ultimate!

CS Theory at it's best.......2000-03-30

I haven't found a better book on the Theoretical foundations of Computer Science. However since this IS theory the text can be a bit cryptic. Still, I'd recomend this book to any PhD Candidate or full Professor. Even a lowly Master's student like myself could use it.

This is a wonderful text about the theory of computation........1999-02-25

It taught me how to think about the theory of computation. The exercises added to the second edition are a big improvement over the first editon.

Logic and Information (Cambridge Tracts in Theoretical Computer Science)

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Logic and Information (Cambridge Tracts in Theoretical Computer Science)
Keith Devlin
Manufacturer: Cambridge University Press
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Binding: Hardcover

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Infosense: Turning Information Into Knowledge
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ASIN: 0521410304

Book Description

In this provocative and ground-breaking book, Keith Devlin argues that in order to obtain a deeper understanding of the nature of intelligence and knowledge acquisition, we must broaden our concept of logic. Classical logic, beginning with the work of Aristotle, has developed into a powerful and rigorous mathematical theory with many applications in mathematics and computer science, but it has proved woefully inadequate in the search for artificial intelligence. The new kind of logic, also mathematically based, outlined by Professor Devlin is the culmination of collaborative research among some of the world's leading logicians, philosophers, linguists, psychologists, and computer scientists. It introduces the concepts of infon, a quantum of information, and situations, a dynamical generalization of sets, and is capable of handlng the issues involved in human communication, thought, speech, and machine information processing.

Theoretical and Experimental DNA Computation (Natural Computing Series)

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Excellent!

Theoretical and Experimental DNA Computation (Natural Computing Series)
Martyn Amos
Manufacturer: Springer
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Binding: Hardcover

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DNA Computing: New Computing Paradigms (Texts in Theoretical Computer Science. An EATCS Series)

ASIN: 3540657738

Book Description

This book provides a broad overview of the entire field of DNA computation, tracing its history and development. It contains detailed descriptions of all major theoretical models and experimental results to date, which are lacking in existing texts, and discusses potential future developments. It also provides a useful reference source for researchers and students, and an accessible introduction for people new to the field.

The field of DNA computation has flourished since the publication of Adleman's seminal article, in which he demonstrated for the first time how a computation may be performed at a molecular level by performing standard operations on a tube of DNA strands. This monograph provides a detailed survey of the field, before describing recent theoretical and experimental developments. It concludes by outlining the challenges faced by researchers in the field and suggests possible future directions.

Customer Reviews:

Excellent!.......2003-05-24

Very well written but not as funny as I'd hoped. Easy to get through nonetheless. I finished it in two days, and I only looked at it when I went to the lav. (I kept it on the cistern behind the spare roll.)

Information Flow: The Logic of Distributed Systems (Cambridge Tracts in Theoretical Computer Science)

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Well worth the effort
A framework for studying information content
Not enough juice for the squeeze
Disappointing among Barwise' other work
new engineering tools

Information Flow: The Logic of Distributed Systems (Cambridge Tracts in Theoretical Computer Science)
Jon Barwise , and Jerry Seligman
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Hardcover

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Topoi: The Categorial Analysis of Logic (Dover Books on Mathematics)

ASIN: 0521583861

Book Description

Information is a central topic in computer science, cognitive science, and philosophy. In spite of its importance in the "information age," there is no consensus on what information is, what makes it possible, and what it means for one medium to carry information about another. Drawing on ideas from mathematics, computer science, and philosophy, this book addresses the definition and place of information in society. The authors, observing that information flow is possible only within a connected distribution system, provide a mathematically rigorous, philosophically sound foundation for a science of information. They illustrate their theory by applying it to a wide range of phenomena, from file transfer to DNA, from quantum mechanics to speech act theory.

Customer Reviews:

Well worth the effort.......2006-08-02

I found the exposition in parts of this book quite poor. The prose is often pretty cryptic, there were some typographical errors that led me down detours while I tried to work through the details, so it took a lot of work (for me at least) to figure out some of the steps. The fact that a lot of the examples are of "toy" problems doesn't help see how the framework developed could be used for anything.

This is one of those books that should say "Some assembly required" on the cover.

I still think this is an important book, and that it deserves considerably more influence in academic philosophy, especially in the literature on causal process theories (developments of the work of Reichenbach and Salmon) and relations between theories in philosophy of science.

This isn't easy, and it's not obvious what it's useful for, but it's still very good. I'm not sure what the theoretical computer scientists make of it.

A framework for studying information content.......2006-04-18

In his book, The Situation in Logic, Jon Barwise wrote, "...it has turned out that one of the most important notions of topology has been that of a homeomorphism, a function from one space to another that respects nearness relations in an appropriate way. I think that something similar is going to happen here. I suspect that we are going to find ourselves studying 'infomorphisms,' maps from one information space to another that preserve information. But before we can define these maps, we need to understand the basic structure they need to preserve....I am currently working on such a framework."

Jon's book with Jerry Seligman presents this framework. Just before its publication I met Jon for the first time at a workshop at Stanford on the business applications of situation theory. I took what I learned there into a subsequent project on intangible assets at the Brookings Institution, wrote a paper listing some possibly relevant requirements on theories of information content, and submitted it for publication at http://www.unifr.ch/econophysics/articoli/articoli-00-jun.html.

Later-- after Jon had passed away-- I found one of the infomorphisms he had told us about in the Born equation of quantum mechanics.(http://www.wexedesignart.com/theBornInfomorphism.html)

It was an honor to know Jon.

Not enough juice for the squeeze.......2001-01-31

Impressive formalization of information flow via mathematics and logic, but in the end it's not clear if this subject holds any practical value. For, in practice, the model of information flow always requires a priori working models of the phenomena under consideration before the principles of information flow can be put to use. For example, consider currency fluctuations as information about some nation's economy. You can model this using Barwise's scheme, but doing so doesn't bring anything to one's understanding that we didn't already have from economics. No real clarification on what information is, either (although there's a lengthy discussion of this at the outset). Requires a fairly good working knowledge of set theory and basic formal logic. Poorly edited.

Disappointing among Barwise' other work.......2000-05-22

There's a lot of informal discussion at the beginning about what information "really is", but very little in terms of innovative content. The book contains a good formal approach that expands on earlier work in the same field, but it hangs on too many ill-fitting concepts as the "Xerox principle" and the notion that knowledge can be quantified. The book also completely ignores the ontological perspective which other researchers have addressed for over a decade.

new engineering tools.......1999-08-17

A mathematical basis for Keith Devlin's book "InfoSense: Turning Information into Knowledge." Buy them both!

Derivation and Computation: Taking the Curry-Howard correspondence seriously (Cambridge Tracts in Theoretical Computer Science)

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Derivation and Computation: Taking the Curry-Howard correspondence seriously (Cambridge Tracts in Theoretical Computer Science)
H. Simmons
Manufacturer: Cambridge University Press
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Binding: Hardcover

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ASIN: 0521771730

Book Description

The two notions of proofs and calculations are intimately related. Proofs can involve calculations, and the algorithm underlying a calculation should be proved correct. This volume explores this key relationship and introduces simple type theory. Starting from the familiar propositional calculus, the author develops the central idea of an applied lambda-calculus. This is illustrated by an account of Gödel's T, a system that codifies number-theoretic function hierarchies. Each of the book's 52 sections ends with a set of exercises, some 200 in total. An appendix contains complete solutions of these exercises.

Elements of Finite Model Theory (Texts in Theoretical Computer Science. An EATCS Series)

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Technical review of this book

Elements of Finite Model Theory (Texts in Theoretical Computer Science. An EATCS Series)
Leonid Libkin
Manufacturer: Springer
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Binding: Hardcover

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DNA Computing: 12th International Meeting on DNA Computing, DNA12, Seoul, Korea, June 5-9, 2006, Revised Selected Papers (Lecture Notes in Computer Science)
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ASIN: 3540212027

Book Description

This book is an introduction to finite model theory which stresses the computer science origins of the area. In addition to presenting the main techniques for analyzing logics over finite models, the book deals extensively with applications in databases, complexity theory, and formal languages, as well as other branches of computer science. It covers Ehrenfeucht-Fraïssé games, locality-based techniques, complexity analysis of logics, including the basics of descriptive complexity, second-order logic and its fragments, connections with finite automata, fixed point logics, finite variable logics, zero-one laws, and embedded finite models, and gives a brief tour of recently discovered applications of finite model theory.

This book can be used both as an introduction to the subject, suitable for a one- or two-semester graduate course, or as reference for researchers who apply techniques from logic in computer science.

Customer Reviews:

Technical review of this book.......2004-09-21

Model theory is the study of the logical properties of mathematical structures. Finite model theory arises when we focus our attention on finite structures, such as finite graphs (graphs with a finite number of nodes). This book presents the most important results of finite model theory in an extremely readable, yet careful and precise manner. Libkin himself is a master of the art, and this shows in his beautiful presentation of the material.
--Ronald Fagin Manager, Foundations of Computer Science, IBM Almaden Research Center, San Jose, CA

Logic and Computation: Interactive Proof with Cambridge LCF (Cambridge Tracts in Theoretical Computer Science)

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Logic and Computation: Interactive Proof with Cambridge LCF (Cambridge Tracts in Theoretical Computer Science)
Lawrence C. Paulson
Manufacturer: Cambridge University Press
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Binding: Paperback

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ASIN: 0521395607

Book Description

Logic and Computation is concerned with techniques for formal theorem-proving, with particular reference to Cambridge LCF (Logic for Computable Functions). Cambridge LCF is a computer program for reasoning about computation. It combines methods of mathematical logic with domain theory, the basis of the denotational approach to specifying the meaning of statements in a programming language. This book consists of two parts. Part I outlines the mathematical preliminaries: elementary logic and domain theory. They are explained at an intuitive level, giving references to more advanced reading. Part II provides enough detail to serve as a reference manual for Cambridge LCF. It will also be a useful guide for implementors of other programs based on the LCF approach.

The Graph Isomorphism Problem: Its Structural Complexity (Progress in Theoretical Computer Science)

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The Graph Isomorphism Problem: Its Structural Complexity (Progress in Theoretical Computer Science)
J. Kobler , U. Schöning , and J. Toran
Manufacturer: Birkhäuser Boston
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Quantum Logic (Discrete Mathematics and Theoretical Computer Science)

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Quantum Logic (Discrete Mathematics and Theoretical Computer Science)
Karl Svozil
Manufacturer: Springer
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Binding: Paperback

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ASIN: 9814021075

Book Description

Quantum Logic deals with the foundations of quantum mechanics and, related to it, the behaviour of finite, discrete deterministic systems. The quantum logical approach is particulalry suitable for the investigation and exclusion of certain hidden parameter models of quantum mechanics. Conversely, it can be used to embed quantum universes into classical ones. It is also highly relevant for the characterization of finite automation. This book has been written with a broad readership in mind. Great care has been given to the motivation of the concepts and to the explicit and detailed discussions of examples.

Epistemic Logic for AI and Computer Science (Cambridge Tracts in Theoretical Computer Science)

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Epistemic Logic for AI and Computer Science (Cambridge Tracts in Theoretical Computer Science)
J.-J. Ch. Meyer , and W. van der Hoek
Manufacturer: Cambridge University Press
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A Mathematical Introduction to Logic, Second Edition

ASIN: 0521602807

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Epistemic logic has grown from its philosophical beginnings to find diverse applications in computer science as a means of reasoning about the knowledge and belief of agents. This book, based on courses taught at universities and summer schools, provides a broad introduction to the subject; many exercises are included together with their solutions. The authors begin by presenting the necessary apparatus from mathematics and logic, including Kripke semantics and the well-known modal logics K, T, S4 and S5. Then they turn to applications in the contexts of distributed systems and artificial intelligence: topics that are addressed include the notions of common knowledge, distributed knowledge, explicit and implicit belief, the interplays between knowledge and time, and knowledge and action, as well as a graded (or numerical) variant of the epistemic operators. The problem of logical omniscience is also discussed extensively. Halpern and Moses’ theory of honest formulae is covered, and a digression is made into the realm of non-monotonic reasoning and preferential entailment. Moore’s autoepistemic logic is discussed, together with Levesque's related logic of ‘all I know’. Furthermore, it is shown how one can base default and counterfactual reasoning on epistemic logic.

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