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Derivation and Computation: Taking the Curry-Howard correspondence seriously (Cambridge Tracts in Theoretical Computer Science)
H. Simmons
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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.
Book Description
This textbook explains the basic principles of categorical type theory and the techniques used to derive categorical semantics for specific type theories. It introduces the reader to ordered set theory, lattices and domains, and this material provides plenty of examples for an introduction to category theory, which covers categories, functors, natural transformations, the Yoneda lemma, cartesian closed categories, limits, adjunctions and indexed categories. Four kinds of formal system are considered in detail, namely algebraic, functional, polymorphic functional, and higher order polymorphic functional type theory. For each of these the categorical semantics are derived and results about the type systems are proved categorically. Issues of soundness and completeness are also considered. Aimed at advanced undergraduates and beginning graduates, this book will be of interest to theoretical computer scientists, logicians and mathematicians specializing in category theory.
Customer Reviews:
Excellent introduction to categories for computer scientists.......1997-11-15
The book gives you all of the cateogry theory you need to study type theory. The examples are from domains that are comfortable for computer scientists. The difficult proofs are given in great detail, while other books often gloss over the details.
Book Description
The Curry-Howard isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance,
minimal propositional logic corresponds to simply typed lambda-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, sequent calculus is related to explicit substitution, etc.
The isomorphism has many aspects, even at the syntactic level:
formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc.
But there is more to the isomorphism than this. For instance, it is an old idea---due to Brouwer, Kolmogorov, and Heyting---that a constructive proof of an implication is a procedure that transforms
proofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. The Curry-Howard isomorphism also provides theoretical foundations for many modern proof-assistant systems (e.g. Coq).
This book give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism. It can serve as an introduction to any or both of typed lambda-calculus and intuitionistic logic.
Key features
- The Curry-Howard Isomorphism treated as common theme
- Reader-friendly introduction to two complementary subjects: Lambda-calculus and constructive logics
- Thorough study of the connection between calculi and logics
- Elaborate study of classical logics and control operators
- Account of dialogue games for classical and intuitionistic logic
- Theoretical foundations of computer-assisted reasoning
· The Curry-Howard Isomorphism treated as the common theme.
· Reader-friendly introduction to two complementary subjects: lambda-calculus and constructive logics
· Thorough study of the connection between calculi and logics.
· Elaborate study of classical logics and control operators.
· Account of dialogue games for classical and intuitionistic logic.
· Theoretical foundations of computer-assisted reasoning
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- for computer scientists, not programmers
- Set high standards
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Comprehensive Mathematics for Computer Scientists 2: Calculus and ODEs, Splines, Probability, Fourier and Wavelet Theory, Fractals and Neural Networks, Categories and Lambda Calculus (Universitext)
Guerino Mazzola ,
Gérard Milmeister , and
Jody Weissmann
Manufacturer: Springer
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Binding: Paperback
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ASIN: 3540208615 |
Book Description
This two-volume textbook
Comprehensive Mathematics for the Working Computer Scientist is a self-contained comprehensive presentation of mathematics including sets, numbers, graphs, algebra, logic, grammars, machines, linear geometry, calculus, ODEs, and special themes such as neural networks, Fourier theory, wavelets, numerical issues, statistics, categories, and manifolds. The concept framework is streamlined but defining and proving virtually everything. The style implicitly follows the spirit of recent topos-oriented theoretical computer science. Despite the theoretical soundness, the material stresses a large number of core computer science subjects, such as, for example, a discussion of floating point arithmetic, Backus-Naur normal forms, L-systems, Chomsky hierarchies, algorithms for data encoding, e.g., the Reed-Solomon code. The numerous course examples are motivated by computer science and bear a generic scientific meaning. This text is complemented by an online university course which covers the same theoretical content, however, in a totally different presentation. The student or working scientist who once gets involved in this text may at any time consult the online interface which comprises applets and other interactive tools.
Customer Reviews:
for computer scientists, not programmers.......2007-01-22
If you are a computer professional and wondering how much maths you need, this book might be a good answer. Much of its content is discrete maths. Like factorising primes. A background in this leads into the theory of public key encryption. Something well worth knowing. Note that the book's discussion of factorising primes is brief. You'd need another text for the public key material.
Also potentially useful is a foray into graph theory. Rings and modules also make their appearance. Predicate logic is another topic explained.
The text also discusses matrices from a viewpoint more formal than that of a typical first course in linear algebra. It generalises to an exposition of vector spaces. This portion of the text is continuum maths.
Do note that the average computer programmer actually uses very little of the material. Emphasis on programmer. But as the title says, the book is directed towards the computer scientist, who needs a more formal grounding.
Set high standards.......2004-08-17
This book is very exciting and motivating, as it sets high standards both with respect to the contents and the style.
It contains the main topics of modern math from the beginning, with applications to Computer Science: Sets, Relations, Logic, Numbers, Groups, Rings, Modules, Linear Algebra, Geometry, Quaternions.
The style is extremely compact, precise and readable, with a professional notation. The book is optimal designed (modularized)
which results in a moderate size (360 p.)
The objective of his approach is to provide a better formal competence for computer scientists by a sound mathematical education:
"Improved formal competence is urged by the object-oriented paradigm which progressively requires a programming style and a design strategy of high abstraction level in conceptual engineering."
The accompanying online course contains illustrations and animations, which at the moment don't go far beyond the book's contents.
Nevertheless, the online part has a great potential for further applications, e.g. programming projects (Moebius-Strip, Raytracing, 3D Stereo, ...)
We hope, this book will establish itself as a standard for ambitious schools and students.
Book Description
The revised edition contains a new chapter which provides an elegant description of the semantics. The various classes of lambda calculus models are described in a uniform manner. Some didactical improvements have been made to this edition. An example of a simple model is given and then the general theory (of categorical models) is developed. Indications are given of those parts of the book which can be used to form a coherent course.
Customer Reviews:
great book, but not available here.......2004-07-03
I have this book checked out from a university library, and it is quite wonderful. Despite the fact that Amazon continues to list it for sale, it is not currently available.
Self-contained Encyclopedia! All you need is your patience!.......2002-01-16
This encyclopedic monograph is now a classic of this field,
lambda-calculus, which is the theoretical basis of practical
functional programming languages such as Standard ML, CAML, Haskell etc.
This book itself is purely theoretical and principally aimed for researchers/students of its field.
This book is very comprehensive. In fact, this book successfully compiles almost all results on type-free lambda-calculus up to the time of its publication (early 1980's).
Surprisingly enough!, however, this very technical encyclopedic monograph is self-contained.
Proofs of all theorems/lemmata are given up to details except for cases that they are intentionally left for excercises.
Therefore, even a novice of its field can follow all of the proofs. The only one thing that such a novice must have is patience. His/her patience will surely be rewarded.
Backgrounds assumed in this encyclopedic monograph is the very beginning level understanding of mathematical logic. If you are not familiar with math logic, you can learn the necessary backgrounds with any introductory textbooks on math logic.
All more technical notions and notations are defined/explained in this book. Many interesting examples are given.
Exercises at the end of each charpter are very helpful and also are very interesting. The author clearly paid much attention and took care on the arrangement of exercises so that readers can tackle easier one at first. Moreover such carefully arranged exercises tell readers more. Readers will understand very delicate but important points during solving exercises by themselves. In other words, the last sentence means the following fact: imagine there are two intuitively similar notions
(it is often the case that very abstract theory has many such pairs of notions) that novices can confuse each other. Solving one exercise tell the novice that one notion is not implied from the other. Also solving another exercise tell vice-versa.
Indices and references are very useful. In fact, indices are carefully designed. Not only the index of technical terms, there are indices for symbols and authors (of references refered in the main text). References are very comprehensive.
There are very few typos (another surprising points! Math books almost always handreds of typos) except for misuses of type-faces which are clearly due to typesetting by the publisher.
This book, as I pointed before, is on pure math logic and its readership is clearly researchers/students of its field.
But, as a computer scientist, I recommend this book to all of the functional programmers, who, at least, are serious about the background of their profession.
If you read this book, you will understand that there is a very beautiful (though abstract) world of theories behind ML/Haskell programming.
If you are a student who wants study lambda-calculus, combinatory logic, type theory, constructive math, etc.,
then, this book is for you, too, of course.
This encyclopedia doubtlessly will give you the basis to become the researcher on such fields.
It's online.......2001-12-30
This is a great book. A must buy for all graduate students in computer science. Because the book is out of print, you can obtain it online at......
Not a classic, but all in all very informative.......2000-07-24
I accidently bought this book thinking I was buying a traditional calculus book in order to prepare for a standardized test I had to take. After a few chapters I realized this was no ordinary Calculus book. None-the-less I got much out of it. It is the kind of book where the author builds up the advanced topics from most basic concepts, so it is pretty much accessible to anyone who is reasonably intelligent. It also has a very nice graphical lay-out with regards to the symbols and such which makes it much easier to read and learn from.
Although it didn't help me do well on the AP Calculus test, I would suggest buying it to anyone who has a good reason to. It is the kind of book that you can use to learn from and use it as a reference book.
Book Description
Language in Action demonstrates the viability of mathematical research into the foundations of categorial grammar, a topic at the border between logic and linguistics. Since its initial publication it has become the classic work in the foundations of categorial grammar. A new introduction to this paperback edition updates the open research problems and records relevant results through pointers to the literature.
Van Benthem presents the categorial processing of syntax and semantics as a central component in a more general dynamic logic of information flow, in tune with computational developments in artificial intelligence and cognitive science. Using the paradigm of categorial grammar, he describes the substructural logics driving the dynamics of natural language syntax and semantics. This is a general type-theoretic approach that lends itself easily to proof-theoretic and semantic studies in tandem with standard logic. The emphasis is on a broad landscape of substructural categorial logics and their proof-theoretical and semantic peculiarities. This provides a systematic theory for natural language understanding, admitting of significant mathematical results. Moreover, the theory makes possible dynamic interpretations that view natural languages as programming formalisms for various cognitive activities.
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Abstract Computing Machines
W. Kluge , and
Werner Kluge
Manufacturer: Springer
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Binding: Hardcover
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ASIN: 3540211462 |
Book Description
The book addresses ways and means of organizing computations, highlighting the relationship between algorithms and the basic mechanisms and runtime structures necessary to execute them using machines. It completely abstracts from concrete programming languages and machine architectures, taking instead the lambda calculus as the basic programming and program execution model to design various abstract machines for its correct implementation. The emphasis is on fully normalizing machines based on a full-fledged beta-reduction as an essential prerequisite for symbolic computations that treat functions and variables truly as first-class objects. Their weakly normalizing counterparts are shown to be functional abstract machines that sacrifice the flavors of full beta-reductions for decidedly simpler runtime structures and improved runtime efficiency. Further downgrading of the lambda calculus leads to classical imperative (von Neumann) machines that permit side-effecting operations on the runtime environment.
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Typed Lambda Calculi and Applications: Second International Conference on Typed Lambda Calculi and Applications, Tlca '95, Edinburgh, United Kingdom, April ... (Lecture Notes in Computer Science)
Manufacturer: Springer
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Binding: Paperback
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ASIN: 354059048X |
Book Description
This volume presents the proceedings of the Second International Conference on Typed Lambda Calculi and Applications, held in Edinburgh, UK in April 1995.The book contains 29 full revised papers selected from 58 submissions and comprehensively reports the state of the art in the field. The following topics are addressed: proof theory of type systems, logic and type systems, typed lambda calculi as models of (higher-order) computation, semantics of type systems, proof verification via type systems, type systems of programming languages, and typed term rewriting systems.
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Adapting Proofs-as-Programs : The Curry-Howard Protocol (Monographs in Computer Science)
Iman Hafiz Poernomo ,
John Newsome Crossley , and
Martin Wirsing
Manufacturer: Springer
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Binding: Hardcover
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ASIN: 0387237593 |
Book Description
This monograph details several important advances in the area known as the proofs-as-programs paradigm, a set of approaches to developing programs from proofs in constructive logic. It serves the dual purpose of providing a state-of-the-art overview of the field and detailing tools and techniques to stimulate further research.
One of the book’s central themes is a general, abstract framework for developing new systems of program synthesis by adapting proofs-as-programs to new contexts, which the authors call the Curry--Howard Protocol. This protocol is used to provide two novel applications for industrial-scale, complex software engineering: contractual imperative program synthesis and structured software synthesis. These applications constitute an exemplary justification for the applicability of the protocol to different contexts.
The book is intended for graduate students in computer science or mathematics who wish to extend their background in logic and type theory as well as gain experience working with logical frameworks and practical proof systems. In addition, the proofs-as-programs research community, and the wider computational logic, formal methods and software engineering communities will benefit. The applications given in the book should be of interest for researchers working in the target problem domains.
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