Proofs and Types PDF Download
Are you looking for read ebook online? Search for your book and save it on your Kindle device, PC, phones or tablets. Download Proofs and Types PDF full book. Access full book title Proofs and Types by Jean-Yves Girard. Download full books in PDF and EPUB format.
Author: Jean-Yves Girard
Publisher: Cambridge University Press
ISBN: 9780521371810
Category : Computers
Languages : en
Pages : 192
Get Book Here
Book Description
This text is an outgrowth of notes prepared by J. Y. Girard for a course at the University of Paris VII. It deals with the mathematical background of the application to computer science of aspects of logic (namely the correspondence between proposition & types). Combined with the conceptual perspectives of Girard's ideas, this sheds light on both the traditional logic material & its prospective applications to computer science. The book covers a very active & exciting research area, & it will be essential reading for all those working in logic & computer science.
Author: Jean-Yves Girard
Publisher: Cambridge University Press
ISBN: 9780521371810
Category : Computers
Languages : en
Pages : 192
Get Book Here
Book Description
This text is an outgrowth of notes prepared by J. Y. Girard for a course at the University of Paris VII. It deals with the mathematical background of the application to computer science of aspects of logic (namely the correspondence between proposition & types). Combined with the conceptual perspectives of Girard's ideas, this sheds light on both the traditional logic material & its prospective applications to computer science. The book covers a very active & exciting research area, & it will be essential reading for all those working in logic & computer science.
Author: Rob Nederpelt
Publisher: Cambridge University Press
ISBN: 1316061086
Category : Computers
Languages : en
Pages : 465
Get Book Here
Book Description
Type theory is a fast-evolving field at the crossroads of logic, computer science and mathematics. This gentle step-by-step introduction is ideal for graduate students and researchers who need to understand the ins and outs of the mathematical machinery, the role of logical rules therein, the essential contribution of definitions and the decisive nature of well-structured proofs. The authors begin with untyped lambda calculus and proceed to several fundamental type systems, including the well-known and powerful Calculus of Constructions. The book also covers the essence of proof checking and proof development, and the use of dependent type theory to formalise mathematics. The only prerequisite is a basic knowledge of undergraduate mathematics. Carefully chosen examples illustrate the theory throughout. Each chapter ends with a summary of the content, some historical context, suggestions for further reading and a selection of exercises to help readers familiarise themselves with the material.
Author: Chris Lavranos
Publisher: Createspace Independent Publishing Platform
ISBN: 9781514634448
Category :
Languages : en
Pages : 122
Get Book Here
Book Description
The scope of Formal Proofs in Maths is to teach students between higher school classes and University undergraduate or postgraduate studies, how to write a formal proof with the true meaning of the concept, of simple theorems in Algebra, particulary in identities concerning equalities, equations and inequalities. This is accomplished by writing four different types of proof namely type(A), type(B), type(C) and type(D) for each theorem or exercise. In TYPE(A) ordinary proofs will be cited in the usual narrative style used by experienced mathematicians. In TYPE(B) a rigorous proof in steps will be introduced to the reader. Each line of that proof will be justified by an appropriate axiom, theorem or definition. In TYPE(C) we will try for a smooth transition from a rigorous proof to a formal proof exposing the way that the laws of logic apply on one or more statements of the proof. In TYPE(D) we will simply write in tabular stepwise form, the results of TYPE(C) mentioning both: 1) Axioms, theorems or definitions. 2) The laws of logic. Hence, finally producing a formal proof according to the definition given in the preface note of the book.
Author: Adam Chlipala
Publisher: MIT Press
ISBN: 0262545748
Category : Computers
Languages : en
Pages : 437
Get Book Here
Book Description
A handbook to the Coq software for writing and checking mathematical proofs, with a practical engineering focus. The technology of mechanized program verification can play a supporting role in many kinds of research projects in computer science, and related tools for formal proof-checking are seeing increasing adoption in mathematics and engineering. This book provides an introduction to the Coq software for writing and checking mathematical proofs. It takes a practical engineering focus throughout, emphasizing techniques that will help users to build, understand, and maintain large Coq developments and minimize the cost of code change over time. Two topics, rarely discussed elsewhere, are covered in detail: effective dependently typed programming (making productive use of a feature at the heart of the Coq system) and construction of domain-specific proof tactics. Almost every subject covered is also relevant to interactive computer theorem proving in general, not just program verification, demonstrated through examples of verified programs applied in many different sorts of formalizations. The book develops a unique automated proof style and applies it throughout; even experienced Coq users may benefit from reading about basic Coq concepts from this novel perspective. The book also offers a library of tactics, or programs that find proofs, designed for use with examples in the book. Readers will acquire the necessary skills to reimplement these tactics in other settings by the end of the book. All of the code appearing in the book is freely available online.
Author: S.R. Buss
Publisher: Elsevier
ISBN: 0080533183
Category : Mathematics
Languages : en
Pages : 823
Get Book Here
Book Description
This volume contains articles covering a broad spectrum of proof theory, with an emphasis on its mathematical aspects. The articles should not only be interesting to specialists of proof theory, but should also be accessible to a diverse audience, including logicians, mathematicians, computer scientists and philosophers. Many of the central topics of proof theory have been included in a self-contained expository of articles, covered in great detail and depth.The chapters are arranged so that the two introductory articles come first; these are then followed by articles from core classical areas of proof theory; the handbook concludes with articles that deal with topics closely related to computer science.
Author: Morten Heine Sørensen
Publisher: Elsevier
ISBN: 0080478921
Category : Mathematics
Languages : en
Pages : 457
Get Book Here
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 transformsproofs 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
Author: Hendrik Pieter Barendregt
Publisher: Springer Science & Business Media
ISBN: 9783540580850
Category : Computers
Languages : en
Pages : 404
Get Book Here
Book Description
This volume contains thoroughly refereed and revised full papers selected from the presentations at the first workshop held under the auspices of the ESPRIT Basic Research Action 6453 Types for Proofs and Programs in Nijmegen, The Netherlands, in May 1993. As the whole ESPRIT BRA 6453, this volume is devoted to the theoretical foundations, design and applications of systems for theory development. Such systems help in designing mathematical axiomatisation, performing computer-aided logical reasoning, and managing databases of mathematical facts; they are also known as proof assistants or proof checkers.
Author: Oscar Levin
Publisher: Createspace Independent Publishing Platform
ISBN: 9781534970748
Category :
Languages : en
Pages : 342
Get Book Here
Book Description
This gentle introduction to discrete mathematics is written for first and second year math majors, especially those who intend to teach. The text began as a set of lecture notes for the discrete mathematics course at the University of Northern Colorado. This course serves both as an introduction to topics in discrete math and as the "introduction to proof" course for math majors. The course is usually taught with a large amount of student inquiry, and this text is written to help facilitate this. Four main topics are covered: counting, sequences, logic, and graph theory. Along the way proofs are introduced, including proofs by contradiction, proofs by induction, and combinatorial proofs. The book contains over 360 exercises, including 230 with solutions and 130 more involved problems suitable for homework. There are also Investigate! activities throughout the text to support active, inquiry based learning. While there are many fine discrete math textbooks available, this text has the following advantages: It is written to be used in an inquiry rich course. It is written to be used in a course for future math teachers. It is open source, with low cost print editions and free electronic editions.
Author: Stefano Berardi
Publisher: Springer
ISBN: 3642024440
Category : Computers
Languages : en
Pages : 331
Get Book Here
Book Description
These proceedings contain a selection of refereed papers presented at or - lated to the Annual Workshop of the TYPES project (EU coordination action 510996), which was held during March 26–29, 2008 in Turin, Italy. The topic of this workshop, and of all previous workshops of the same project, was f- mal reasoning and computer programming based on type theory: languages and computerized tools for reasoning, and applications in several domains such as analysis of programming languages, certi?ed software, mobile code, formali- tion of mathematics, mathematics education. The workshop was attended by more than 100 researchers and included more than 40 presentations. We also had three invited lectures, from A. Asperti (University of Bologna), G. Dowek (LIX, Ecole polytechnique, France) and J. W. Klop (Vrije Universiteit, A- terdam, The Netherlands). From 27 submitted papers, 19 were selected after a reviewing process. Each submitted paper was reviewed by three referees; the ?nal decisions were made by the editors. This workshop is the last of a series of meetings of the TYPES working group funded by the European Union (IST project 29001, ESPRIT Working Group 21900, ESPRIT BRA 6435).
Author: Thomas Piecha
Publisher: Springer
ISBN: 331922686X
Category : Philosophy
Languages : en
Pages : 281
Get Book Here
Book Description
This volume is the first ever collection devoted to the field of proof-theoretic semantics. Contributions address topics including the systematics of introduction and elimination rules and proofs of normalization, the categorial characterization of deductions, the relation between Heyting's and Gentzen's approaches to meaning, knowability paradoxes, proof-theoretic foundations of set theory, Dummett's justification of logical laws, Kreisel's theory of constructions, paradoxical reasoning, and the defence of model theory. The field of proof-theoretic semantics has existed for almost 50 years, but the term itself was proposed by Schroeder-Heister in the 1980s. Proof-theoretic semantics explains the meaning of linguistic expressions in general and of logical constants in particular in terms of the notion of proof. This volume emerges from presentations at the Second International Conference on Proof-Theoretic Semantics in Tübingen in 2013, where contributing authors were asked to provide a self-contained description and analysis of a significant research question in this area. The contributions are representative of the field and should be of interest to logicians, philosophers, and mathematicians alike.
