Basic Proof Theory

Basic Proof Theory

Introduction to proof theory and its applications in mathematical logic, theoretical computer science and artificial intelligence.

Author: A. S. Troelstra

Publisher: Cambridge University Press

ISBN: 0521779111

Category: Computers

Page: 417

View: 717

Introduction to proof theory and its applications in mathematical logic, theoretical computer science and artificial intelligence.
Categories: Computers

Ordinal Analysis with an Introduction to Proof Theory

Ordinal Analysis with an Introduction to Proof Theory

This book provides readers with a guide to both ordinal analysis, and to proof theory.

Author: Toshiyasu Arai

Publisher: Springer Nature

ISBN: 9789811564598

Category: Electronic books

Page: 327

View: 876

This book provides readers with a guide to both ordinal analysis, and to proof theory. It mainly focuses on ordinal analysis, a research topic in proof theory that is concerned with the ordinal theoretic content of formal theories. However, the book also addresses ordinal analysis and basic materials in proof theory of first-order or omega logic, presenting some new results and new proofs of known ones. Primarily intended for graduate students and researchers in mathematics, especially in mathematical logic, the book also includes numerous exercises and answers for selected exercises, designed to help readers grasp and apply the main results and techniques discussed.
Categories: Electronic books

Mathematical Intuitionism

Mathematical Intuitionism

This book intends to present the most important methods of proof theory in intuitionistic logic and to acquaint the reader with the principal axiomatic theories based on intuitionistic logic.

Author: Al'bert Grigor'evi_ Dragalin

Publisher: American Mathematical Soc.

ISBN: 9780821845202

Category: Mathematics

Page: 228

View: 336

In the area of mathematical logic, a great deal of attention is now being devoted to the study of nonclassical logics. This book intends to present the most important methods of proof theory in intuitionistic logic and to acquaint the reader with the principal axiomatic theories based on intuitionistic logic.
Categories: Mathematics

Proof Theory and Automated Deduction

Proof Theory and Automated Deduction

(Gore only uses -1, A and D as basic connectives.) These rules add a non-
negligible amount of non-determinism and of branching: finding proofs in such
logics is in fact not easy. 3 OTHER MODAL LOGICS OF INTEREST IN
COMPUTER ...

Author: Jean Goubault-Larrecq

Publisher: Springer Science & Business Media

ISBN: 1402003684

Category: Computers

Page: 444

View: 959

Proof Theory and Automated Deduction is written for final-year undergraduate and first-year post-graduate students. It should also serve as a valuable reference for researchers in logic and computer science. It covers basic notions in logic, with a particular stress on proof theory, as opposed to, for example, model theory or set theory; and shows how they are applied in computer science, and especially the particular field of automated deduction, i.e. the automated search for proofs of mathematical propositions. We have chosen to give an in-depth analysis of the basic notions, instead of giving a mere sufficient analysis of basic and less basic notions. We often derive the same theorem by different methods, showing how different mathematical tools can be used to get at the very nature of the objects at hand, and how these tools relate to each other. Instead of presenting a linear collection of results, we have tried to show that all results and methods are tightly interwoven. We believe that understanding how to travel along this web of relations between concepts is more important than just learning the basic theorems and techniques by rote. Audience: The book is a valuable reference for researchers in logic and computer science.
Categories: Computers

Natural Deduction

Natural Deduction

Gerhard Gentzen invented proof-theoretic semantics in the early 1930s, and Dag Prawitz, the author of this study, extended its analytic proofs to systems of natural deduction.

Author: Dag Prawitz

Publisher: Courier Dover Publications

ISBN: 9780486446554

Category: Mathematics

Page: 113

View: 292

An innovative approach to the semantics of logic, proof-theoretic semantics seeks the meaning of propositions and logical connectives within a system of inference. Gerhard Gentzen invented proof-theoretic semantics in the early 1930s, and Dag Prawitz, the author of this study, extended its analytic proofs to systems of natural deduction. Prawitz's theories form the basis of intuitionistic type theory, and his inversion principle constitutes the foundation of most modern accounts of proof-theoretic semantics. The concept of natural deduction follows a truly natural progression, establishing the relationship between a noteworthy systematization and the interpretation of logical signs. As this survey explains, the deduction's principles allow it to proceed in a direct fashion — a manner that permits every natural deduction's transformation into the equivalent of normal form theorem. A basic result in proof theory, the normal form theorem was established by Gentzen for the calculi of sequents. The proof of this result for systems of natural deduction is in many ways simpler and more illuminating than alternative methods. This study offers clear illustrations of the proof and numerous examples of its advantages.
Categories: Mathematics

Proof Theory

Proof Theory

A Selection of Papers from the Leeds Proof Theory Programme 1990 Leeds
Proof Theory Programme (1990 Leeds ... The original BASIC axioms ( the
BBASIC axioms ) were defined to serve as a base theory for a number of theories
of ...

Author: Leeds Proof Theory Programme (1990 Leeds University)

Publisher: Cambridge University Press

ISBN: 052141413X

Category: Computers

Page: 306

View: 972

A collection of expository and research articles derived from the SERC 'Logic for IT' Summer School and Conference on Proof Theory.
Categories: Computers

Proof Theory

Proof Theory

Archiv f ̈ur Mathematische Logik und Grundlagenforschung 22, 69–87 (1982)
Pohlers, W.: Proof Theory. An Introduction. No. 1407 in Lecture ... 867–895
Schwichtenberg, H., Troelstra, A.: Basic Proof Theory. No. 43 in Cambridge
Tracts in ...

Author: Wolfram Pohlers

Publisher: Springer Science & Business Media

ISBN: 354069319X

Category: Mathematics

Page: 374

View: 327

The kernel of this book consists of a series of lectures on in?nitary proof theory which I gave during my time at the Westfalische ̈ Wilhelms–Universitat ̈ in Munster ̈ . It was planned as a successor of Springer Lecture Notes in Mathematics 1407. H- ever, when preparing it, I decided to also include material which has not been treated in SLN 1407. Since the appearance of SLN 1407 many innovations in the area of - dinal analysis have taken place. Just to mention those of them which are addressed in this book: Buchholz simpli?ed local predicativity by the invention of operator controlled derivations (cf. Chapter 9, Chapter 11); Weiermann detected applications of methods of impredicative proof theory to the characterization of the provable recursive functions of predicative theories (cf. Chapter 10); Beckmann improved Gentzen’s boundedness theorem (which appears as Stage Theorem (Theorem 6. 6. 1) in this book) to Theorem 6. 6. 9, a theorem which is very satisfying in itself - though its real importance lies in the ordinal analysis of systems, weaker than those treated here. Besides these innovations I also decided to include the analysis of the theory (? –REF) as an example of a subtheory of set theory whose ordinal analysis only 2 0 requires a ?rst step into impredicativity. The ordinal analysis of(? –FXP) of non- 0 1 0 monotone? –de?nable inductive de?nitions in Chapter 13 is an application of the 1 analysis of(? –REF).
Categories: Mathematics

Advances in Proof Theory

Advances in Proof Theory

As far as we know, the proofs for S4 and S4.3 are new, and avoid the
complexities of previous proofs for these logics. ... We assume the reader is
familiar with basic proof-theory and higher-order logic, but assume that the
reader is a novice in ...

Author: Reinhard Kahle

Publisher: Birkhäuser

ISBN: 9783319291987

Category: Mathematics

Page: 425

View: 858

The aim of this volume is to collect original contributions by the best specialists from the area of proof theory, constructivity, and computation and discuss recent trends and results in these areas. Some emphasis will be put on ordinal analysis, reductive proof theory, explicit mathematics and type-theoretic formalisms, and abstract computations. The volume is dedicated to the 60th birthday of Professor Gerhard Jäger, who has been instrumental in shaping and promoting logic in Switzerland for the last 25 years. It comprises contributions from the symposium “Advances in Proof Theory”, which was held in Bern in December 2013. ​Proof theory came into being in the twenties of the last century, when it was inaugurated by David Hilbert in order to secure the foundations of mathematics. It was substantially influenced by Gödel's famous incompleteness theorems of 1930 and Gentzen's new consistency proof for the axiom system of first order number theory in 1936. Today, proof theory is a well-established branch of mathematical and philosophical logic and one of the pillars of the foundations of mathematics. Proof theory explores constructive and computational aspects of mathematical reasoning; it is particularly suitable for dealing with various questions in computer science.
Categories: Mathematics

An Introduction to Proofs with Set Theory

An Introduction to Proofs with Set Theory

This text is intended as an introduction to mathematical proofs for students.

Author: Daniel Ashlock

Publisher: Morgan & Claypool Publishers

ISBN: 9781681738802

Category: Mathematics

Page: 249

View: 539

This text is intended as an introduction to mathematical proofs for students. It is distilled from the lecture notes for a course focused on set theory subject matter as a means of teaching proofs. Chapter 1 contains an introduction and provides a brief summary of some background material students may be unfamiliar with. Chapters 2 and 3 introduce the basics of logic for students not yet familiar with these topics. Included is material on Boolean logic, propositions and predicates, logical operations, truth tables, tautologies and contradictions, rules of inference and logical arguments. Chapter 4 introduces mathematical proofs, including proof conventions, direct proofs, proof-by-contradiction, and proof-by-contraposition. Chapter 5 introduces the basics of naive set theory, including Venn diagrams and operations on sets. Chapter 6 introduces mathematical induction and recurrence relations. Chapter 7 introduces set-theoretic functions and covers injective, surjective, and bijective functions, as well as permutations. Chapter 8 covers the fundamental properties of the integers including primes, unique factorization, and Euclid's algorithm. Chapter 9 is an introduction to combinatorics; topics included are combinatorial proofs, binomial and multinomial coefficients, the Inclusion-Exclusion principle, and counting the number of surjective functions between finite sets. Chapter 10 introduces relations and covers equivalence relations and partial orders. Chapter 11 covers number bases, number systems, and operations. Chapter 12 covers cardinality, including basic results on countable and uncountable infinities, and introduces cardinal numbers. Chapter 13 expands on partial orders and introduces ordinal numbers. Chapter 14 examines the paradoxes of naive set theory and introduces and discusses axiomatic set theory. This chapter also includes Cantor's Paradox, Russel's Paradox, a discussion of axiomatic theories, an exposition on Zermelo‒Fraenkel Set Theory with the Axiom of Choice, and a brief explanation of Gödel's Incompleteness Theorems.
Categories: Mathematics

Proof Theory in Computer Science

Proof Theory in Computer Science

Jesse Alt made a major contribution to finding the proof of the strong
normalization theorem λ∞ and wrote most of sections 2.2, 3. ... 407-474, 1998
A.S. Troelstra and H. Schwichtenberg, Basic Proof Theory, Cambridge University
Press, 1996.

Author: Reinhard Kahle

Publisher: Springer

ISBN: 9783540455042

Category: Computers

Page: 246

View: 662

Proof theory has long been established as a basic discipline of mathematical logic. It has recently become increasingly relevant to computer science. The - ductive apparatus provided by proof theory has proved useful for metatheoretical purposes as well as for practical applications. Thus it seemed to us most natural to bring researchers together to assess both the role proof theory already plays in computer science and the role it might play in the future. The form of a Dagstuhl seminar is most suitable for purposes like this, as Schloß Dagstuhl provides a very convenient and stimulating environment to - scuss new ideas and developments. To accompany the conference with a proc- dings volume appeared to us equally appropriate. Such a volume not only ?xes basic results of the subject and makes them available to a broader audience, but also signals to the scienti?c community that Proof Theory in Computer Science (PTCS) is a major research branch within the wider ?eld of logic in computer science.
Categories: Computers

Type Theory and Formal Proof

Type Theory and Formal Proof

A gentle introduction for graduate students and researchers in the art of formalizing mathematics on the basis of type theory.

Author: Rob Nederpelt

Publisher: Cambridge University Press

ISBN: 9781107036505

Category: Computers

Page: 490

View: 886

A gentle introduction for graduate students and researchers in the art of formalizing mathematics on the basis of type theory.
Categories: Computers

Bounded Arithmetic Propositional Logic and Complexity Theory

Bounded Arithmetic  Propositional Logic and Complexity Theory

Discusses the deep connections between logic and complexity theory, and lists a number of intriguing open problems.

Author: Jan Krajicek

Publisher: Cambridge University Press

ISBN: 0521452058

Category: Mathematics

Page: 343

View: 143

Discusses the deep connections between logic and complexity theory, and lists a number of intriguing open problems.
Categories: Mathematics

Proofs and Fundamentals

Proofs and Fundamentals

Reviews of the first edition: This is a well-written book, based on very sound pedagogical ideas. It would be an excellent choice as a textbook for a 'transition' course. —Zentralblatt Math 'Proofs and Fundamentals' has many strengths.

Author: Ethan D. Bloch

Publisher: Springer Science & Business Media

ISBN: 1441971270

Category: Mathematics

Page: 358

View: 559

“Proofs and Fundamentals: A First Course in Abstract Mathematics” 2nd edition is designed as a "transition" course to introduce undergraduates to the writing of rigorous mathematical proofs, and to such fundamental mathematical ideas as sets, functions, relations, and cardinality. The text serves as a bridge between computational courses such as calculus, and more theoretical, proofs-oriented courses such as linear algebra, abstract algebra and real analysis. This 3-part work carefully balances Proofs, Fundamentals, and Extras. Part 1 presents logic and basic proof techniques; Part 2 thoroughly covers fundamental material such as sets, functions and relations; and Part 3 introduces a variety of extra topics such as groups, combinatorics and sequences. A gentle, friendly style is used, in which motivation and informal discussion play a key role, and yet high standards in rigor and in writing are never compromised. New to the second edition: 1) A new section about the foundations of set theory has been added at the end of the chapter about sets. This section includes a very informal discussion of the Zermelo– Fraenkel Axioms for set theory. We do not make use of these axioms subsequently in the text, but it is valuable for any mathematician to be aware that an axiomatic basis for set theory exists. Also included in this new section is a slightly expanded discussion of the Axiom of Choice, and new discussion of Zorn's Lemma, which is used later in the text. 2) The chapter about the cardinality of sets has been rearranged and expanded. There is a new section at the start of the chapter that summarizes various properties of the set of natural numbers; these properties play important roles subsequently in the chapter. The sections on induction and recursion have been slightly expanded, and have been relocated to an earlier place in the chapter (following the new section), both because they are more concrete than the material found in the other sections of the chapter, and because ideas from the sections on induction and recursion are used in the other sections. Next comes the section on the cardinality of sets (which was originally the first section of the chapter); this section gained proofs of the Schroeder–Bernstein theorem and the Trichotomy Law for Sets, and lost most of the material about finite and countable sets, which has now been moved to a new section devoted to those two types of sets. The chapter concludes with the section on the cardinality of the number systems. 3) The chapter on the construction of the natural numbers, integers and rational numbers from the Peano Postulates was removed entirely. That material was originally included to provide the needed background about the number systems, particularly for the discussion of the cardinality of sets, but it was always somewhat out of place given the level and scope of this text. The background material about the natural numbers needed for the cardinality of sets has now been summarized in a new section at the start of that chapter, making the chapter both self-contained and more accessible than it previously was. 4) The section on families of sets has been thoroughly revised, with the focus being on families of sets in general, not necessarily thought of as indexed. 5) A new section about the convergence of sequences has been added to the chapter on selected topics. This new section, which treats a topic from real analysis, adds some diversity to the chapter, which had hitherto contained selected topics of only an algebraic or combinatorial nature. 6) A new section called ``You Are the Professor'' has been added to the end of the last chapter. This new section, which includes a number of attempted proofs taken from actual homework exercises submitted by students, offers the reader the opportunity to solidify her facility for writing proofs by critiquing these submissions as if she were the instructor for the course. 7) All known errors have been corrected. 8) Many minor adjustments of wording have been made throughout the text, with the hope of improving the exposition.
Categories: Mathematics

Notes on Logic and Set Theory

Notes on Logic and Set Theory

Consequently the book, while making an attractive first textbook for those who plan to specialise in logic, will be particularly valuable for mathematics and computer scientists whose primary interests lie elsewhere.

Author: P. T. Johnstone

Publisher: Cambridge University Press

ISBN: 0521336929

Category: Mathematics

Page: 110

View: 453

This short textbook provides a succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. It will be suitable for all mathematics undergraduates coming to the subject for the first time. The book is based on lectures given at the University of Cambridge and covers the basic concepts of logic: first order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. There are also chapters on recursive functions, the axiom of choice, ordinal and cardinal arithmetic and the incompleteness theorems. Dr Johnstone has included numerous exercises designed to illustrate the key elements of the theory and to provide applications of basic logical concepts to other areas of mathematics. Consequently the book, while making an attractive first textbook for those who plan to specialise in logic, will be particularly valuable for mathematics and computer scientists whose primary interests lie elsewhere.
Categories: Mathematics

Structural Proof Theory

Structural Proof Theory

It works both as a proof editor and as an automatic theorem prover. Proofs
constructed in PESCA can both be seen on the terminal and printed into ETEX
files. The user of PESCA can choose among different versions of classical and ...

Author: Sara Negri

Publisher: Cambridge University Press

ISBN: 0521068428

Category: Mathematics

Page: 276

View: 521

A concise introduction to structural proof theory, a branch of logic studying the general structure of logical and mathematical proofs.
Categories: Mathematics

Popular Lectures on Mathematical Logic

Popular Lectures on Mathematical Logic

Noted logician discusses both theoretical underpinnings and practical applications, exploring set theory, model theory, recursion theory and constructivism, proof theory, logic's relation to computer science, and other subjects. 1981 ...

Author: Hao Wang

Publisher: Courier Corporation

ISBN: 9780486171043

Category: Mathematics

Page: 292

View: 945

Noted logician discusses both theoretical underpinnings and practical applications, exploring set theory, model theory, recursion theory and constructivism, proof theory, logic's relation to computer science, and other subjects. 1981 edition, reissued by Dover in 1993 with a new Postscript by the author.
Categories: Mathematics

The Art of Proof

The Art of Proof

The book ends with short essays on further topics suitable for seminar-style presentation by small teams of students, either in class or in a mathematics club setting.

Author: Matthias Beck

Publisher: Springer Science & Business Media

ISBN: 1441970231

Category: Mathematics

Page: 182

View: 661

The Art of Proof is designed for a one-semester or two-quarter course. A typical student will have studied calculus (perhaps also linear algebra) with reasonable success. With an artful mixture of chatty style and interesting examples, the student's previous intuitive knowledge is placed on solid intellectual ground. The topics covered include: integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, and uncountable sets. Methods, such as axiom, theorem and proof, are taught while discussing the mathematics rather than in abstract isolation. The book ends with short essays on further topics suitable for seminar-style presentation by small teams of students, either in class or in a mathematics club setting. These include: continuity, cryptography, groups, complex numbers, ordinal number, and generating functions.
Categories: Mathematics

A Proof Theory for Description Logics

A Proof Theory for Description Logics

The first motivation for developing such systems is the extraction of computational
content of ALC and ALCQI proofs. More precisely, these systems were developed
to allow the use of natural language to render a Natural Deduction proof.

Author: Alexandre Rademaker

Publisher: Springer Science & Business Media

ISBN: 9781447140023

Category: Mathematics

Page: 106

View: 543

Description Logics (DLs) is a family of formalisms used to represent knowledge of a domain. They are equipped with a formal logic-based semantics. Knowledge representation systems based on description logics provide various inference capabilities that deduce implicit knowledge from the explicitly represented knowledge. A Proof Theory for Description Logics introduces Sequent Calculi and Natural Deduction for some DLs (ALC, ALCQ). Cut-elimination and Normalization are proved for the calculi. The author argues that such systems can improve the extraction of computational content from DLs proofs for explanation purposes.
Categories: Mathematics

Logics of Proofs and Justifications

Logics of Proofs and Justifications

Its position at the junction of mathematics, philosophy, and computer science makes it of interest to a wide audience. This book provides a rigorous introduction to justification logic.

Author: Roman Kuznets

Publisher:

ISBN: 1848901682

Category:

Page: 246

View: 373

Justification logics are closely related to modal logics and can be viewed as a refinement of the latter with machinery for justification manipulation. Justifications are represented directly in the language by terms, which can be interpreted as formal proofs in a deductive system, evidence for knowledge, winning strategy in a game, etc. This more expressive language proved beneficial in both proof theory and epistemology and helped investigate problems ranging from a classical provability semantics for intuitionistic logic to the logical omniscience problem. Justification logic is a new and fast evolving field that offers unexpected new approaches and insights into old problems. Its position at the junction of mathematics, philosophy, and computer science makes it of interest to a wide audience. This book provides a rigorous introduction to justification logic. It covers the basic constructions of justification logic as well as epistemic models and provability semantics. Further it includes chapters on decidability and complexity of justification logics as well as a chapter on self-referentiality. It also contains detailed historic remarks on the subject.
Categories:

Proof Theory for Fuzzy Logics

Proof Theory for Fuzzy Logics

This book provides an accessible and up-to-date introduction to this fast-growing and increasingly popular area.

Author: George Metcalfe

Publisher: Springer Science & Business Media

ISBN: 9781402094095

Category: Mathematics

Page: 276

View: 821

Fuzzy logics are many-valued logics that are well suited to reasoning in the context of vagueness. They provide the basis for the wider field of Fuzzy Logic, encompassing diverse areas such as fuzzy control, fuzzy databases, and fuzzy mathematics. This book provides an accessible and up-to-date introduction to this fast-growing and increasingly popular area. It focuses in particular on the development and applications of "proof-theoretic" presentations of fuzzy logics; the result of more than ten years of intensive work by researchers in the area, including the authors. In addition to providing alternative elegant presentations of fuzzy logics, proof-theoretic methods are useful for addressing theoretical problems (including key standard completeness results) and developing efficient deduction and decision algorithms. Proof-theoretic presentations also place fuzzy logics in the broader landscape of non-classical logics, revealing deep relations with other logics studied in Computer Science, Mathematics, and Philosophy. The book builds methodically from the semantic origins of fuzzy logics to proof-theoretic presentations such as Hilbert and Gentzen systems, introducing both theoretical and practical applications of these presentations.
Categories: Mathematics