Aspects of the Computational Content of Proofs PDF Download

Author: Judith Lynne Underwood
Publisher:
ISBN:
Category :
Languages : en
Pages : 154

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Author: Pavol Safarik
Publisher: Logos Verlag Berlin GmbH
ISBN: 3832537651
Category : Computers
Languages : en
Pages : 164

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The Fields medalist, Terence Tao, recently emphasized the importance of ''hard'' (or finitary) analysis and connected the finitisation to the methods we will employ in this thesis: ... The main advantage of working in a finitary setting ... is that the underlying dynamical system becomes extremely explicit. ... In proof theory, this finitisation is known as Gödel functional interpretation ... For convergence theorems Tao calls the finitary formulation metastability and the corresponding explicit content its rate(s). In the case of the mean ergodic theorem such a rate can be used to obtain even an effective bound on the number of fluctuations. We introduce effective learnability and three other natural kinds of such finitary information and analyze the corresponding proof-theoretic conditions. Effective learnability not only provides means to know when to expect a bound on the number of fluctuations but also explains a very common pattern in the realizers for strong ergodic theorems. Moreover, we will see how a most natural example for a non-learnable convergence theorem closely relates to a notable exception to this pattern, the strong nonlinear ergodic theorem due to Wittmann. Finally, we show how can computational content be extracted in the context of non-standard analysis.

Author: Richard H. Hammack
Publisher:
ISBN: 9780989472111
Category : Mathematics
Languages : en
Pages : 314

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This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.

Author: Sanjeev Arora
Publisher: Cambridge University Press
ISBN: 0521424267
Category : Computers
Languages : en
Pages : 609

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New and classical results in computational complexity, including interactive proofs, PCP, derandomization, and quantum computation. Ideal for graduate students.

Author: Helmut Schwichtenberg
Publisher: Cambridge University Press
ISBN: 1139504169
Category : Mathematics
Languages : en
Pages : 480

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Book Description
Driven by the question, 'What is the computational content of a (formal) proof?', this book studies fundamental interactions between proof theory and computability. It provides a unique self-contained text for advanced students and researchers in mathematical logic and computer science. Part I covers basic proof theory, computability and Gödel's theorems. Part II studies and classifies provable recursion in classical systems, from fragments of Peano arithmetic up to Π11–CA0. Ordinal analysis and the (Schwichtenberg–Wainer) subrecursive hierarchies play a central role and are used in proving the 'modified finite Ramsey' and 'extended Kruskal' independence results for PA and Π11–CA0. Part III develops the theoretical underpinnings of the first author's proof assistant MINLOG. Three chapters cover higher-type computability via information systems, a constructive theory TCF of computable functionals, realizability, Dialectica interpretation, computationally significant quantifiers and connectives and polytime complexity in a two-sorted, higher-type arithmetic with linear logic.

Author: Joel David Hamkins
Publisher: MIT Press
ISBN: 0262542234
Category : Mathematics
Languages : en
Pages : 350

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An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, and set theory--that give rise again and again to philosophical considerations.

Author: Morten Heine Sørensen
Publisher: Elsevier
ISBN: 0080478921
Category : Mathematics
Languages : en
Pages : 457

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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: Helmut Schwichtenberg
Publisher: Springer Science & Business Media
ISBN: 3642590489
Category : Computers
Languages : en
Pages : 405

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The Marktoberdorf Summer School 1995 'Logic of Computation' was the 16th in a series of Advanced Study Institutes under the sponsorship of the NATO Scientific Affairs Division held in Marktoberdorf. Its scientific goal was to survey recent progress on the impact of logical methods in software development. The courses dealt with many different aspects of this interplay, where major progress has been made. Of particular importance were the following. • The proofs-as-programs paradigm, which makes it possible to extract verified programs directly from proofs. Here a higher order logic or type theoretic setup of the underlying language has developed into a standard. • Extensions of logic programming, e.g. by allowing more general formulas and/or higher order languages. • Proof theoretic methods, which provide tools to deal with questions of feasibility of computations and also to develop a general mathematical understanding of complexity questions. • Rewrite systems and unification, again in a higher order context. Closely related is the now well-established Grabner basis theory, which recently has found interesting applications. • Category theoretic and more generally algebraic methods and techniques to analyze the semantics of programming languages. All these issues were covered by a team of leading researchers. Their courses were grouped under the following headings.

Author: Martin Aigner
Publisher: Springer Science & Business Media
ISBN: 3662223430
Category : Mathematics
Languages : en
Pages : 194

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According to the great mathematician Paul Erdös, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such "perfect proofs," those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, analysis, combinatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics.

Author: Neil V. Murray
Publisher: Springer
ISBN: 3540487549
Category : Computers
Languages : en
Pages : 343

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This book constitutes the refereed proceedings of the International Conference on Analytic Tableaux and Related Methods, TABLEAUX'99, held in Saratoga Springs, NY, USA, in June 1999. The volume presents 18 revised full papers and three system descriptions selected from 41 submissions. Also included are system comparisons and abstracts of an invited paper and of two tutorials. All current issues surrounding mechanization of reasoning with tableaux and similar methods are addressed - ranging from theoretical foundations to implementation and systems development and applications, as well as covering a broad variety of logic calculi. As application areas, formal verification of software and computer systems, deductive databases, knowledge representation, and systems diagnosis are covered.