Proof And Computation Ii From Proof Theory And Univalent Mathematics To Program Extraction And Verification

Author : Klaus Mainzer,Helmut Schwichtenberg,Peter Michael Schuster
Publisher : World Scientific
Page : 425 pages
File Size : 46,7 Mb
Release : 2021-07-27
Category : Mathematics
ISBN : 9789811236495

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Proof And Computation Ii: From Proof Theory And Univalent Mathematics To Program Extraction And Verification by Klaus Mainzer,Helmut Schwichtenberg,Peter Michael Schuster Pdf

This book is for graduate students and researchers, introducing modern foundational research in mathematics, computer science, and philosophy from an interdisciplinary point of view. Its scope includes proof theory, constructive mathematics and type theory, univalent mathematics and point-free approaches to topology, extraction of certified programs from proofs, automated proofs in the automotive industry, as well as the philosophical and historical background of proof theory. By filling the gap between (under-)graduate level textbooks and advanced research papers, the book gives a scholarly account of recent developments and emerging branches of the aforementioned fields.

Author : Klaus Mainzer,Stefania Centrone
Publisher : World Scientific
Page : 221 pages
File Size : 42,5 Mb
Release : 2023-05-12
Category : Mathematics
ISBN : 9789811268557

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Temporal Logic: From Philosophy And Proof Theory To Artificial Intelligence And Quantum Computing by Klaus Mainzer,Stefania Centrone Pdf

Calculi of temporal logic are widely used in modern computer science. The temporal organization of information flows in the different architectures of laptops, the Internet, or supercomputers would not be possible without appropriate temporal calculi. In the age of digitalization and High-Tech applications, people are often not aware that temporal logic is deeply rooted in the philosophy of modalities. A deep understanding of these roots opens avenues to the modern calculi of temporal logic which have emerged by extension of modal logic with temporal operators. Computationally, temporal operators can be introduced in different formalisms with increasing complexity such as Basic Modal Logic (BML), Linear-Time Temporal Logic (LTL), Computation Tree Logic (CTL), and Full Computation Tree Logic (CTL*). Proof-theoretically, these formalisms of temporal logic can be interpreted by the sequent calculus of Gentzen, the tableau-based calculus, automata-based calculus, game-based calculus, and dialogue-based calculus with different advantages for different purposes, especially in computer science.The book culminates in an outlook on trendsetting applications of temporal logics in future technologies such as artificial intelligence and quantum technology. However, it will not be sufficient, as in traditional temporal logic, to start from the everyday understanding of time. Since the 20th century, physics has fundamentally changed the modern understanding of time, which now also determines technology. In temporal logic, we are only just beginning to grasp these differences in proof theory which needs interdisciplinary cooperation of proof theory, computer science, physics, technology, and philosophy.

Author : Mainzer Klaus,Schwichtenberg Helmut,Schuster Peter Michael
Publisher : World Scientific
Page : 300 pages
File Size : 44,7 Mb
Release : 2018-05-30
Category : Mathematics
ISBN : 9789813270954

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Proof And Computation: Digitization In Mathematics, Computer Science And Philosophy by Mainzer Klaus,Schwichtenberg Helmut,Schuster Peter Michael Pdf

This book is for graduate students and researchers, introducing modern foundational research in mathematics, computer science, and philosophy from an interdisciplinary point of view. Its scope includes Predicative Foundations, Constructive Mathematics and Type Theory, Computation in Higher Types, Extraction of Programs from Proofs, and Algorithmic Aspects in Financial Mathematics. By filling the gap between (under-)graduate level textbooks and advanced research papers, the book gives a scholarly account of recent developments and emerging branches of the aforementioned fields. Contents: Proof and Computation (K Mainzer) Constructive Convex Programming (J Berger and G Svindland) Exploring Predicativity (L Crosilla) Constructive Functional Analysis: An Introduction (H Ishihara) Program Extraction (K Miyamoto) The Data Structures of the Lambda Terms (M Sato) Provable (and Unprovable) Computability (S Wainer) Introduction to Minlog (F Wiesnet) Readership: Graduate students, researchers, and professionals in Mathematics and Computer Science. Keywords: Proof Theory;Computability Theory;Program Extraction;Constructive Analysis;PredicativityReview: Key Features: This book gathers recent contributions of distinguished experts It makes emerging fields accessible to a wider audience, appealing to a broad readership with diverse backgrounds It fills a gap between (under-)graduate level textbooks and state-of-the-art research papers

Author : Dieter Probst,Peter Schuster
Publisher : Walter de Gruyter GmbH & Co KG
Page : 384 pages
File Size : 42,6 Mb
Release : 2016-07-25
Category : Philosophy
ISBN : 9781501502620

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Concepts of Proof in Mathematics, Philosophy, and Computer Science by Dieter Probst,Peter Schuster Pdf

A proof is a successful demonstration that a conclusion necessarily follows by logical reasoning from axioms which are considered evident for the given context and agreed upon by the community. It is this concept that sets mathematics apart from other disciplines and distinguishes it as the prototype of a deductive science. Proofs thus are utterly relevant for research, teaching and communication in mathematics and of particular interest for the philosophy of mathematics. In computer science, moreover, proofs have proved to be a rich source for already certified algorithms. This book provides the reader with a collection of articles covering relevant current research topics circled around the concept 'proof'. It tries to give due consideration to the depth and breadth of the subject by discussing its philosophical and methodological aspects, addressing foundational issues induced by Hilbert's Programme and the benefits of the arising formal notions of proof, without neglecting reasoning in natural language proofs and applications in computer science such as program extraction.

Author : Helmut Schwichtenberg,Stanley S. Wainer
Publisher : Cambridge University Press
Page : 480 pages
File Size : 47,5 Mb
Release : 2011-12-15
Category : Mathematics
ISBN : 9781139504164

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Proofs and Computations by Helmut Schwichtenberg,Stanley S. Wainer Pdf

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 : Jacob T. Schwartz,Domenico Cantone,Eugenio G. Omodeo
Publisher : Springer Science & Business Media
Page : 426 pages
File Size : 46,6 Mb
Release : 2011-07-16
Category : Computers
ISBN : 9780857298089

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Computational Logic and Set Theory by Jacob T. Schwartz,Domenico Cantone,Eugenio G. Omodeo Pdf

This must-read text presents the pioneering work of the late Professor Jacob (Jack) T. Schwartz on computational logic and set theory and its application to proof verification techniques, culminating in the ÆtnaNova system, a prototype computer program designed to verify the correctness of mathematical proofs presented in the language of set theory. Topics and features: describes in depth how a specific first-order theory can be exploited to model and carry out reasoning in branches of computer science and mathematics; presents an unique system for automated proof verification in large-scale software systems; integrates important proof-engineering issues, reflecting the goals of large-scale verifiers; includes an appendix showing formalized proofs of ordinals, of various properties of the transitive closure operation, of finite and transfinite induction principles, and of Zorn’s lemma.

Author : Gila Hanna,David Alexander Reid,Michael D. De Villiers
Publisher : Unknown
Page : 379 pages
File Size : 47,8 Mb
Release : 2019
Category : Electronic books
ISBN : 3030284840

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Proof Technology in Mathematics Research and Teaching by Gila Hanna,David Alexander Reid,Michael D. De Villiers Pdf

This book presents chapters exploring the most recent developments in the role of technology in proving. The full range of topics related to this theme are explored, including computer proving, digital collaboration among mathematicians, mathematics teaching in schools and universities, and the use of the internet as a site of proof learning. Proving is sometimes thought to be the aspect of mathematical activity most resistant to the influence of technological change. While computational methods are well known to have a huge importance in applied mathematics, there is a perception that mathematicians seeking to derive new mathematical results are unaffected by the digital era. The reality is quite different. Digital technologies have transformed how mathematicians work together, how proof is taught in schools and universities, and even the nature of proof itself. Checking billions of cases in extremely large but finite sets, impossible a few decades ago, has now become a standard method of proof. Distributed proving, by teams of mathematicians working independently on sections of a problem, has become very much easier as digital communication facilitates the sharing and comparison of results. Proof assistants and dynamic proof environments have influenced the verification or refutation of conjectures, and ultimately how and why proof is taught in schools. And techniques from computer science for checking the validity of programs are being used to verify mathematical proofs. Chapters in this book include not only research reports and case studies, but also theoretical essays, reviews of the state of the art in selected areas, and historical studies. The authors are experts in the field.

Author : Wolfram Pohlers
Publisher : Unknown
Page : 228 pages
File Size : 44,6 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 3662178974

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Proof Theory by Wolfram Pohlers Pdf

Author : Michael M. Richter
Publisher : Springer
Page : 492 pages
File Size : 40,7 Mb
Release : 1984
Category : Mathematics
ISBN : PSU:000021096631

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Computation and Proof Theory by Michael M. Richter Pdf

Author : Vincent F. Hendricks,Stig Andur Pedersen,Klaus Frovin Jorgensen
Publisher : Unknown
Page : 272 pages
File Size : 50,5 Mb
Release : 2014-01-15
Category : Electronic
ISBN : 940172797X

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Proof Theory by Vincent F. Hendricks,Stig Andur Pedersen,Klaus Frovin Jorgensen Pdf

Author : Anonim
Publisher : Univalent Foundations
Page : 484 pages
File Size : 51,9 Mb
Release : 2024-06-28
Category : Electronic
ISBN : 8210379456XXX

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Homotopy Type Theory: Univalent Foundations of Mathematics by Anonim Pdf

Author : Grigorii E. Mints
Publisher : North-Holland
Page : 294 pages
File Size : 49,7 Mb
Release : 1992-01-01
Category : Electronic
ISBN : 0444896198

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Selected Papers in Proof Theory by Grigorii E. Mints Pdf

This collection includes papers devoted to the structural theory of proofs, which was born in the framework of Hilbert's program and is applied now in connection with various projects using the effective contents of formalized proofs. The main tool and unifying topic here is normalization, i.e. putting proofs into a normal form. The book presupposes some familiarity with the definition and elementary properties of Gentzen-type systems but little more. The first three papers introduce various normalization procedures different from popular ones. The next group deals with unwinding proofs, that is the extraction of an explicit realization from the proof of existential theorems. Normalization (or more precisely, normal form theorems) is applied to the solution of some problems in the following two papers. A separate group is formed by three papers dealing with applications of the theory of proofs to algebra, more specifically to coherence theorems in category theory. The last paper of the volume is a survey of proof theory and elementary model theory for modal logic up to the year 1974.

Author : Samuel Mimram
Publisher : Unknown
Page : 539 pages
File Size : 53,5 Mb
Release : 2020-07-03
Category : Electronic
ISBN : 9798615591839

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Program = Proof by Samuel Mimram Pdf

This course provides a first introduction to the Curry-Howard correspondence between programs and proofs, from a theoretical programmer's perspective: we want to understand the theory behind logic and programming languages, but also to write concrete programs (in OCaml) and proofs (in Agda). After an introduction to functional programming languages, we present propositional logic, λ-calculus, the Curry-Howard correspondence, first-order logic, Agda, dependent types and homotopy type theory.

Author : Talia Ringer,Karl Palmskog,Ilya Sergey,Milos Gligoric,Zachary Tatlock
Publisher : Unknown
Page : 194 pages
File Size : 44,8 Mb
Release : 2019-09-03
Category : Electronic
ISBN : 1680835947

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QED at Large by Talia Ringer,Karl Palmskog,Ilya Sergey,Milos Gligoric,Zachary Tatlock Pdf

Development of formal proofs of correctness of programs can increase actual and perceived reliability and facilitate better understanding of program specifications and their underlying assumptions. Tools supporting such development have been available for over 40 years but have only recently seen wide practical use. Projects based on construction of machine-checked formal proofs are now reaching an unprecedented scale, comparable to large software projects, which leads to new challenges in proof development and maintenance. Despite its increasing importance, the field of proof engineering is seldom considered in its own right; related theories, techniques, and tools span many fields and venues. QED at Large covers the timeline and research literature concerning proof development for program verification, including theories, languages, and tools. It emphasizes challenges and breakthroughs at each stage in history and highlights challenges that are currently present due to the increasing scale of proof developments. This monograph is intended for use by researchers and students who are new to the field. It provides the reader with an insightful overview of the work that has led to modern-day techniques for formally verifying software. In times of increasing automation, this underpins many software systems so future trends are also highlighted.

Author : Michael Harris
Publisher : Princeton University Press
Page : 468 pages
File Size : 51,9 Mb
Release : 2017-05-30
Category : Mathematics
ISBN : 9780691175836

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Mathematics without Apologies by Michael Harris Pdf

An insightful reflection on the mathematical soul What do pure mathematicians do, and why do they do it? Looking beyond the conventional answers—for the sake of truth, beauty, and practical applications—this book offers an eclectic panorama of the lives and values and hopes and fears of mathematicians in the twenty-first century, assembling material from a startlingly diverse assortment of scholarly, journalistic, and pop culture sources. Drawing on his personal experiences and obsessions as well as the thoughts and opinions of mathematicians from Archimedes and Omar Khayyám to such contemporary giants as Alexander Grothendieck and Robert Langlands, Michael Harris reveals the charisma and romance of mathematics as well as its darker side. In this portrait of mathematics as a community united around a set of common intellectual, ethical, and existential challenges, he touches on a wide variety of questions, such as: Are mathematicians to blame for the 2008 financial crisis? How can we talk about the ideas we were born too soon to understand? And how should you react if you are asked to explain number theory at a dinner party? Disarmingly candid, relentlessly intelligent, and richly entertaining, Mathematics without Apologies takes readers on an unapologetic guided tour of the mathematical life, from the philosophy and sociology of mathematics to its reflections in film and popular music, with detours through the mathematical and mystical traditions of Russia, India, medieval Islam, the Bronx, and beyond.