A. S. Troelstra

Bio: A. S. Troelstra is an academic researcher from University of Amsterdam. The author has contributed to research in topics: Higher-order logic & Intuitionistic logic. The author has an hindex of 23, co-authored 63 publications receiving 4446 citations.

Metamathematical investigation of intuitionistic arithmetic and analysis

Abstract: Intuitionistic formal systems.- Models and computability.- Realizability and functional interpretations.- Normalization theorems for systems of natural deduction.- Applications of Kripke models.- Iterated inductive definitions, trees and ordinals.- Erratum.

Basic proof theory

Abstract: This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and then apply them elsewhere in much more complex settings. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to selected exercises have been included.

Constructivism In Mathematics An Introduction

Abstract: 7. The Topology of Metric Spaces. 8. Algebra. 9. Finite Type Arithmetic and Theories of Operators. 10. Proof Theory of Intuitionistic Logic. 11. The Theory of Types and Constructive Set Theory. 12. Choice Sequences. 13. Semantical Completeness. 14. Sheaves, Sites and Higher Order Logic. 15. Applications of Sheaf Models. 16. Epilogue. Bibliography. Index.

Lectures on linear logic

Abstract: 1. Introduction 2. Sequent calculus for linear logic 3. Some elementary syntactic results 4. The calculus of two implications: a digression 5. Embeddings and approximations 6. Natural deduction systems for linear logic 7. Hilbert-type systems 8. Algebraic semantics 9. Combinatorial linear logic 10. Girard domains 11. Coherence in symmetric monoidal categories 12. The storage operator as a coffee comonoid 13. Evaluation in typed calculi 14. Computation by lazy evaluation in CCC's 15. Computation by lazy evaluation in SMC's and ILC's 16. The categorical and linear machine 17. Proofnets for the multiplicative fragment 18. The algorithm of cut elimination for proof nets 19. Multiplicative operators 20. The undecidability of linear logic 21. Cut elimination and strong normalization References Index.

The collected works

Abstract: This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

Implementing Mathematics with The Nuprl Proof Development System

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The Coq proof assistant : reference manual, version 6.1

Abstract: Coq is a proof assistant based on a higher-order logic allowing powerful definitions of functions. Coq V6.1 is available by anonymous ftp at ftp.inria.fr:/INRIA/Projects/coq/V6.1 and ftp.ens-lyon.fr:/pub/LIP/COQ/V6.1

Lambda Calculus with Types

Abstract: This handbook with exercises reveals in formalisms, hitherto mainly used for hardware and software design and verification, unexpected mathematical beauty. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to Lisp, and was treated in the first author's classic The Lambda Calculus (1984). The formalism has since been extended with types and used in functional programming (Haskell, Clean) and proof assistants (Coq, Isabelle, HOL), used in designing and verifying IT products and mathematical proofs. In this book, the authors focus on three classes of typing for lambda terms: simple types, recursive types and intersection types. It is in these three formalisms of terms and types that the unexpected mathematical beauty is revealed. The treatment is authoritative and comprehensive, complemented by an exhaustive bibliography, and numerous exercises are provided to deepen the readers' understanding and increase their confidence using types.