Lectures on the Curry-Howard isomorphism / Morten Heine Srensen, Pawe Urzyczyn.

Format:

Book

Author/Creator:

Srensen, Morten Heine.

Contributor:

Urzyczyn, Pawe.

Series:

Studies in logic and the foundations of mathematics ; v. 149.

Studies in logic and the foundations of mathematics, 0049-237X ; v. 149

Language:

English

Subjects (All):

Curry-Howard isomorphism.

Lambda calculus.

Proof theory.

Physical Description:

1 online resource (457 p.)

Edition:

1st ed.

Place of Publication:

Amsterdam ; Boston [MA] : Elsevier, 2006.

Language Note:

English

Summary:

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,

Contents:

Cover; Title Page; Copyright Page; Table of Contents; Chapter 1 Type-free Lambda-calculus; 1.1 A gentle introduction; 1.2 Pre-terms and Lambda-terms; 1.3 Reduction; 1.4 The Church-Rosser theorem; 1.5 Leftmost reductions are normalizing; 1.6 Perpetual reductions and the conservation theorem; 1.7 Expressibility and undeeidability; 1.8 Notes; 1.9 Exercises; Chapter 2 Intultionistic logic; 2.1 The BHK interpretation; 2.2 Natural deduction; 2.3 Algebraic semantics of classical logic; 2.4 Heyting algebras; 2.5 Kripke semantics; 2.6 The implicational fragment; 2.7 Notes; 2.8 Exercises

Chapter 3 Simply typed Lambda-calcuIus3.1 Simply typed Lambda-ealculus a la Curry; 3.2 Type reconstruction algorithm; 3.3 Simply typed Lambda-calculus a la Church; 3.4 Church versus Curry typing; 3.5 Normalization; 3.6 Church-Rosser property; 3.7 Expressibility; 3.8 Notes; 3.9 Exercises; Chapter 4 The Curry-Howard isomorphism; 4.1 Proofs and terms; 4.2 Type inhabitation; 4.3 Not an exact isomorphism; 4.4 Proof normalization; 4.5 Sums and products; 4.6 Prover-skeptic dialogues; 4.7 Prover-skeptic dialogues with absurdity; 4.8 Notes; 4.9 Exercises; Chapter 5 Proofs as combinators

5.1 Hubert style proofs5.2 Combinatory logic; 5.3 Typed combinators; 5.4 Combinators versus lambda terms; 5.5 Extensionality; 5.6 Relevance and linearity; 5.7 Notes; 5.8 Exercises; Chapter 6 Classical logic and control operators; 6.1 Classical prepositional lope; 6.2 The Lambda meu-calculus; 6.3 Subject reduction, confluence, strong normalization; 6.4 Logical embedding and CPS translation; 6.5 Classical prover-skeptic dialogues; 6.6 The pure implicational fragment; 6.7 Conjunction and disjunction; 6.8 Notes; 6.9 Exercises; Chapter 7 Sequent calculus; 7.1 Gentzen's sequent calculus LK

7.2 Fragments of LK versus natural deduction7.3 Gentzen's Hauptaatz; 7.4 Cut elimination versus normalization; 7.5 Lorenzen dialogues; 7.6 Notes; 7.7 Exercises; Chapter 8 First-order logic; 8.1 Syntax of first-order logic; 8.2 Informal semantics; 8.3 Proof systems; 8.4 Classical semantics; 8.5 Algebraic semantics of intuitionistie logic; 8.6 Kripke semantics; 8.7 Lambda-calculus; 8.8 Undeddability; 8.9 Notes; 8.10 Exercises; Chapter 9 First-order arithmetic; 9.1 The language of arithmetic; 9.2 Peano Arithmetic; 9.3 Godel's theorems; 9.4 Representable and provably recursive functions

9.5 Heyting Arithmetic9.6 Kleene's realfeability interpretation; 9.7 Notes; 9.8 Exercises; Chapter 10 Godel's system T; 10.1 From Heyting Arithmetic to system T; 10.2 Syntax; 10.3 Strong normalization; 10.4 Modified realizability; 10.5 Notes; 10.6 Exercises; Chapter 11 Second-order logic and polymorphism; 11.1 Prepositional second-order logic; 11.2 Polymorphic lambda-calculus (system F); 11.3 Expressive power; 11.4 Gurry-style polymorphism; 11.5 Strong normalization; 11.6 The inhabitation problem; 11.7 Higher-order polymorphism; 11.8 Notes; 11.9 Exercises

Chapter 12 Second-order arithmetic

Notes:

Description based upon print version of record.

Includes bibliographical references (p. 403-430) and index.

ISBN:

1-281-05105-5

9786611051051

0-08-047892-1

OCLC:

469589380