Derivation and Computation: Taking the Curry-Howard Correspondence Seriously

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Cambridge University Press, May 18, 2000 - Mathematics - 384 pages
Mathematics is about proofs, that is the derivation of correct statements; and calculations, that is the production of results according to well-defined sets of rules. The two notions are intimately related. Proofs can involve calculations, and the algorithm underlying a calculation should be proved correct. The aim of the author is to explore this relationship. The book itself forms an introduction to simple type theory. Starting from the familiar propositional calculus the author develops the central idea of an applied lambda-calculus. This is illustrated by an account of Gödel's T, a system which codifies number-theoretic function hierarchies. Each of the book's 52 sections ends with a set of exercises, some 200 in total. These are designed to help the reader get to grips with the subject, and develop a further understanding. An appendix contains complete solutions of these exercises.
 

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Contents

DERIVATION SYSTEMS
11
THE TYPED XCALCULUS
67
SUBSTITUTION ALGORITHMS
82
APPLIED XCALCULI
100
A DERIVATION SYSTEMS
215
COMPUTATION MECHANISMS
234
Exercises
245
THE TYPED COMBINATOR CALCULUS
249
E SUBSTITUTION ALGORITHMS
279
F APPLIED XCALCULI
290
MULTIRECURSIVE ARITHMETIC
318
ORDINALS AND ORDINAL NOTATIONS
341
HIGHER ORDER RECURSION
355
POSTVIEW
371
COMMONLY USED SYMBOLS
377
Copyright

THE TYPED XCALCULUS
264

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