Derivation and Computation: Taking the Curry-Howard Correspondence Seriously

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Cambridge University Press, May 18, 2000 - Computers - 384 pages
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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
3
COMPUTATION MECHANISMS
27
THE TYPED COMBINATOR CALCULUS
48
THE TYPED ACALCULUS
67
SUBSTITUTION ALGORITHMS
82
APPLIED ACALCULI
100
MULTIRECURSIVE ARITHMETIC
137
ORDINALS AND ORDINAL NOTATIONS
170
THE TYPED COMBINATOR CALCULUS
249
THE TYPED ACALCULUS
264
E SUBSTITUTION ALGORITHMS
279
F APPLIED ACALCULI
290
G MULTIRECURSIVE ARITHMETIC
318
H ORDINALS AND ORDINAL NOTATIONS
341
HIGHER ORDER RECURSION
355
POSTVIEW
371

HIGHER ORDER RECURSION
189
A DERIVATION SYSTEMS
215
B COMPUTATION MECHANISMS
234
COMMONLY USED SYMBOLS
377
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About the author (2000)

Harold Simmons is officially retired but still active in research. He also teaches postgraduate courses in the School of Mathematics at the University of Manchester.

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