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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The two notions of proofs and calculations are intimately related. Proofs can involve calculations, and the algorithm underlying a calculation should be proved correct. This volume explores this key relationship and introduces 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 that codifies number-theoretic function hierarchies. Each of the book's 52 sections ends with a set of exercises, some 200 in total. 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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