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CSE-321 Programming Languages Curry-Howard Isomorphism

CSE-321 Programming Languages Curry-Howard Isomorphism. 박성우. POSTECH June 4, 2006. ?. Truth Table. T = True, F = False. ???. Tautology. The computational meaning of. ) Truth of A is not affected by truth of B. Tautology. But what is the computational meaning of.

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CSE-321 Programming Languages Curry-Howard Isomorphism

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  1. CSE-321 Programming LanguagesCurry-Howard Isomorphism 박성우 POSTECH June 4, 2006

  2. ? Truth Table • T = True, F = False ???

  3. Tautology The computational meaning of ) Truth of A is not affected by truth of B.

  4. Tautology But what is the computational meaning of

  5. law of excluded middle law of double-negation Peirce's law Classical Logic • Concerned with: • "whether a given proposition is true or not." • Every proposition is either true or false. • Logic from omniscient God's point of view • Logic for mathematics • Tautologies in classical logic

  6. Constructive Logic • Logic from a human's point of view • we know only what we can prove. • Eg. P = NP • Logic for computer science • Not provable in constructive logic

  7. Classical vs. Constructive • Proposition: there exists a rational number absuch that a and bare both irrational numbers. • Proof in classical logic • let c = p2p2 • if c = p2p2 is rational, let a = b = p2 = irrational. then ab = p2p2 = rational. • if c = p2p2 is irrational, let a = c =p2p2 = irrational, b = p2 = irrational. then ab = (p2p2)p2 = p22 = 2 = rational. • But what are a and b at all? • Proof in constructive logic • more involved, but provides a and b.

  8. Constructive Conjunction

  9. Constructive Implication

  10. Disjunction ?

  11. Outline • Classical logic vs Constructive logic V • Curry-Howard isomorphism • Summary

  12. Curry-Howard Isomorphism

  13. Conjunction vs. Product Type

  14. Logic vs. Type System

  15. -Reduction

  16. -Reduction

  17. Curry-Howard Isomorphism ?

  18. 5 4 to be taught in LICS Fall 2007! Four Color Problem • "Every map can be colored with no more than 4 colors." • Proposed in 1852 • Computer-based solution published in 1976 • Formalization of the proof in 1994 • theorem prover Coq

  19. Outline • Classical logic vs Constructive logic V • Curry-Howard isomorphism V • Summary

  20. Curry

  21. Howard

  22. Curry-Howard Isomorphism =

  23. Finally we will have ...

  24. Final Exam • 3 hours • Closed book • Covers all chapters. Su Mo Tu We Th Fr Sa 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

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