CSE321 Programming Languages CurryHoward Isomorphism

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CSE-321 Programming LanguagesCurry-Howard
Isomorphism
???

• POSTECH
• June 4, 2009

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Truth Table

• T True, F False

???
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Tautology
The computational meaning of
) Truth of A is not affected by truth of B.
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Tautology
But what is the computational meaning of
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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

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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

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Classical vs. Constructive

• Proposition there exists a rational number ab
  such that a and b are 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.

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Constructive Conjunction
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Constructive Implication
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Disjunction
?
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Outline

• Classical logic vs Constructive logic V
• Curry-Howard isomorphism
• Summary

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Curry-Howard Isomorphism
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Conjunction vs. Product Type
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Logic vs. Type System
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?-Reduction
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?-Reduction
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Curry-Howard Isomorphism
?
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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

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Final Assignment in LICS 2009

• Proof of the Pythagoras' theorem
• Theorem even_is_even_times_even forall (n
  nat), even (n n) -gt even n.
• Theorem main_thm forall (n p nat), n n
  double (p p) -gt p 0.
• Definition irrational (x R) Prop
• forall (p Z) (q nat), q ltgt 0 -gt x ltgt (p /
  q)R.
• Theorem irrational_sqrt_2 irrational (sqrt
  2nat).

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Outline

• Classical logic vs Constructive logic V
• Curry-Howard isomorphism V
• Summary

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Curry
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Howard
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Curry-Howard Isomorphism

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Finally we will have ...
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Final Exam

• 1.5 hours
• Closed book
• Covers all chapters.
• June 2009
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