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CSE-321 Programming Languages
Curry-Howard Isomorphism
박성우
POSTECH
June 4, 2009
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
law of excluded middle
law of double-negation
Peirce's law
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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
Logic
Type System
proposition
type
proof
proof term (expression)
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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
Logic
Type System
propositional logic
simply-type -calculus
first-order logic
dependent types
second-order logic
polymorphism
modal logic
modal type system
classical logic
continuations
?
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Four Color Problem
• "Every map can be colored with no more than 4 colors."
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• 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)
-> even n.
Theorem main_thm: forall (n p : nat),
n * n = double (p * p) -> p = 0.
Definition irrational (x : R) : Prop :=
forall (p : Z) (q : nat), q <> 0 -> x <> (p / q)%R.
Theorem irrational_sqrt_2: irrational (sqrt 2%nat).
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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 ...
Final Exam
• 1.5 hours
• Closed book
• Covers all chapters.
June 2009
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