CSE-321 Programming Languages

Curry-Howard Isomorphism

POSTECH

June 4, 2009

박성우

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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

law of double-negation

Peirce's law

law of excluded middle

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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."

• Proposed in 1852• Computer-based solution published in 1976• Formalization of the proof in 1994

– theorem prover Coq

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to be taught in LICS Fall 2009!

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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

=

Finally we will have ...

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Final Exam• 1.5 hours• Closed book• Covers all chapters.

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