The Curry-Howard-Lambek Correspondence

Transcript

1. the curry-howard-lambek correspondence
   Denisa Diaconescu
   Bucharest FP #38 · 23 January 2019
   

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2. change perspectives
   Roger Antonsen
   University of Oslo
   TED Talk: Math is the hidden secret to understanding the world
   ... understanding has to do with the ability to change your perspective
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3. overview
   We will look at some elements from three different perspectives:
   ∙ Type Theory
   ∙ Logic
   ∙ Category Theory
   Disclamer: We will not go into depth in any of the above fields.
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4. some type theory
   

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5. a simple program in haskell
   data Point = Point Int Int
   makePoint :: Int → Int → Point
   makePoint x y = Point x y
   getX :: Point → Int
   getX (Point x y) = x
   getY :: Point → Int
   getY (Point x y) = y
   origin :: Point
   origin = makePoint 0 0
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6. a simple program in haskell
   Let’s change perspective!
   data Point = Point Int Int
   makePoint :: Int → Int → Point
   makePoint x y = Point x y
   x : Int y : Int
   makePoint x y : Point (PointI)
   getX :: Point → Int
   getX (Point x y) = x
   p : Point
   getX p : Int (PointE1
   )
   getY :: Point → Int
   getY (Point x y) = y
   p : Point
   getY p : Int (PointE2
   )
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7. generalising
   x : Int y : Int
   makePoint x y : Point (PointI)
   a : A b : B
   ⟨a, b⟩ : A × B (×I)
   p : Point
   getX p : Int (PointE1
   )
   p : A × B
   fst p : A (×E1
   )
   p : Point
   getY p : Int (PointE2
   )
   p : A × B
   snd p : B (×E2
   )
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8. another simple example
   > let f = (\x → x * 3) :: Int → Int
   [x : Int]
   .
   .
   .
   x ∗ 3 : Int
   λx.x ∗ 3 : Int → Int (funI)
   > f 5
   15
   f : Int → Int 5 : Int
   f 5 : Int (funE)
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9. generalising
   [x : Int]
   .
   .
   .
   x ∗ 3 : Int
   λx.x ∗ 3 : Int → Int (funI)
   [x : A]
   .
   .
   .
   b : B
   λx.b : A → B (→I)
   (abstraction)
   f : Int → Int 5 : Int
   f 5 : Int (funE)
   f : A → B x : A
   f x : B (→E)
   (application)
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10. summing up
    A Typed λ-Calculus
    a : A b : B
    ⟨a, b⟩ : A × B (×I)
    p : A × B
    fst p : A (×E1
    )
    p : A × B
    snd p : B (×E2
    )
    [x : A]
    .
    .
    .
    b : B
    λx.b : A → B (→I)
    f : A → B x : A
    f x : B (→E)
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11. some logic
    

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12. what’s true and what’s false?
    If it is dark outside then if pigs fly then it is dark outside.
    A = it is dark outside
    A ⊃ (B ⊃ A)
    B = pigs fly
    Is this proposition true?
    A B B ⊃ A A ⊃ (B ⊃ A)
    false false true true
    false true false true
    true false true true
    true true true true
    This proposition is always true!
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13. what’s true and what’s false?
    If it is dark outside then if pigs fly then it is dark outside.
    A = it is dark outside
    A ⊃ (B ⊃ A)
    B = pigs fly
    Is this proposition true?
    A B B ⊃ A A ⊃ (B ⊃ A)
    0 0 1 1
    0 1 0 1
    1 0 1 1
    1 1 1 1
    This proposition is always true!
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14. truth-value semantics
    We give values to atoms in the set {0, 1}, i.e.
    define an evaluation e : Atoms → {0, 1}.
    Given an evaluation, we extend it to propositions using the truth tables:
    & : {0, 1} × {0, 1} → {0, 1} ⊃: {0, 1} × {0, 1} → {0, 1}
    A B A&B
    0 0 0
    0 1 0
    1 0 0
    1 1 1
    A B A ⊃ B
    0 0 1
    0 1 1
    1 0 0
    1 1 1
    If for all evaluations, a proposition takes the value 1 then we say it is
    always true (a tautology).
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15. the faces of a logic
    Syntax of a logic
    ∙ Deals with the notion of provability
    ∙ Gives a method to manipulate the symbols from the logic (i.e, atoms,
    ⊃, &) in order to establish when a proposition is provable (a theorem)
    Completeness = syntax and semantics coincide
    Soundness = syntax implies semantics
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16. a natural deduction system
    ∙ Rules for handling each logical connectives
    ∙ Rules for introducing and eliminating connectives
    ∙ Rules are of the form
    Assumptions
    Conclusion
    A B
    A&B (&I)
    A&B
    A (&E1
    )
    A&B
    B (&E2
    )
    [A]
    .
    .
    .
    B
    A ⊃ B (⊃I)
    A ⊃ B A
    B (⊃E)
    Does it look familiar?
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17. what we have seen so far
    A Typed λ-Calculus A Natural Deduction System
    a : A b : B
    ⟨a, b⟩ : A × B (×I)
    A B
    A&B (&I)
    p : A × B
    fst p : A (×E1
    )
    A&B
    A (&E1
    )
    p : A × B
    snd p : B (×E2
    )
    A&B
    B (&E2
    )
    [x : A]
    .
    .
    .
    b : B
    λx.n : A → B (→I)
    [A]
    .
    .
    .
    B
    A ⊃ B (⊃I)
    f : A → B x : A
    f x : B (→E)
    A ⊃ B A
    B (⊃E)
    Propositions are types! ♡ 16
    

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18. let’s dig further
    A Typed λ-Calculus A Natural Deduction System
    a : A b : B
    ⟨a, b⟩ : A × B (×I)
    A B
    A&B (&I)
    Inhabitation of type A means that A is provable! ♡
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19. let’s dig further
    A Typed λ-Calculus A Natural Deduction System
    λx.x : A → A (id)
    [A]
    A
    A ⊃ A
    λx.(λy.x) : A → (B → A) (const)
    [A]
    [B]
    A
    B ⊃ A
    A ⊃ (B ⊃ A)
    Proofs are Terms! ♡
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20. the curry-howard correspondence
    Type Theory Logic
    types propositions
    terms proofs
    inhabitation of type A proof of proposition A
    product type conjunction
    function type implication
    sum type disjunction
    void type (empty type) falsity
    unit type (singleton type) truth
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21. natural deduction for classical (propositional) logic
    [A]
    .
    .
    .
    B
    A ⊃ B (⊃I)
    A ⊃ B A
    B (⊃E)
    A B
    A&B (&I)
    A&B
    A (&E1
    )
    A&B
    B (&E2
    )
    A
    A ∨ B (∨I1
    )
    B
    A ∨ B (∨I2
    )
    A ∨ B A ⊃ C B ⊃ C
    C (∨E)
    ⊥
    A (ex falso quodlibet) ¬¬A
    A (reductio ad absurdum)
    (¬A is an abbreviation for A ⊃⊥)
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22. natural deduction for intuitionistic (propositional) logic
    [A]
    .
    .
    .
    B
    A ⊃ B (⊃I)
    A ⊃ B A
    B (⊃E)
    A B
    A&B (&I)
    A&B
    A (&E1
    )
    A&B
    B (&E2
    )
    A
    A ∨ B (∨I1
    )
    B
    A ∨ B (∨I2
    )
    A ∨ B A ⊃ C B ⊃ C
    C (∨E)
    ⊥
    A (ex falso quodlibet)
    (¬A is an abbreviation for A ⊃⊥)
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23. intuitionistic logic
    ∙ Constructive logic
    ∙ Based on the notion of proof
    ∙ Useful because proofs are executable and produce examples
    ∙ The following equivalent propositions are unprovable
    in intuitionistic logic:
    ∙ double negation law: ¬¬A ⊃ A
    ∙ excluded middle law: A ∨ ¬A
    ∙ Pierce’s law: ((A ⊃ B) ⊃ A) ⊃ A
    ∙ There is no truth-values semantics for intuitionistic logic!
    It has alternative semantics (e.g., Kripke semantics)
    The original Curry-Howard correspondence is between
    Church’s simply typed
    and
    Gentzen’s natural deduction
    λ-calculus for intuitionistic logic.
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24. go beyond original case
    Type Theory Logic
    types propositions
    terms proofs
    inhabitation of type A proof of proposition A
    product type conjunction
    function type implication
    sum type disjunction
    void type (empty type) falsity
    unit type (singleton type) truth
    dependent types quantifiers
    call/cc operator Peirce’s law
    monads a modal logic
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25. why care?
    ∙ This is just fascinating
    ∙ Don’t think of logic and computing as distinct fields
    ∙ Thinking from a different perspective can help you know what is
    possible/impossible
    ∙ Type systems should not be ad hoc piles of rules!
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26. some category theory
    

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27. a category
    ∙ A category is an embarrassingly simple concept.
    – Bartosz Milewski, Category Theory for Programmers
    ∙ Category = objects + arrows
    ∙ Key ingredient: composition of arrows
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28. a category
    credits: Bartosz Milewski
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29. a category
    A category C consists of
    ∙ Objects: denoted A, B, C, . . .
    ∙ Arrows: for any objects A and B, there is a set of arrows C(A, B)
    ∙ we denote f ∈ C(A, B) with f : A → B or A f
    −→ B
    ∙ Composition: for any arrows f : A → B and g : B → C there is an arrow
    g ◦ f : A → C
    A B
    C
    f
    g◦f
    g
    ∙ Identity: for any object A there is an arrow idA : A → A
    ∙ Axioms: for any arrows f : A → B, g : B → C, and h : C → D
    h ◦ (g ◦ f) = (h ◦ g) ◦ f f ◦ idA = f = idB ◦ f
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30. example - the category of sets
    The category Set has
    ∙ Objects: sets
    ∙ Arrows: functions
    ∙ Composition: composition of functions
    ∙ Identity: for each set A, the identity function idA : A → A, idA(a) = a
    ∙ Axioms: ✓
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31. monoids
    A monoid M is a structure ⟨M, +, e⟩ such that
    ∙ M is a set
    ∙ + : M × M → M is associative
    (aka (a + b) + c = a + (b + c) for any a, b, c ∈ M)
    ∙ e ∈ M is an identity for +
    (aka e + a = a + e = a for any a ∈ M)
    Monoids are an amazingly powerful concept:
    ∙ They are behind basic arithmetics
    ∙ both addition and multiplication form a monoid
    ∙ They are ubiquitous in programming
    ∙ strings, lists, foldable data structures, . . .
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32. example - the category of monoids
    The category Mon has
    ∙ Objects: monoids
    ∙ Arrows: monoid morphisms
    (aka functions not messing with the monoid operation)
    ∙ Composition: composition of monoid morphisms
    ∙ Identity: for each object M, idM : M → M, idM(m) = m
    ∙ Axioms: ✓
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33. example - a monoid is a category
    Any monoid M = ⟨M, +, e⟩ is a category with
    ∙ Objects: just one object □
    ∙ Arrows: the elements of the set M (i.e, M(□, □) = M)
    ∙ Composition: the monoid operation +
    ∙ Identity: the monoid identity e
    ∙ Axioms:
    h ◦ (g ◦ f) = (h ◦ g) ◦ f f ◦ idA = f = idB ◦ f
    a + (b + c) = (a + b) + c a + e = a = e + a
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34. initial and terminal object
    In a category C
    ∙ an object T is called terminal if for every object A there exists a unique
    arrow
    τA : A → T
    ∙ an object I is called initial if for every object A there exists a unique
    arrow
    ιA : I → A
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35. why the terminal object?
    We can generalise our rules. For example:
    A B
    A&B (&I) Γ ⊢ A Γ ⊢ B
    Γ ⊢ A&B (&I)
    Deduction from no assumptions Deduction from assumptions Γ
    Γ = ∅ (Deduction from truth)
    Let C be a category with a terminal object T. We have the following
    interpretations:
    ∙ Propositions as objects of C
    ∙ Truth as the terminal object T
    ∙ A proof of A as an arrow f : T → A
    ∙ A proof of A from assumptions B as an arrow f : B → A
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36. products
    Let A and B be objects in a category C.
    We say that
    A π1
    ←− A × B π2
    −→ B
    is a product of A and B if for every
    A f
    ←− C g
    −→ B
    there exists a unique arrow
    ⟨f, g⟩ : C −→ A × B
    such that
    π1 ◦ ⟨f, g⟩ = f π2 ◦ ⟨f, g⟩ = g
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37. products
    A A × B B
    C
    π1 π2
    f g
    ⟨f,g⟩
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38. let’s change the perspective
    Let C be a category with a terminal object T and products.
    Let A, B be two objects in C.
    A A × B B
    T
    π1 π2
    f g
    ⟨f,g⟩
    f : T → A g : T → B
    ⟨f, g⟩ : T → A × B
    ⟨f, g⟩ : T → A × B
    π1 ◦ ⟨f, g⟩ : T → A
    ⟨f, g⟩ : T → A × B
    π2 ◦ ⟨f, g⟩ : T → B
    Does it look familiar?
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39. what we have seen so far
    A Typed λ-Calculus A Natural Ded. Syst. A Category∗
    a : A b : B
    ⟨a, b⟩ : A × B (×I)
    A B
    A&B (&I)
    f : T → A g : T → B
    ⟨f, g⟩ : T → A × B
    p : A × B
    fst p : A (×E1
    )
    A&B
    A (&E1
    )
    ⟨f, g⟩ : T → A × B
    π1 ◦ ⟨f, g⟩ : T → A
    p : A × B
    snd p : B (×E2
    )
    A&B
    B (&E2
    )
    ⟨f, g⟩ : T → A × B
    π2 ◦ ⟨f, g⟩ : T → B
    * A category with a terminal object T and products
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40. the curry-howard-lambek correspondence
    Type Theory Logic Category Theory
    types propositions objects
    terms proofs arrows
    inhabitation of type A proof of proposition A arrow f : T → A
    function type implication ?
    product type conjunction product
    sum type disjunction coproduct
    void type (empty type) falsity initial object
    unit type (singleton type) truth terminal object T
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41. towards an interpretation for implication
    In Set, given two sets A and B, we can form the set of functions
    A ⇒ B = Set(A, B) which is again a set!
    How can we axiomatise this situation?
    What can we do with it operationally?
    Apply functions to their arguments! That is, there is an arrow
    ApA,B : (A ⇒ B) × A → B ApA,B(f, a) = f(a)
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42. towards an interpretation for implication
    For any g : C × A → B, there is an unique arrow Λ(g) : C → (A ⇒ B) such
    that
    A ⇒ B (A ⇒ B) × A B
    C C × A
    ApA,B
    Λ(g) Λ(g)×idA g
    In Set, this is defined by
    Λ(g)(c) = f where f : A → B with f(a) = g(c, a)
    This process of transforming a function of two arguments into a function
    with one argument is known as Currying.
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43. exponentials in a category
    Let C be a category with a terminal object T and products.
    We say that C has exponentials if for any objects A and B,
    there are an object A ⇒ B and an arrow
    ApA,B : (A ⇒ B) × A → B
    such that for any arrow g : C × A → B, there is a unique arrow
    Λ(g) : C → (A ⇒ B) such that
    ApA,B ◦ (Λ(g) × idA) = g
    A ⇒ B (A ⇒ B) × A B
    C C × A
    ApA,B
    Λ(g) Λ(g)×idA g
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44. cartesian closed categories
    A category with
    ∙ a terminal object
    ∙ products
    ∙ exponentials
    is called a Cartesian Closed Category (CCC)
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45. implication
    Let C be a CCC with a terminal object T.
    A Typed λ-Calculus A Natural Ded. Syst. A CCC
    [x : A]
    .
    .
    .
    b : B
    λx.n : A → B (→I)
    [A]
    .
    .
    .
    B
    A ⊃ B (⊃I)
    g : T×A → B
    Λ(g) : T → (A ⇒ B)
    f : A → B x : A
    f x : B (→E)
    A ⊃ B A
    B (⊃E)
    f : T → (A ⇒ B) g : T → A
    ApA,B ◦ ⟨f, g⟩ : T → B
    A ⇒ B (A ⇒ B) × A B
    T T × A
    ApA,B
    Λ(g) Λ(g)×idA g
    B
    A ⇒ B (A ⇒ B) × A A
    T
    π1 π2
    ApA,B
    f g
    ⟨f,g⟩
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46. what we have seen so far
    A Typed λ-Calculus A Natural Ded. Syst. A CCC
    a : A b : B
    ⟨a, b⟩ : A × B (×I)
    A B
    A&B (&I)
    f : T → A g : T → B
    ⟨f, g⟩ : T → A × B
    p : A × B
    fst p : A (×E1
    )
    A&B
    A (&E1
    )
    ⟨f, g⟩ : T → A × B
    π1 ◦ ⟨f, g⟩ : T → A
    p : A × B
    snd p : B (×E2
    )
    A&B
    B (&E2
    )
    ⟨f, g⟩ : T → A × B
    π2 ◦ ⟨f, g⟩ : T → B
    [x : A]
    .
    .
    .
    b : B
    λx.n : A → B (→I)
    [A]
    .
    .
    .
    B
    A ⊃ B (⊃I)
    g : T×A → B
    Λ(g) : T → (A ⇒ B)
    f : A → B x : A
    f x : B (→E)
    A ⊃ B A
    B (⊃E)
    f : T → (A ⇒ B) g : T → A
    ApA,B ◦ ⟨f, g⟩ : T → B
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47. the curry-howard-lambek correspondence
    Type Theory Logic Category Theory
    types propositions objects
    terms proofs arrows
    inhabitation of type A proof of proposition A arrow f : T → A
    function type implication exponential
    product type conjunction product
    sum type disjunction coproduct
    void type (empty type) falsity initial object
    unit type (singleton type) truth terminal object T
    46
    

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48. Thanks for your attention!
    

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49. references
    ∙ Roger Antonsen, TED Talk: Math is the hidden secret to understanding
    the world
    https://www.ted.com/talks/roger_antonsen_math_is_the_hidden_secret_to_
    understanding_the_world
    ∙ Samson Abramsky, Categories, Proofs and Processed Lecture III - The
    Curry-Howard-Lambek Correspondence
    http://www.math.helsinki.fi/logic/sellc-2010/course/LectureIII.pdf
    ∙ Philip Wadler, Propositions as Types
    https://homepages.inf.ed.ac.uk/wadler/papers/propositions-as-types/
    propositions-as-types.pdf
    ∙ Dan Grossman, Lecture notes on The Curry-Howard Isomorphism
    https://courses.cs.washington.edu/courses/cse505/12au/lec12_6up.pdf
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