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

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 1 

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. 2 

some type theory 

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 4 

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 ) 5 

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 ) 6 

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) 7 

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) 8 

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) 9 

some logic 

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! 11 

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! 12 

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). 13 

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 14 

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? 15 

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 

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! ♡ 17 

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! ♡ 18 

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 19 

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 ⊃⊥) 20 

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 ⊃⊥) 21 

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. 22 

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 23 

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! 24 

some category theory 

a category ∙ A category is an embarrassingly simple concept. – Bartosz Milewski, Category Theory for Programmers ∙ Category = objects + arrows ∙ Key ingredient: composition of arrows 26 

a category credits: Bartosz Milewski 27 

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 28 

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: ✓ 29 

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, . . . 30 

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: ✓ 31 

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 32 

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 33 

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 34 

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 35 

products A A × B B C π1 π2 f g ⟨f,g⟩ 36 

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? 37 

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 38 

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 39 

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) 40 

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. 41 

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 42 

cartesian closed categories A category with ∙ a terminal object ∙ products ∙ exponentials is called a Cartesian Closed Category (CCC) 43 

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⟩ 44 

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 45 

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 

Thanks for your attention! 

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 47