The Curry-Howard-isomorphism and Cartesian Closed Categories Florian Rabe Part of the course on Computational Logic by Michael Kohlhase Fall 2007, Jacobs University Bremen 1 06.11.2007 2 Review: Categories of Contexts I A recurring pattern in logic I Essentially the same for FOL and STT; in the following ConSTT(Σ) as an example I Objects: Contexts x1 : A1;:::; xn : An 0 I Morphisms from Γ = x1 : A1;:::; xm : Am to Γ = y1 : B1;:::; yn : Bn: tuples (t1;:::; tn) such that x1 : A1;:::; xm : Am `Σ ti : Bi for i = 1;:::; n 0 I Morphism (t1;:::; tn) from Γ to Γ can be seen as substitutions σ = [t1=y1;:::; tn=yn] (where, intutitively, the substitutions go against the morphism direction) I Identity idx1:A1;:::;xm:Am = (x1;:::; xm) I Composition σ • (t1;:::; tn) = (σ(t1); : : : ; σ(tn)) 3 References: Curry-Howard-correspondence @Bookf c u r r y , author = "H. Curry and R. Feys", title = "Combinatory Logic", publisher = "North−Holland " , address = "Amsterdam", year = "1958", g @InCollection fhoward , author = "W. Howard", title = "The formulas −as−types notion of construction ", booktitle = "To H.B. Curry: Essays on Combinatory Logic , Lambda−Calculus and Formalism", editors = "J. Seldin and J. Hindley", publisher = "Academic Press", year = "1980", pages = "479−−490", g 4 Review: First-order logic Summary: I Inference rules for FOL-fragment consisting of true, conjunction and implication I Natural deduction with context, e.g., Γ ` A for Γ = A1;:::; An I Rules: truth introduction, conjunction introduction and elimination, implication introduction and elimination 5 Review: Simply-typed lambda calculus Summary: I Typing of terms for lambda calculus with unit type, product type, and function type I Type inference rules with context, e.g., Γ ` t : A for Γ = x1 : A1;:::; xn : An I Rules: unit element, pairing, projections, lambda abstraction, application 6 The Curry-Howard-correspondence I Observation: Rules look almost alike I formulas correspond to types I context (= list of hypotheses) corresponds to context (= list of variable declarations) I inference rules correspond to typing rules I What FOL-entity corresponds to STT-terms? Answer: Proofs. I Thus: Proof-checking corresppnds to type-checking; proof-search corresponds to type-inhabitation (where a type is inhabited if there is a term of it). 7 Proof term assignment Summary: I In the FOL-inference rules, replace A1;:::; An ` A with x1 : A1;:::; xn : An ` p : A where I xi : names for the assumed proofs of the Ai I p proof of A possibly using the assumed proofs xi , i.e., p is a (proof) term with variables xi I Give every rule a name, so that a rule becomes a constructor for proof terms. For example, call modus ponens MP. I Then every proof is a term, for example MP(p1; p2) is a proof of B if p1 is a proof of A ! B and p2 is a proof of A. I Then the rules for FOL with proofs and for STT become the same. 8 07.11.2007 9 Review: Constructions in Categories I Pairs terminal/initial, product/coproduct, pullback/pushout, equalizer/coequalizer of dual constructions I All constructions exist in Set. I Results: 0 0 I If U is an X , then: U is an X iff U =∼ U , where X is one of the above constructions. op I If C has all X , then C has all duality partners X . I If C has a terminal object, then having all pullbacks is equivalent to having all products and all equalizers. I Dually: If C has an initial object, then having all pushouts is equivalent to having all coproducts and all coequalizers. I I I I I Categories relevant for logic: Sig , Th , ConΣ, Mod (Σ) I Candidates for constructions in the relevant categories I I I Sig and Th : initial - empty, coproduct - union, pushout - union with sharing I I ConΣ: terminal - empty, product - concatenation I I Mod (Σ): terminal - singleton, product - cartesian product, initial - term model 10 Reference: Cartesian Closed Categories @bookf ccc , a ut h o r = fJ. Lambek and P. Scott g , p u b l i s h e r = f Cambridge University Press g , s e r i e s = f Cambridge Studies in Advanced Mathematics g , t i t l e = f Introduction to Higher−Order Categorical L o g i c g , volume = f7g , y e a r = f1986g g Recommended read: Chapter 6.1 in Steve Awodey's notes 11 Cartesian Closed Categories A1;A2 A1;A2 I Assume a category C with products (A1 × A2; π1 ; π2 ) for all A1; A2 2 jCj. I Then for all fi 2 C (O; Ai ) for i = 1; 2, there is a unique morphism hf1; f2i : C (O; A1 × A2). I For gi 2 C (Ai ; Bi ) for i = 1; 2, we put A1;A2 A1;A2 g1 × g2 := hπ1 • f1; π2 • f2i 2 C (A1 × A2; B1 × B2). A A;B A;B A I For A; B 2 jCj,(B ; eval ) where eval 2 C B × A; B is called an exponential of A and B iff: C (Γ × A; B) =∼ C Γ; BA ; f 7! f¯ A;B for all Γ 2 jCj and (f¯ × idA) • eval = f for all f 2 C (Γ × A; B). I A category is called cartesian closed if it has a terminal object, products for all pairs of objects, and exponentials for all pairs of objects. 12 Cartesian Closed Categories (2) I CCC is the category with cartesian closed categories as objects. 0 0 I CCC-morphisms from C to C : Functors F from C to C that preserve CCC-structure, i.e., 0 I F maps the terminal object of C to the terminal object of C , A;B A;B 0 I (F (A × B); F (π1 ); F (π2 )) is a product in C of F (A) and F (B), A A;B 0 I (F (B ); F (eval )) is an exponential in C of F (A) and F (B). I Identity and composition are as in Cat. 13 Example: Set I Terminal: Singletons I Product: Cartesian product with projections I Exponential of A and B: A I B = Set (A; B) A;B I eval :('; α) 7! '(α) for ' 2 Set (A; B) and α 2 A. 14 STT Example: ConΣ I Here: STT with unit type T , product types A × B, and function types A ! B I Only the case with one variable needed because ∼ x1 : A1;:::; xm : Am = x : A1 × ::: × Am. I Then: Morphisms from x : A to x : B are simply terms t such that x : A ` t : B. I Terminal object is x : T (or isomorphically: empty context). I Product object of x : A and x : B is x : A × B (or isomorphically: x : A; y : B). I Exponential object of x : A and x : B is x : A ! B. I Still missing: projection morphisms for the product, evaluation morphism for the exponential I Needed for the proof: unique morphisms into the terminal and product object and the isomorphism from the definition of exponential 15 Example: FOL I Use the fragment of FOL with truth, conjunction, implication; call it P. I This is even a fragment of propositional logic, i.e., no quantifiers, no terms; signatures contain only nullary predicate symbols. 0 I Let Σ be the STT-signature containing one constant for every inference rule of P. We use one base type for every nullary predicate symbol of Σ. P I Build a category PfΣ I objects are named lists of hypotheses, i.e., x1 : A1;:::; xn : Am P for formulas A1;:::; Am 2 Sen (Σ) I morphisms from x1 : A1;:::; xn : Am to y1 : B1;:::; yn : Bn are tuples (p1;:::; pn) where pi is a proof of Bi using the assumptions A1;:::; Am STT I identity and composition just like for ConΣ0 P ∼ STT Then clearly: PfΣ = ConΣ0 in CCC. 16.