Categories, Proofs and Programs

Transcription

1 Categories, Proofs and Programs Samson Abramsky and Nikos Tzevelekos Lecture 4: Curry-Howard Correspondence and Cartesian Closed Categories

2 In A Nutshell Logic Computation Categories The upper link (Logic Computation) is usually attributed to H. B. Curry and W. A. Howard, although the idea is related to the operational interpretation of intuitionistic logic given in various formulations by Brouwer, Heyting and Kolmogorov. The link to Categories is mainly due to the pioneering work of J.Lambek in the 1970s. 2/36

3 Logic

4 What is Logic about? One main kind of answer focuses on Truth. Another, on Proof. We shall focus mainly on Proof: Formal proofs What follows from what? Traditional introductions to logic focus on Hilbert-style proof systems: generating the set of theorems of a system from a set of axioms by applying rules of inference (e.g. Modus Ponens). A key step in logic took place in the 1930 s, with the advent of Gentzen-style systems. Instead of focusing on theorems, look more generally and symmetrically at What follows from what? 4/36

5 Proof from Assumptions To capture this formally: Proof of formula A from assumptions A 1,...,A n : (cf. open vs. closed Systems) A 1,...,A n A We use Γ, to range over finite sets of formulas, writing Γ A etc. We write Γ,A as short for Γ {A}. We call each such Γ A a sequent. We look at the (, )-fragment of propositional logic. That is: A,B ::= p A B A B where p ranges over some set of atomic propositions. 5/36

6 Natural Deduction system for, Identity Γ,A A Id Conjunction Γ A Γ B Γ A B -intro Γ A B Γ A -elim 1 Γ A B Γ B -elim 2 Implication Γ,A B Γ A B -intro Γ A B Γ A Γ B -elim 6/36

7 Examples A proof of a sequent Γ A is simply a tree built using the proof rules and such that Γ A is at its root. For example, here is a proof of A B,B C,A A C (transitivity of ): Notes: A B,B C,A B C Id Id A B,B C,A A B A B,B C,A A Id A B,B C,A B A B,B C,A C A B,B C A C I The same sequent may have several proofs e.g: A,A A A or A B A B. There are rules for transforming proofs. Standard rules like Cut are admissible. E E 7/36

8 Example: Cut admissibility Consider the rule: Γ A A, B Γ, B Cut Seeing the proof of A, B as a procedure that produces B given A and, the rule means feeding Γ A in that procedure. We can prove that the rule is admissible: whenever we have proofs for Γ A and A, B, then we can obtain a proof of Γ, B. This can be done by replacing all leaves, i.e. axioms, of the form,a A inside the proof of A, B by a proof Γ, A. Alternatively, we can see that it can be (almost) derived using introduction and elimination of. 8/36

9 Structural Proof Theory The idea is to study the space of formal proofs as a mathematical structure in its own right, rather than to focus only on Provability Truth (i.e. the usual notions of soundness and completeness). Why? One motivation comes from trying to understand and use the computational content of proofs. To make this precise, we shall look at the Curry-Howard correspondence. 9/36

10 Computation

11 λ-calculus The λ-calculus is a pure calculus of functions, defined as follows. Assuming a set of Variables, ranged over by x,y,z,..., we define Terms: t,u ::= x tu }{{} application λx.t }{{} abstraction Examples λx. x identity function λx.x+1 successor function (via an encoding) λf. λx. f x application λf. λx. f(f x) double application λf.λg.λx.g(f(x)) composition g f Note: Terms are identified up to α-equivalence, that is we identify two terms if they differ solely in their choice of bound variables (i.e. variables in binding position: immediately following a λ). 11/36

12 Conversion and Reduction The basic equation governing this calculus is β-conversion: (λx.t)u = t[u/x] where substitution is given by: y[t/x] { t if y = x y if x y (λz.u)[t/x] λz.(u[t/x]) (uv)[t/x] (u[t/x])(v[t/x]) ( ) The equation is applied in sub-terms (i.e. we extend it to a congruence). E.g. (λf.λx.f(fx))(λx.x+1)0 = 2. By orienting this equation, we get a dynamics β-reduction: (λx.t)u t[u/x] 12/36

13 Examples Projection functions: λx.λy.x and λx.λy.y. For any pair of terms u,t: (λx.λy.x)ut (λy.u)t u whereas (λx.λy.y)ut t. Nested application operators: λf.λx.fx, λf.λx.f(fx),... For any terms u,t: (λf.λx.fx)ut (λx.ux)t ut whereas (λf.λx.fx)ut u(ut), etc. If-then-else: λc.λx.λy. c x y. For any terms u,t: (λc.λx.λy.cxy)(λx.λy.x)ut u projections can be used as booleans! Recursion: Y λf.(λx.f(xx))(λx.f(xx)). Yt (λx.t(xx))(λx.t(xx)) t((λx.t(xx))(λx.t(xx))) = t(yt). Historically, Curry extracted Y from an analysis of Russell s Paradox. 13/36

14 Expressive power What kind of programs can we write? What kind of functions can we define? Via an encoding of integers, we can express all (total) computable functions, i.e. the λ-calculus is Turing complete. n λf.λx.f(f( (f x) )) }{{} n times E.g. how can we define the successor function? While this is great in theory, in practice it means that any meaningful property about λ-calculus reduction is undecidable: termination, equality, etc. 14/36

15 From type-free to typed Pure λ-calculus is very unconstrained. For example, it allows terms like ω λx.xx self-application. Hence Ω ωω, which diverges: Ω Ω While divergence is natural in programming, the fact that any term can be applied to any other term is not. Solution: introduce Types. Types are there to stop you doing (bad) things Types impose the intuitive distinction between a function and its argument (and data-type disciplines in general). It has turned out that types constitute one of the most fruitful positive ideas in Computer Science! 15/36

16 Simply-Typed λ-calculus The simply-typed λ-calculus is defined as follows. Types B ::= ι... θ,θ ::= B θ θ θ θ ( e.g. ι ι ι first-order function type ) base types general types (ι ι) ι second-order function type Terms t,u ::= x tu λx.t t,u π 1 u π 2 u Typed terms x 1 : θ 1,...,x k : θ k t : θ (typing judgement) ( ) the term t has type θ in context Γ = {x1 : θ 1,...,x k : θ k }, i.e. under the assumption that x 1 has type θ 1,..., and x k has type θ k. 16/36

17 Typing rules Terms are typed by use of typing rules: Variable Γ,x : θ x : θ Product Γ t : θ Γ u : θ Γ v : θ θ Γ t,u : θ θ Γ π 1 v : θ Γ v : θ θ Γ π 2 v : θ Function Γ,x : θ t : θ Γ λx.t : θ θ Γ t : θ θ Γ u : θ Γ tu : θ 17/36

18 The power of types We can still type the terms: λx.x λf.λx.fx λf.λx.f(fx) λf.λg.λx.g(f(x)) Also typable: λx.λy. x,y λx.λg. (π 1 g)x,(π 2 g)x However, we cannot type λx.xx, and hence neither Y. 18/36

19 Reduction rules Computation rules (β-reductions): (λx.t)u t[u/x] π 1 t,u t π 2 t,u u Also, extensionality principles (η-laws): t = λx.tx u = π 1 u,π 2 u x not free in t, at function types at product types 19/36

20 Strong Normalisation A λ-term is called a redex if it is in one of forms of the left-hand-side of the reduction rules: (λx.t)u t[u/x] π 1 t,u t π 2 t,u u and therefore β-reduction can be applied to it. A term is in normal form if it contains no redexes as subterms. All reductions in the simply-typed λ-calculus lead to a normal form. Fact (Strong Normalisation) For every term t, there is no infinite sequence of β-reductions t t 0 t 1 t 2... The complexity of reducing to normal form is nonelementary. 20/36

21 The Curry-Howard Correspondence Simple Type System for, Γ,x : t x : θ Natural Deduction System for, Γ,A A Id Γ t : θ Γ u : θ Γ t,u : θ θ Γ A Γ B Γ A B I Γ v : θ θ Γ π 1 v : θ Γ,x : θ t : θ Γ λx.t : θ θ Γ A B Γ A E 1 Γ,A B Γ A B I Γ t : θ θ Γ u : θ Γ tu : θ Γ A B Γ B Γ A E 21/36

22 The Curry-Howard Correspondence ctd If we equate they are the same! This is the Curry-Howard correspondence (sometimes: Curry-Howard isomorphism ), and it works on three levels: Formulas Types Proofs Terms Proof transformations Term reductions Essentially, it shows that the simply-typed λ-calculus is an internal language (a formal syntax) for proofs in natural deduction. Reduction results in a normal form: a proof in which all lemmas have been eliminated, resulting in an explicit but much longer expression. 22/36

23 Constructive reading of formulas The Brouwer-Heyting-Kolmogorov interpretation of intuitionistic logic: A proof of an implication A B is a construction which transforms any proof of A into a proof of B. A proof of A B is a pair consisting of a proof of A and a proof of B. These readings motivate identifying A B with A B, and A B with A B. The correspondence has strong connections to programming: the λ-calculus can be seen as the pure core of functional programming languages such as Haskell and OCaml. So we get a reading of Proofs as Programs 23/36

24 The link to Categories

25 The Connection to Categories We now have our link Logic Computation We next complete the triangle by showing the connection to Categories: Logic Computation Categories This is sometimes called the Curry-Howard-Lambek correspondence. 25/36

26 Exponentials In Set, given sets A,B, we can form the set of functions from A to B: which is again a set. B A = Set(A,B) This closure of Set under forming function spaces is one of its most important properties. How can we axiomatise/generalise this situation? Once again, rather than asking what the elements of a function space are, we ask rather what can we do with it operationally? Answer: apply functions to their arguments. That is, there is a map ev A,B : B A A B ev A,B (f,a) = f(a) Think of the function as a black box: we can feed it inputs and observe the outputs. 26/36

27 Couniversal property of evaluation For any g : C A B, there is a unique map Λ(g) : C B A such that commutes. In Set, this is defined by Λ(g) id A B A A ev A,B B g 1 C A Λ(g)(c) : A B := a g(c,a). This process of transforming a function of two arguments into a function-valued function of one argument is known as Currying, after H. B. Curry. It is an algebraic form of λ-abstraction. 27/36

28 General definition of exponentials Let C be a category with binary products. For each object A of C, we can define a functor A : C C We say that C has exponentials if for all objects A and B of C there is a couniversal arrow from A to B, i.e. there is an object B A of C and a morphism ev A,B : B A A B such that for every g : C A B there is a unique morphism Λ(g) : C B A such that the following diagram commutes: Λ(g) id A B A A ev A,B B g 1 C A 28/36

29 Uniqueness equationally The uniqueness of Λ(g) can be expressed equationally. Proposition. C has exponentials if for all objects A and B of C there is: an object B A of C, a morphism ev A,B : B A A B, and for each object C, a mapping Λ : C(C A,B) C(C,B A ) such that, for all g : C A B: ev A,B (Λ(g) id A ) = g and, for all h : C B A : Λ(ev A,B (h id A )) = h 29/36

30 The exponential functor From the existence of exponentials we obtain, for each A in C, a functor: A : C C which maps each B to the exponential B A and each f : B C to: ( f A : B A C A := Λ B A A ev ) A,B B f C Lemma. Suppose f : B A C, h : B B and g : C C. Then: g A Λ(f) h = Λ(g f (h id)). Proof: Hence, Λ(f) h = Λ(ev ((Λ(f) h) id)) = Λ(ev ((Λ(f) id) (h id))) = Λ(f (h id)). g A Λ(f) h = Λ(g ev) Λ(f (h id)) = Λ(g ev (Λ(f (h id)) id)) = Λ(g f (h id)) 30/36

31 Exponentials via right adjoints We can show that A is right adjoint to A. As we saw in the previous lecture, this is no coincidence. Proposition. C has exponentials iff, for each object A, the functor A : C C has a right adjoint (i.e. A ). Equivalently, if there is a functor A : C C and a family of (set) bijections: Λ B,C : C(B A,C) = C(B,C A ) natural in B and C. (what is naturality? what is ev A,B?) 31/36

32 Cartesian Closed Categories A category with a terminal object, products and exponentials is called a Cartesian Closed Category (CCC). Equivalently: a CCC is a category with (all) finite products and with exponentials. This notion is fundamental in understanding functional types, models of λ-calculus, and the structure of proofs. Notation The notation of B A for exponential objects is standard in the category theory literature. However, for our purposes, it will be more convenient to write A B. 32/36

33 Example: Boolean Algebras A Boolean algebra, seen as a preorder category, is a CCC. (E.g. a powerset P(X).) Products are given by conjunctions A B. We define exponentials as implications: A B = A B Evaluation is just Modus Ponens: (A B) A B Couniversality is the Deduction Theorem : C A B C A B. 33/36

34 The Connection To Logic Let C be a category. We shall interpret Formulas (or Types) as Objects of C. A morphism f : A B will then correspond to a proof of B from assumption A, i.e. a proof of A B. Note that the bare structure of a category only supports proofs from a single assumption. Now suppose C has finite products. A proof of will correspond to a morphism This time, we map to, and to. A 1,...,A k A f : A 1 A k A. 34/36

35 Axiom, Conjunction Axiom Conjunction Γ,A A Id π 2 : Γ A A Γ A Γ B Γ A B I f : Γ A g : Γ B f,g : Γ A B Γ A B Γ A E 1 Γ A B Γ B E 2 f : Γ A B π 1 f : Γ A f : Γ A B π 2 f : Γ B 35/36

36 Implication Now let C be cartesian closed. Γ,A B Γ A B I f : Γ A B Λ(f) : Γ (A B) Γ A B Γ B Γ A E f : Γ (A B) g : Γ A ev A,B f,g : Γ B Moreover, the β- and η-equations are all then derivable from the equations of cartesian closed categories. So cartesian closed categories are models of (, )-logic, at the level of proofs and proof transformations, and of simply typed λ-calculus, at the level of terms and equations between terms. 36/36