VU Logic and Computation Agata Ciabattoni Institute fur¨ Computersprachen, Theory and Logic group [email protected] VU Logic and Computation – p. 1 Lection 4 – Computability Theory • Combinatory Logic • Recursive functions • (Hints on the) Equivalence between λ-calculus, recursive functions and TM • Computable and non computable functions VU Logic and Computation – p. 2 Combinatory Logic VU Logic and Computation – p. 3 Combinatory Logic: Motivation Example: commutative law of addition in arithmetic (∀xy) x + y = y + x VU Logic and Computation – p. 4 Combinatory Logic: Motivation Example: commutative law of addition in arithmetic (∀xy) x + y = y + x it can be expressed without the bound variables x and y: • first define A(x,y) = x + y (for all x and y) • then introduce an operator C defined by (Cf)(x,y) = f(y,x) (for all f,x and y) • then the commutative law becomes simply A = CA VU Logic and Computation – p. 4 Combinatory Logic: Motivation Example: commutative law of addition in arithmetic (∀xy) x + y = y + x it can be expressed without the bound variables x and y: • first define A(x,y) = x + y (for all x and y) • then introduce an operator C defined by (Cf)(x,y) = f(y,x) (for all f,x and y) • then the commutative law becomes simply A = CA C is called combinator VU Logic and Computation – p. 4 Combinatory Logic • only two constants S and K • only application (MN) VU Logic and Computation – p. 5 Combinatory Logic • only two constants S and K • only application (MN) (KMN) → M (SMNP ) → (MP )(NP ) with M,N and P terms of CL (combinators). VU Logic and Computation – p. 5 Equivalence with λ calculus K and S are expressible in λ calculus VU Logic and Computation – p. 6 Equivalence with λ calculus K and S are expressible in λ calculus (Vice versa) For each combinator M, there exist λ∗x.M which behaves as the λ abstraction, that is ∗ (λ x.MN) → M[N/x] Proof: We define λ∗x.M by structural induction: If M = x, then λ∗x.x = I =(SKK) If M does not contain x, then λ∗x.M =(KM) If M =(P Q), then λ∗x.(P Q)=(Sλ∗x.P λ∗x.Q) VU Logic and Computation – p. 6 Recursive Functions • An abstract description of computable functions – Gödel and Kleene (1936) • useful to study recursive programs VU Logic and Computation – p. 7 Recursive Functions • An abstract description of computable functions – Gödel and Kleene (1936) • useful to study recursive programs They form the smallest collection of partial functions containing some basic functions and closed under some fundamental operations for forming new functions from old – composition, primitive recursion and minimization. VU Logic and Computation – p. 7 Basic Functions • n n Projection functions, proji : N →N : n proji (x1,...xn) = xi • Constant funtions with value 0, zeron : N n →N : n zero (x1,...xn) = 0 • Successor function, succ : N →N : succ(x) = x + 1 VU Logic and Computation – p. 8 Composition n m Given f : N →N and g1,...gn : N →N , their composition is m the function f ◦ [g1,...gn] ∈N →N such that f ◦ [g1,...gn](x1,...xm) ≃ f(g1(x1,...xm),...gn(x1,...xm)) where ≃ is the equivalence of possibly undefined expressions VU Logic and Computation – p. 9 Recursion A function f is recursive when the value of f on some arguments depends on the value of f on "simpler" arguments. VU Logic and Computation – p. 10 Recursion A function f is recursive when the value of f on some arguments depends on the value of f on "simpler" arguments. • f1(0) = 0 f1(x + 1) = f1(x)+(x + 1) • f2(0) = 1 f2(x + 1) = (x + 1)f2(x) VU Logic and Computation – p. 10 Primitive Recursive Functions k Is the class of functions containing zero, succ and proji , closed under compositions and primitive recursion: Let g be a k-ary PRF and h be a k + 2-ary PRF. Then the k + 1-ary function f: f(0, x) = g(x) f(y + 1, x) = h(y,f(y, x), x) is PR. VU Logic and Computation – p. 11 PRF – Example 1 Addition: 1 sum(0,x) = proj1 (x) sum(y + 1,x) = h(y, sum(y,x),x) where 3 h := λxyz.succ(proj2 (x,y,z)) VU Logic and Computation – p. 12 PRF – Example 2 Product P (0,x) = zero1(x) P (y + 1,x) = h′(y, P (y,x),x) where ′ 3 3 h := λxyz.sum(proj2 (x,y,z),proj3 (x,y,z)) VU Logic and Computation – p. 13 Elementary Functions Is the subclass of primitive recursive functions containing k zero, succ, subtr and proji , closed under compositions and • m bounded summation f(m, x1,...,xn) = Pi=0 g(i, x1,...,xn) • m bounded product f(m, x1,...,xn) = Qi=0 g(i, x1,...,xn) (where g is elementary recursive) VU Logic and Computation – p. 14 Primitive Recursive Functions k Is the class of functions containing zero, succ and proji , closed under compositions and primitive recursion: f(0, x) = g(x) f(y + 1, x) = h(y,f(y, x), x) • total functions • all total computable function? VU Logic and Computation – p. 15 Ackermann Function f(0, 0,y) = y f(0,x + 1,y) = f(0,x,y) + 1 f(1, 0,y) = 0 f(z + 2, 0,y) = 1 f(z + 1,x + 1,y) = f(z,f(z + 1,x,y),y)) • f is computable • f grows faster than any PRF! (i.e., for any PRF F, there is an n such that f(n,x,y) >F (x,y)) VU Logic and Computation – p. 16 Extending the PRF g(x) = min y such that f(x,y) = 0 is g computable, if so is f? VU Logic and Computation – p. 17 Recursive Functions Is the smallest collection of functions containing all primitive k recursive functions (zero, succ and proji , closed under compositions and primitive recursion) and closed under minimalization: g(x) = µy(f(x,y) = 0) = min y such that f(x,y) = 0 and (f(x, z) ↓ for all z ≤ y and f(x, z) =6 0) ↑ otherwise Theorem If f is computable, so is g. VU Logic and Computation – p. 18 Example x D(x,y) = y if ∃z such that yz = x, and it is ↑ otherwise. VU Logic and Computation – p. 19 Example x D(x,y) = y if ∃z such that yz = x, and it is ↑ otherwise. D(x,y) := µz[eq(yz,x) = 0] where • Pred(0) = 0, and Pred(succ(x)) = x • x −⋆ 0 = x and x −⋆ succ(y) = Pred(x −⋆ y) • eq(x,y) = x −⋆ y + y −⋆ x VU Logic and Computation – p. 19 Church-Turing Thesis Formal notions appeared, starting in 1936: λ-calculus of Alonzo Church Turing machines of Alan Turing Recursive functions of Gödel and Kleene . These definitions look very different, but are provably equivalent. VU Logic and Computation – p. 20 Model of computations: equivalences • Recursive functions • λ-calculus • Combinatory logic • Turing machines • ... VU Logic and Computation – p. 21 Encoding of recursive functions in λ calculus Definition λ-definable functions f : N n →N is λ-definable if there is a closed λ-term F that n represents it: that is for all (x1,...xn) ∈N and y ∈N β • if f(x1,...xn) = y then Fx1 ...xn = y • if f(x1,...xn) ↑, then Fx1 ...xn has no β-nf VU Logic and Computation – p. 22 Church’s numerals n := λxy.x(x(. (x y) . ) | n times{z } Example 0 = λxy.y 1 = λxy.xy 2 = λxy.x(xy) (nM)N =β M(M(. (M N) . ) | n times{z } examples of functions on the blackboard ... VU Logic and Computation – p. 23 λ calculus and recursion Example (factorial function fact) λn. if n = 0 then 0 else n · ??(n − 1) VU Logic and Computation – p. 24 λ calculus and recursion Example (factorial function fact) λn. if n = 0 then 0 else n · ??(n − 1) Fixed point of a function f is a value x for which fx = x VU Logic and Computation – p. 24 λ calculus and recursion Example (factorial function fact) λn. if n = 0 then 0 else n · ??(n − 1) Fixed point of a function f is a value x for which fx = x Proposition Every λ term has a fixpoint, i.e. a term θ s.t. for all M: θM =β M(θM) 1. N := λf.λn. if n = 0 then 1 else n · f(n − 1) 2. fact := θN VU Logic and Computation – p. 24 Fixed Point and µ-recursion Fixed Point Is a term θ s.t. for all M: θM = M(θM). µ-recursion g(x) = µy(f(x,y) = 0) = min y such that f(x,y) = 0 and (f(x, z) ↓ for all z ≤ y and f(x, z) =6 0) ↑ otherwise VU Logic and Computation – p. 25 λ-definable ⇒ Turing Computable Theorem A λ-definable function is Turing Computable (T-Computable). VU Logic and Computation – p. 26 λ-definable ⇒ Turing Computable Theorem A λ-definable function is Turing Computable (T-Computable). To show this one can • code λ-terms as numbers (ensuring that operations for constructing and deconstructing terms are given by T-computable functions on codes) • write a T-interpreter for β-reductions VU Logic and Computation – p. 26 T-Computable ⇒ Recursive function Theorem A Turing Computable function is definable as a recursive function. VU Logic and Computation – p. 27 T-Computable ⇒ Recursive function Theorem A Turing Computable function is definable as a recursive function. To show this one can • code tape configurations and states of a Turing machine during a computation by natural numbers • the operation of the machine is represented by recursive functions of these numbers VU Logic and Computation – p.