David Evanshttp://www.cs.virginia.edu/evans

CS200: Computer ScienceUniversity of VirginiaComputer Science

Class 33:Making Recursion

M.C. Escher, Ascending and

Descending also known as

f. (( x.f (xx)) ( x. f (xx))

16 April 2003 CS 200 Spring 2003 2

Menu

• Recap: Proving Lambda Calculus is Universal

• How to make recursive definitions without define

• Fixed points

16 April 2003 CS 200 Spring 2003 3

Lambda Calculus Review

( f. (( x.f (xx)) ( x. f (xx)))) (z.z)

term ::= variable |term term | (term)| variable . term

-reduction (renaming)

y. M v. (M [y v])

where v does not occur in M.

-reduction (substitution)

(x. M)N M [ x N ]

Parens are just for grouping, not like in Scheme

16 April 2003 CS 200 Spring 2003 4

Universal Language

• Is Lambda Calculus a universal language?– Can we compute any computable algorithm

using Lambda Calculus?

• To prove it isn’t:– Find some Turing Machine that cannot be

simulated with Lambda Calculus

• To prove it is:– Show you can simulate every Turing Machine

using Lambda Calculus

16 April 2003 CS 200 Spring 2003 5

Simulating Computationz z z z z z z z z z z z z z z zz z z z

1

Start

HALT

), X, L

2: look for (

#, 1, -

), #, R

(, #, L

(, X, R

#, 0, -

Finite State Machine

• Lambda expression corresponds to a computation: input on the tape is transformed into a lambda expression

• Normal form is that value of that computation: output is the normal form

• How do we simulate the FSM?

16 April 2003 CS 200 Spring 2003 6

Lambda Calculus is a Universal Computer

z z z z z z z z z z z z z z z zz z z z

1

Start

HALT

), X, L

2: look for (

#, 1, -

), #, R

(, #, L

(, X, R

#, 0, -

Finite State Machine

• Read/Write Infinite Tape Mutable Lists• Finite State Machine Numbers to keep track of state• Processing Way of making decisions (if) Way to keep going

We have this, butwe cheated using to make recursive definitions!

16 April 2003 CS 200 Spring 2003 7

Alyssa P. Hacker’s Answer

( f. (( x.f (xx)) ( x. f (xx)))) (z.z)

(x.(z.z)(xx)) ( x. (z.z)(xx))

(z.z) ( x.(z.z)(xx)) ( x.(z.z)(xx))

(x.(z.z)(xx)) ( x.(z.z)(xx))

(z.z) ( x.(z.z)(xx)) ( x.(z.z)(xx))

(x.(z.z)(xx)) ( x.(z.z)(xx))

...

16 April 2003 CS 200 Spring 2003 8

Recursive Definitions

Which of these make you uncomfortable?

x = 1 + xx = 4 – xx = 9 / xx = x

x = 1 / (16x3)

No solutions over integers1 integer solution, x = 22 integer solutions, x = 3, x = -3Infinitely many integer solutions

No integer solutions, two rational solutions (1/2 and –1/2), four complex solutions (1/2, -1/2, i/2, -i/2)

16 April 2003 CS 200 Spring 2003 9

Equations Functions

x = 4 – x

f x. sub 4 xFind an x such that (f x) = x.

Same as solve for x.

sub x . y. if (zero? y) x

(sub (pred x) (pred y))

16 April 2003 CS 200 Spring 2003 10

What’s a function?

f x. sub 4 x

f: Nat Nat

{ <0, 4>, <1, 3>, <2, 2>,

<3, 1>, <4, 0>,

<5, ???>, ... }

16 April 2003 CS 200 Spring 2003 11

Domains

• Set: unordered collection of values

Nat = { 0, 2, 1, 4, 3, 6, 5, 8, 7, ... }

• Domains: like a set, but has structure

Nat = { 0, 1, 2, 3, 4, 5, 6, 7, 8, ... }

where 0 is the least element, and there is an order of the elements.

16 April 2003 CS 200 Spring 2003 12

Making Domains

• Primitive domains can be combined to make new domains

• Product domain: D1 x D2

– Pairs of values

Nat x Nat =

{ (tupleNat,Nat 0 0), (tupleNat,Nat 0 1),

(tupleNat,Nat 1 0), (tupleNat,Nat 1 1),

(tupleNat,Nat 0 2), ... }

16 April 2003 CS 200 Spring 2003 13

Functions• A set of input-output pairs

– The inputs (Di) and outputs (Do) are elements of a domain

• The function is an element of the domain Di Do.

• It is a (completely defined) function if and only if for every element d Di, the set of input-output pairs has one member whose first member matches d.

16 April 2003 CS 200 Spring 2003 14

Functions?

f: Nat Nat z. subNat Nat 4 z

f: Nat Int z. subNat Int 4 z

f: Nat Nat z. addNat Nat z 1

Not a function since there is no value in output domain for z > 4.

16 April 2003 CS 200 Spring 2003 15

Functions

f: (Nat Nat) n. add 1 ( f n))

f: (Nat Nat) n. f (add 1 n))

No solutions (over natural numbers) – would require x = 1 + x.

Infinitely many solutions – e.g., f(x) = x. 3{ <0, 3>, <1, 3>, <2, 3>, ... }

16 April 2003 CS 200 Spring 2003 16

Fixed Points

• A fixed point of a function f: (D D) is an element d D such that

(f d) = d.

• Examples:

f: Nat Int z. sub 4 z

f: Nat Nat z. mul 2 z

f: Nat Nat z.z

2

0

infinitely many

16 April 2003 CS 200 Spring 2003 17

Generating Functions

• Any recursive definition can be encoded with a (non-recursive) generating function

f: (Nat Nat) n. (add 1 ( f n)) g: (Nat Nat) (Nat Nat)

f. n. (add 1 ( f n))• Solution to a recursive definition is a fixed

point of its associated generating function

16 April 2003 CS 200 Spring 2003 18

Example fact: (Nat Nat) n. if (= n 0) 1 (mul n (fact (sub n 1)))

gfact: (Nat Nat) (Nat Nat) f.n. if (= n 0) 1 (mul (n (f (sub n 1)))

16 April 2003 CS 200 Spring 2003 19

gfact I

gfact (z.z)

n. if (= n 0) 1

(mul (n ((z.z) (sub n 1)))

What is (gfact I) 5?

gfact: (Nat Nat) (Nat Nat) f.n. if (= n 0) 1 (mul (n (f (sub n 1)))

16 April 2003 CS 200 Spring 2003 20

gfact fact

gfact fact

n. if (= n 0) 1

(mul n (fact (sub n 1)))

fact

fact is a fixed point of gfact

gfact: (Nat Nat) (Nat Nat) f.n. if (= n 0) 1 (mul (n (f (sub n 1)))fact : (Nat Nat) n. if (= n 0) 1 (mul (n (fact (sub n 1)))

16 April 2003 CS 200 Spring 2003 21

Unsettling Questions

• Is the factorial function the only fixed point

of gfact?

• Given an arbitrary function, how does one find a fixed point?

• If there is more than one fixed point, how do you know which is the right one?

16 April 2003 CS 200 Spring 2003 22

Iterative Fixed Point Technique

To find a fixed point of g: D D

• Start with some element d D • Calculate g(d), g(g(d)), g(g(g(d))), ...

until you get g(g(v)) = v

then v is a fixed point.

16 April 2003 CS 200 Spring 2003 23

Where to start?

• If you start with D you get the least fixed point (which is the “best” one)

(pronounced “bottom”) is the element of

D such that for any element d D, d.

• means “has less information than” or “is weaker than”

• Not all domains have a .

16 April 2003 CS 200 Spring 2003 24

Pointed Partial Order

• A partial order (D, ) is pointed if it has a bottom element u D such that u d for all elements d D.

• Bottom of (Nat, <=)?• Bottom of (Nat, =)?• Bottom of (Int, <=)?• Bottom of ({Nat}, <=)?

0

Not a pointed partial order

Not a pointed partial order

{}

16 April 2003 CS 200 Spring 2003 25

• Think of bottom as the element with the least information, or the “worst” possible approximation.

• Bottom of Nat Nat is a function that is undefined for all inputs. That is, the function with the graph {}.

• To find the least fixed point in a function domain, start with the function whose graph is {} and iterate.

Getting to the of things

16 April 2003 CS 200 Spring 2003 26

Least Fixed Point of gfact

gfact: (Nat Nat) (Nat Nat) f. n. if (= n 0) 1

mul n ( f (sub n 1)))

gfactn (function with graph {})

= fact as n approaches infinity.

16 April 2003 CS 200 Spring 2003 27

Fixed Point Theorem

Do all -calculus terms have a fixed point?

16 April 2003 CS 200 Spring 2003 28

Fixed Point Theorem

F , X such that FX = X

• Proof:Let W = x.F(xx) and X = WW.

X = WW = ( x.F(xx))W

F (WW) = FX

16 April 2003 CS 200 Spring 2003 29

Why of Y?• Y is f. WW:

Y f. ( x.f (xx)) ( x. f (xx))

• Y calculates a fixed point of any lambda term!

• Hence: we don’t need define to do recursion!– Works in Scheme too - check the

“lecture” from the Adventure Game

16 April 2003 CS 200 Spring 2003 30

Fixed PointThe fixed point of our Turing Machine simulator on some input is the result of running the TM on that input. If there is no fixed point, the TM doesn’t halt!

fixed-point TM input

result of running TM on input

16 April 2003 CS 200 Spring 2003 31

Lambda Calculus is a Universal Computer!

z z z z z z z z z z z z z z z zz z z z

1

Start

HALT

), X, L

2: look for (

#, 1, -

), #, R

(, #, L

(, X, R

#, 0, -

Finite State Machine

• Read/Write Infinite Tape Mutable Lists• Finite State Machine Numbers to keep track of state• Processing Way of making decisions (if) Way to keep going

16 April 2003 CS 200 Spring 2003 32

Mystery Function (Teaser)

p xy. pca.pca (x.x xy.x) x) y (p ((x.x xy.y) x) (x. z.z (xy.y) y)

m xy. pca.pca (x.x xy.x) x) x.x (p y (m ((x.x xy.y) x) y))f x. pca.pca ((x.x xy.x) x) (z.z (xy.y) (x.x)) (m x (f ((x.x xy.y) x)))

16 April 2003 CS 200 Spring 2003 33

QED

• Lambda calculus is a Universal Programming language

• All you need is beta-reduction and you can compute anything

• Integers, booleans, if, while, +, *, =, <, classes, define, inheritance, etc. are for wimps! Real programmers only use and beta-reduction.

16 April 2003 CS 200 Spring 2003 34

Charge

• Friday: chance to ask questions before Exam 2 is handed out – Email questions before if you want to make

sure they are covered– Today’s lecture is not covered by Exam 2

• Office hours this week: – Wednesday, after class - 3:45– Thursday 10am-10:45am; 4-5pm.