CMSC 330: Organization of Programming Languages

Lambda Calculus Introduction

λ

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Review of CMSC 330

• Syntax– Regular expressions– Finite automata– Context-free grammars

• Semantics– Operational semantics– Lambda calculus

• Implementation– Concurrency

– Generics

– Argument passing

– Scoping

– Exceptions

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Introduction

• We’ve seen that several standard language “conveniences” aren’t strictly necessary– Multi-argument functions: use currying or tuples– Loops: use recursion– Side-effects: we don't need them either

• Goal: come up with a “core” language that’s as small as possible and still Turing complete– This will give a way of illustrating important language

features and algorithms

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Lambda Calculus

• A lambda calculus expression is defined as

e ::= x variable

| λx.e function

| e e function application

• λx.e is like (fun x -> e) in OCaml

• That’s it! Only higher-order functions

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Intuitive Understanding

• Before we work more with the mathematical notation of lambda calculus, we’re going to play a puzzle game!

http://worrydream.com/AlligatorEggs/

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Puzzle Pieces

• Hungry alligators: eat and guard family

• Old alligators: guard family

• Eggs: hatch into new family

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Example Families

• Families are shown in columns• Alligators guard families below them

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Puzzle Rule 1: The Eating Rule

• If families are side-by-side:– Top left alligator eats

the entire family to her right

– The top left alligator dies

– Any eggs she was guarding of the same color hatch into what she just ate

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Eating Rule Practice

• What happens to these alligators?

Puzzle 1:

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Eating Rule Practice

• What happens to these alligators?

Puzzle 1: Puzzle 2:

Answer 1: Answer 2:

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Puzzle Rule 2: The Color Rule

• If an alligator is about to eat a family and a color appears in both families then we need to change that color in one of the families.– In the picture below, green and red appear in both the first and second

families. So, in the second family, we switch all of the greens to cyan, and all of the reds to blue.

• If a color appears in both families, but only as an egg, no color change is made.

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Puzzle Rule 3: The Old Alligator Rule

• An old alligator that is guarding only one family dies.

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Challenging Puzzles!

• Try to reduce these groups of alligators as much as possible using the three puzzle rules:

• More links on website (see “Resources” page)

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Naming Families

• When family Not eats family True it becomes family False and when Not eats False it becomes True

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Lambda Calculus

• A lambda calculus expression is defined as

e ::= x variable (x: egg)

| λx.e function (λx: alligator)

| e e function application

(adjacency of families)

• Use parentheses to group expressions

(old alligators)

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Precedence and Associativity

• The scope of λ extends as far to the right as possible– λx.λy.x y is λx.(λy.(x y))

• Function application is left-associative– x y z is (x y) z– Same rule as OCaml

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Operational Semantics

• All we’ve got are functions, so all we can do is call them

• To evaluate (λx.e1) e2– Evaluate e1 with x bound to e2

• This application is called “beta-reduction”– (λx.e1) e2 → e1[x/e2] (the eating rule)

• e1[x/e2] is e1 where occurrences of x are replaced by e2• Slightly different than the environments we saw for Ocaml

– Substitutions instead of environments

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Examples

• (λx.x) z →• (λx.y) z →• (λx.x y) z →

– A function that applies its argument to y

• (λx.x y) (λz.z) →• (λx.λy.x y) z →

– A curried function of two arguments that applies its first argument to its second

• (λx.λy.x y) (λz.z z) x →

zy

(λz.z) y → y

z y

λy.z y

λy.((λz.z z)y)x → (λz.z z)x → x x

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Static Scoping and Alpha Conversion

• Lambda calculus uses static scoping

• Consider the following– (λx.x (λx.x)) z → ?

• The rightmost “x” refers to the second binding

– This is a function that takes its argument and applies it to the identity function

• This function is “the same” as (λx.x (λy.y))– Renaming bound variables consistently is allowed

• This is called alpha-renaming or alpha conversion (color rule)

– Ex. λx.x = λy.y = λz.z λy.λx.y = λz.λx.z

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Static Scoping (cont’d)

• How about the following?– (λx.λy.x y) y → ?– When we replace y inside, we don’t want it to be

“captured” by the inner binding of y

• This function is “the same” as (λx.λz.x z)

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Church-Turing Thesis

• Informally: If an algorithm exists to carry out a particular calculation, then– the same calculation can also be computed by a

Turing machine, and– the same calculation can also be represented by a

λ-function.

• More of a definition than a thesis– Cannot be proven– However, is nearly universally accepted

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Church-Turing Thesis

All algorithms can be represented by lambda calculus and executed on a Turing machine.

Turing MachinesAlan Turing

(1936)

Lambda CalculusAlonzo Church

(1940)

Church-Turing ThesisStephen Kleene

(1952)

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What does this mean?

• The Church-Turing thesis implies that we can encode any computation we want in lambda calculus– ... but ONLY if we’re sufficiently clever...

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Booleans

true = λx.λy.x

false = λx.λy.y

if a then b else c is defined to be the λ expression: a b c

• Examples:– if true then b else c → (λx.λy.x) b c → (λy.b) c → b– if false then b else c → (λx.λy.y) b c → (λy.y) c → c

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Booleans

Other Boolean operations:

not = λx.((x false) true)

and = λx.λy.((x y) false)

or = λx.λy.((x true) y)

• Given these operations, we can build up a logical inference system

• Exercise: Show that not, and and or have the desired properties

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Pairs

(a,b) = λx.if x then a else bfst = λf.f truesnd = λf.f false

• Examples:– fst (a,b) = (λf.f true) (λx.if x then a else b) → (λx.if x then a else b) true → if true then a else b → a– snd (a,b) = (λf.f false) (λx.if x then a else b) → (λx.if x then a else b) false → if false then a else b → b

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Natural Numbers (Church*)

0 = λf.λy.y1 = λf.λy.f y2 = λf.λy.f (f y)3 = λf.λy.f (f (f y))

i.e., n = λf.λy.<apply f n times to y>

succ = λz.λf.λy.f (z f y)iszero = λg.g (λy.false) true

– Recall that this is equivalent to λg.((g (λy.false)) true)

*(Named after Alonzo Church, developer of lambda calculus)

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Natural Numbers (cont’d)

• Examples:succ 0 =

(λz.λf.λy.f (z f y)) (λf.λy.y) →

λf.λy.f ((λf.λy.y) f y) →

λf.λy.f y = 1

iszero 0 =

(λz.z (λy.false) true) (λf.λy.y) →

(λf.λy.y) (λy.false) true →

(λy.y) true →

true

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Arithmetic defined• Addition, if M and N are integers (as λ expressions):

M + N = λx.λy.(M x)((N x) y)Equivalently: + = λM.λN.λx.λy.(M x)((N x) y)

• Multiplication: M * N = λx.(M (N x))• Prove 1+1 = 2.

1+1 = λx.λy.(1 x)((1 x) y) →λx.λy.((λx.λy.x y) x)(((λx.λy.x y) x) y) → λx.λy.(λy.x y)(((λx.λy.x y) x) y) → λx.λy.(λy.x y)((λy.x y) y) →λx.λy.x ((λy.x y) y) →λx.λy.x (x y) = λf.λy.f (f y) = 2

• With these definitions, can build a theory of integer arithmetic.

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Looping

• Define D = λx.x x• Then

– D D = (λx.x x) (λx.x x) → (λx.x x) (λx.x x) = D D

• So D D is an infinite loop– In general, self application is how we get looping

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The “Paradoxical” Combinator

Y = λf.(λx.f (x x)) (λx.f (x x))• Then

Y g =

(λf.(λx.f (x x)) (λx.f (x x))) g →

(λx.g (x x)) (λx.g (x x)) →

g ((λx.g (x x)) (λx.g (x x)))

= g (Y g)

• Thus Y g = g (Y g) = g (g (Y g)) = ...

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Example

fact = λf. λn.if n = 0 then 1 else n * (f (n-1))– The second argument to fact is the integer– The first argument is the function to call in the body

• We’ll use Y to make this recursively call fact

(Y fact) 1 = (fact (Y fact)) 1

→ if 1 = 0 then 1 else 1 * ((Y fact) 0)

→ 1 * ((Y fact) 0)

→ 1 * (fact (Y fact) 0)

→ 1 * (if 0 = 0 then 1 else 0 * ((Y fact) (-1))

→ 1 * 1 → 1

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Discussion

• Using encodings we can represent pretty much anything we have in a “real” language– But programs would be pretty slow if we really

implemented things this way– In practice, we use richer languages that include built-

in primitives

• Lambda calculus shows all the issues with scoping and higher-order functions

• It's useful for understanding how languages work