lambda calculus slides

8/8/2019 Lambda Calculus Slides

1/33

Introductory Courseon Logic and Automata Theory

Introduction to the lambda calculus

[email protected]

Based on slides by Jeff Foster, UMD

Introduction to lambda calculus . 1/


2/33

History

Formal mathematical systemSimplest programming language

Intended for studying functions, recursion

Invented in 1936 by Alonzo Church (1903-1995)Churchs Thesis:

Every effectively calculable function (effectively decidable predicate) is general recursive i.e. can be computed by lambda calculus

Churchs Theorem:

First order logic is undecidable



3/33

Syntax

Simple syntax:

e ::= x Variables

| x.e Functions| e e Function applications

Pure lambda calculus: only functionsArguments are functionsReturned value is functionA function on functions is higher-order



4/33

Semantics

Evaluating function application: ( x.e1) e2Replace every x in e1 with e2Evaluate the resulting term

Return the result of the evaluationFormally: -reduction

( x.e1) e2 e1[e2/ x]A term that can be -reduced is a redex We omit when obvious



5/33

Convenient assumptions

Syntactic sugar for declarationslet x = e1 in e2

Scope of extends as far to the right as possible

x. y. x y is x.( y.( x y))Function application is left-associative

x y zmeans ( x y) z



6/33

Scoping and parameter passing

-reduction is not yet well-dened:( x.e1) e2 e1[e2/ x]There might be many x dened in e1

ExampleConsider the programlet x = a in

let y = z. x inlet x = b in y xWhich x is bound to a , and which to b?



7/33

Lexical scoping

Variable refers to closest denitionWe can rename variables to avoid confusion:let x = a in

let y = z. x inlet w = b in y w



8/33

Free/bound variables

The set of free variables of a term is

FV ( x) = xFV ( x.e) = FV (e) \ { x}FV (e1 e2) = FV (e1)FV (e2)

A term e is closed if FV (e) = /0A variable that is not free is bound



9/33

-conversion

Terms are equivalent up to renaming of bound variables x.e = y.e[ y/ x] if y /FV (e)Renaming of bound variables is called -conversion

Used to avoid having duplicate variables, capturingduring substitution

Introduction to lambda calculus . /


10/33

Substitution

Formal denition

x[e/ x] = e y[e/ x] = y when x = y

(e1 e2)[e/ x] = (e1[e/ x] e2[e/ x])( y.e1)[e/ x] = y.(e1[e/ x]) when y = x and y /FV (e)

Example( x. y x) x = (w. y w) x y xWe omit writing -conversion



11/33

Functions with many arguments

We cant yet write functions with many argumentsFor example, two arguments: ( x, y).e

Solution: take the arguments, one at a time

x. y.eA function that takes x and returns another function thattakes y and returns e

( x. y.e) a b ( y.e[a / x]) b e[a / x][b/ y]This is called Currying Can represent any number of arguments



12/33

Representing booleans

true = x. y. xfalse = x. y. yif a then b else c = a b cFor example:

if true then b else c ( x. y. x) b c ( y.b) c bif false then b else c ( x. y. y) b c ( y. y) c c



13/33

Combinators

Any closed term is also called a combinator true and false are combinators

Other popular combinators:

I = x. xK = x. y. xS = x. y. z. x z ( y z)

We can dene calculi in terms of combinatorsThe SKI-calculusSKI-calculus is also Turing-complete



14/33

Encoding pairs

(a , b) = x.if x then a else bfst = p. p truesnd = p. p falseThen

fst (a , b) ... asnd (a , b) ... b



15/33

Natural numbers (Church)

0 = x. y. y1 = x. y. xy2 = x. y. x ( x y)i.e. n = x. y. apply x n times to ysucc = z. x. y. x ( z x y)

iszero = z. z ( y.false ) true



16/33

Natural numbers (Scott)

0 = x. y. x1 = x. y. y 02 = x. y. y 1i.e. n = x. y. y (n 1)succ = z. x. y. yzpred

= z

. z

0 ( x

. x

)iszero = z. z true ( x.false )



17/33

Nondeterministic semantics

( x.e1) e2 e1[e2/ x]e e

( x.e) ( x.e )

e1 e1

e1 e2 e 1 e2e2 e

2

e1 e2 e1 e 2Question: why is this semantics non-deterministic?



18/33

Example

We can apply reduction anywhere in the term( x.( y. y) x (( z.w) x) x.( x (( z.w) x) x. x w( x.( y. y) x (( z.w) x) x.( y. y) x w x. x w

Does the order of evaluation matter?



19/33

The Church-Rosser Theorem

Lemma (The Diamond Property):If a b and a c, then there exists d such that b d and c d

Church-Rosser theorem:If a b and a c, then there exists d such thatb d and c d

Proof by diamond propertyChurch-Rosser also called conuence

Introduction to lambda calculus . 1 /


20/33

Normal form

A term is in normal form if it cannot be reducedExamples: x. x, x. y. z

By the Church-Rosser theorem, every term reduces to at

most one normal formOnly for pure lambda calculus with non-deterministicevaluation

Notice that for function application, the argument need not bein normal form


i l


21/33

-equivalence

Let = be the reexive, symmetric, transitive closure of E.g., ( x. x) y y ( z.w. z) y y so all three are-equivalent

If a = b, then there exists c such that a

c and b

cFollows from Church-Rosser theorem

In particular, if a = b and both are normal forms, then they

are equal


N t t h l f


22/33

Not every term has a normal form

Consider = x. x xThen

In general, self application leads to loops. . . which is good if we want recursion


Fi i t bi t


23/33

Fixpoint combinator

Also called a paradoxical combinatorY = f .( x. f ( x x)) ( x. f ( x x))There are many versions of the Y combinator

Then, Y F = F (Y F )Y F = ( f .( x. f ( x x)) ( x. f ( x x))) F ( x.F ( x x)) ( x.F ( x x))

F (( x.F ( x x)) ( x.F ( x x))) F (Y F )


E ample


24/33

Example

fact (n) = if (n = 0) then 1 else nfact (n 1)Let G = f .n.if (n = 0) then 1 else nf (n 1)Y G 1 = G (Y G) 1

= ( f .n.if (n = 0) then 1 else nf (n 1)) (Y G) 1= if (1 = 0) then 1 else 1((Y G) 0)

= if (1 = 0) then 1 else 1(G (Y G) 0)

= if (1 = 0) then 1 else 1( f .n.if (n = 0) then 1 else nf (n 1) (Y G) 0)

= if (1 = 0) then 1 else 1(if (0 = 0) then 1 else 0((Y G) 0))

= 11 = 1


I th d


25/33

In other words

The Y combinator unrolls or unfolds its argument aninnite number of timesY G = G (Y G) = G (G (Y G)) = G (G (G (Y G))) = . . .G needs to have a base case to ensure termination

But, only works because we follow call-by-nameDifferent combinator(s) for call-by-value

Z = f .( x. f ( y. x x y)) ( x. f ( y. x x y))Why is this a xed-point combinator? How does itsdifference from Y work for call-by-value?


Why encodings


26/33

Why encodings

Its fun!Shows that the language is expressive

In practice, we add constructs as languages primitives

More efcientMuch easier to analyze the program, avoid mistakesOur encodings of 0 and true are the same, we may want

to avoid mixing them, for clarity


Lazy and eager evaluation


27/33

Lazy and eager evaluation

Our non-deterministic reduction rule is ne for theory, butawkward to implement

Two deterministic strategies:

Lazy : Given ( x.e1) e2 , do not evaluate e2 if e1 does notneed x anywhereAlso called left-most, call-by-name, call-by-need,applicative, normal-order evaluation (with slightlydifferent meanings)

Eager : Given ( x.e1) e2 , always evaluate e2 to a normalform, before applying the function

Also called call-by-value


Lazy operational semantics


28/33

Lazy operational semantics

( x.e1) l ( x.e1)

e1 l x.e e[e2/ x] l e

e1 e2 l e

The rules are deterministic, big-step The right-hand side is reduced all the way

The rules do not reduce under The rules are normalizing:

If a is closed and there is a normal form b such thata b, then a l d for some d


Eager (big-step) semantics


29/33

Eager (big-step) semantics

( x.e1) e ( x.e1)

e1 e x.e e2 e e e[e / x] e e

e1 e2 e e This big-step semantics is also deterministic and does notreduce under But is not normalizing!

Example: let x = in ( y. y)

Introduction to lambda calculus . 2 /

Lazy vs eager in practice


30/33

Lazy vs eager in practice

Lazy evaluation (call by name, call by need)Has some nice theoretical propertiesTerminates more often

Lets you play some tricks with innite objectsMain example: Haskell

Eager evaluation (call by value)

Is generally easier to implement efcientlyBlends more easily with side-effectsMain examples: Most languages (C, Java, ML, . . . )


Functional programming


31/33

Functional programming

The calculus is a prototypical functional programminglanguageHigher-order functions (lots!)No side-effects

In practice, many functional programming languages are notpure: they permit side-effects

But youre supposed to avoid them. . .


Functional programming today


32/33

Functional programming today

Two main campsHaskell Pure, lazy functional language; no side-effectsML (SML, OCaml) Call-by-value, with side-effects

Old, still around: Lisp, SchemeDisadvantage/feature: no static typing


Inuence of functional programming


33/33

Inuence of functional programming

Functional ideas move to other langaugesGarbage collection was designed for Lisp; now most newlanguages use GCGenerics in C++/Java come from ML polymorphism, orHaskell type classesHigher-order functions and closures (used in Ruby, existin C#, proposed to be in Java soon) are everywhere in

functional languagesMany object-oriented abstraction principles come fromMLs module system

. . .

Introduction to lambda calculus 33/

lambda calculus slides

Documents