Lambda Calculuscs-gw.utcluj.ro/~adrian/pf/pf_13.pdf · 2012-01-10 · λ-calculus Operational...

Lambda Calculus

Adrian Groza

Department of Computer ScienceTechnical University of Cluj-Napoca

January 3, 2011

λ-calculus Operational Semantics Lambda Calculus as a Functional Language Food for Tought

Outline

1 λ-calculus

2 Operational SemanticsSyntaxConversionsNormal Form

3 Lambda Calculus as a Functional LanguageAritmethic OperationsBoolean ExpressionsTuples

4 Food for Tought

What is Computable?

Computation is usually modelled as a mapping from inputs tooutputs, carried out by a formal ”machine” or program, whichprocesses its input in a sequence of steps.

An ”effectively computable” function is one that can becomputed in a finite amount of time using finite resources.

Computation Models

Turing Machine (Turing, 1936)modifies the content of the memory and of aninput tape whose movement is controllablecorresponds to imperative languages where thestate of the computation process is representedand explicitly processed

Lambda Calculus (Church, 1932)works on unary functionsrelies on function application ⇔ the substitutionin the function body of the formal parameter withthe effective parameterprocessing without state

(1912-1954)

(1903-1995)

Church-Turing thesis

If an algorithm (a procedure that terminates) exists then there is anequivalent Turing Machine or applicable λ-function for that algorithm.

Lambda Calculus

1 universal: any computable function can be expressed andevaluated using this formalism

2 to directly express higher-order functions3 to study evaluation orders, termination, uniqueness of

answers4 to serve as a kernel language for functional languages5 the smallest universal programming language of the world

Outline

1 λ-calculus

4 Food for Tought

Syntax

E = x | λx . E | E1 E2

x - variableλx.E - functional abstraction (x - bound variable, E - body)E1 E2 - function application (E1 function, E2 argument)

Examples:x λ-expression reduced to a variableλx.x is a function taking an argument x, and returning x (identity fun)λx. λy.x function which selects the first value from a pair of valuesf x is a function f applied to an argument x (same as f(x))(λx.x) y applying the function λx.x on the argument y

Syntax

Examples

(λx. x x) (λy. y) ≡ = (λy. y)(λx.(x + 5)) 3 = 3+5 = 8((λx.(λy.x + y)) 3) 5 = (λy.3 + y)) 5 = 3+5=8(λf.λx.f(f x))(λy.y+1) = λx.(λy.y+1)((λy.y+1) x)

= λx.(λy.y+1)(x+1)= λx.x+1+1

Syntax

Parsing Lambda Expressions

1 Lambda extends as far as possible to the rightλf.x y≡ λ f.(x y)λx.E1 E2 E3≡ λ x.(E1 E2 E3)λx.x λz.x z x ≡λx.(x λz.(x z x))

2 Application is left-associativex y z ≡ (x y) z

3 Parantheses are needed for (λx.E1 E2) E3 where E3 is anargument to the function λx.E1 E2, not part of the body ofthe function

4 Multiple lambdas may be suppressedλ f g . x ≡ λ f . λ g . x

Syntax

Functions with Many Arguments

We can’t yet write functions with many arguments - forexample, two arguments: λ(x,y).eSolution: take the arguments, one at a time

λx.λy.eA function that takes x and returns another function thattakes y and returns e(λx.λy.e) a b → (λy.e[a/x]) b → e[a/x][b/y]This is called CurryingCan represent any number of arguments

Syntax

Free and Bound Variables

In function λx.t , the variable x is bound; t is the body of thefunctionA variable which is not bound is free.

Examples:(λx . x y) x is bound, y is free(λx . x)(λy . y x ) x is bound in the first expression, x is independent of x(λz . (λx . (y x))) x is bound, y is free

Observations:Name of bound var does not matter: λx.(3+x) = λy.(3+y)Name of free var does matter: λx.(x+y) 6= λx.(x+z)Same identifier can occur free and bound in the sameexpression: (λx.xy)(λy.y)

Conversions

α-conversion

Conversion = rule for transforming a λ-expression into aanother one with the same meaning.

α-conversion: renaming bound variables(λx .t) ≡α (λy .t [y/x ])

Example(λx .ax) ≡α (λy .ay)(λx .(x(λy .(yx)))) ≡α (λz.(z(λy .(yz))))[z/x ](λx .x(λx .ax))b ≡α (λy .y(λx .ax))b[y/x ]

Conversions

β-conversion

β-conversion: applying abstractions to arguments((λx .t)u) ≡β (t [u/x ])

Example((λx .((fx)x)(ga) ≡β ((f (ga))(ga))((λz.(za)(λx .x) ≡β (λx .x)a ≡β a((λx .λy .y)a)b) ≡β ((λy .y))b

≡ (λy .y)b≡β b

β-conversion represents the computationalengine of the lambda calculus.

Conversions

η-conversion

η-conversion: eliminates redundant lambda-expressions(λx .fx)y ≡η fy

Every time when x is not free in f, we can write:

(λ x . f x ) = f

Example(λ b . y b) ≡ y(λ z . (λ x . x y) z) ≡ λ x . x y

Conversions

Applying Conversions

Conversions1 (λx .t) ≡α (λy .t [y/x ])

2 ((λx .t)u) ≡β (t [u/x ])

3 (λx .fx)y ≡η fy

(λ x y . x y) (λ x . x y) (λ a b . a b) NB: left assoc≡α (λ x z . x z) (λ x . x y) (λ a b . a b) α-conversion [z/x]≡β (λ z . (λ x . x y) z) (λ a b . a b) β-reduction [(λ x . x y)/x]≡η (λ x . x y) (λ a b . a b) η-reduction (λ z . f z )= f≡β (λ a b . a b) y β-reduction [(λ a b . a b)/x]≡β (λ b . y b) β-reduction [(y/a]≡η y η-reduction(λ z . f z)= f

Conversions

Nested expressions

Syntax simplification:

(λx1 . (λx2 . ...(λxn . t)...)) ≡ (λx1x2...xn. t)

(λx y.y x) y b ≡ (λx . (λy. y x)) y b≡α (λx . (λz. z x)) y b≡β (λz . z y) b≡β b y

(λx y.y x) y b ≡ (λx . (λy. y x)) y b≡β(λy . y y) b≡β b b

Don’t accidentally bind a free variable!

Normal Form

A λ-expression is said to be in the normal form if no furtherreduction steps could be applied to it.

Example

(λ x. a x)((λ y. b y) c) ≡α (λ z. a z)((λ y. b y) c)≡β a ((λ y. b y) c)≡ a (b c)

Not all expressions have a normal form(λ x . x x) (λ x . x x) ≡β (x x)[(λ x . x x)/x]

≡ (λ x . x x) (λ x . x x)≡ ... ⇔ the program never ends.

Normal Form

Church-Rosser Thesis

ThesisTwo sequences of reductions applied to a λ-expressions willalways result in equivalent normal forms.

Thus the normal form does not depend on the order thereductions are performed.(λ x. a x)((λ y. b y) c) ≡α (λ z. a z)((λ y. b y) c)

≡β a ((λ y. b y) c)≡ a (b c)

(λ x. a x)((λ y. b y) c) ≡β (λx. a x)(b c)≡βa (b c)

Normal Form

Call by name and call by value in λ-calculus

Call by name(λ y . a) ((λ x . x x) (λ x . x x))≡β aargument is not evaluated and it is substitued in the body of thefunction.

Call by value(λ y . a) ((λ x . x x) (λ x . x x)) ≡β (λ y . a) ((λ x . x x) (λ x . x x))

This type of substitution always finds the normal form, providedthat such a form actually exists.

(\x -> \y -> x) 1 (5/0) = 1

Outline

1 λ-calculus

4 Food for Tought

Aritmethic Operations

λ-Calculus Expressivity

In the lambda calculus we can only define new functions.How can we program with only functions?

IdeaEncode the behavior of values and not their structure.

Introducing variableslet x = e1 in e2 ⇔ (λx.e1) e2

Aritmethic

Can we represent numbers in λ-calculus?Numbers will be defined as ... functions:0 := λsz.z1 := λsz.s(z)2 := λsz.s(s(z))...

DefinitionA natural number is a function that given an operation s and astarting value z, applies s a number of times to z.

Successor Function

Our first interesting function...

S ≡ λwyx.y(wyx)

S 0 ≡ λw y x . y(wyx) (λsz . z)≡β λy x . y ((λsz . z) y x)≡β λy x . y ((λz . z) x)≡β λy x . y (x)≡α λs x . s(x)≡1

Summing numbers

Our second interesting function...

PLUS ≡ λ mnfx . nf (mfx)

PLUS 1 2 ≡ (λ mnfx.nf (mfx))1 2≡ββ λ fx.2f (1fx)≡ λ fx.2f ((λ fx.f (x)) fx)≡η λ fx.2f (f (x))≡ λ fx.(λ fx.f (f (x)))f (f (x))≡ λ fx.f (f (f (x)))≡α 3

Boolean Expressions

What is ”true”?True is something that used as the first parameter of if makesthe result of if to be its second parameter

if T M N ⇒ M

if ≡ λ p. λthen.λelse. p then elseif ≡ λp . (λ c . (λ a . pca))) ≡ λpca . pca

A boolean is a function that given two choices selects one of themT ≡ λx . (λy. x) ≡ λx y . xF ≡ λx . (λy. y) ≡ λx y . y

if T M N ≡ λp c a . p c a (λx y . x) M N≡β (λc a . (λx y . x) c a) M N≡β (λa . (λx y . x) a) M N≡β (λx y . x) M N≡β (λy . M) N ≡βM

Boolean Expressions

Logical Operators

And, Or, Notand ≡ λx (λy if x y F)≡ λxy. x y For ≡ λx (λy if x T y) ≡ λxy. x T ynot ≡ λx . x F T

1 (λx .t) ≡α (λy .t [y/x ])

2 ((λx .t)u) ≡β (t [u/x ])

and T F ≡ (λx y. x y F) T F≡β (λy. T y F) F≡β T F F≡β (λx y . x) F F≡β (λy . F) F≡β F

Boolean Expressions

Logical Operators

And, Or, Notand ≡ λx (λy if x y F)≡... λxy. x y For ≡ λx (λy if x T y) ≡ λxy. x T ynot ≡ λx . x F T

1 (λx .t) ≡α (λy .t [y/x ])

2 ((λx .t)u) ≡β (t [u/x ])

and T F ≡ (λx y. x y F) T F≡β (λy. T y F) F≡β T F F≡β (λx y . x) F F≡β (λy . F) F≡β F

not T ≡ (λ b . b F T ) (λ x y . x )≡β (λ x y . x) F T≡β F

Boolean Expressions

Testing if a number is zero

Another interesting function ...

Z ≡ λx.xF¬F

Z 0 ≡ λx.xF¬F 0≡ 0F¬F≡ (λs z . z) F¬F≡ ¬F≡ T

Boolean Expressions

Testing if a number is zero

Z ≡ λx.xF¬F

Z 0 ≡ λx.xF¬F 0≡ 0F¬F≡ (λs z . z) F¬F≡ ¬F≡ T

Z 1 ≡ λx.xF¬F 1≡ 1F¬F≡ (λsz.s(z))F¬F≡ F(¬)F≡ (λxy.y)(¬)F≡ F

Tuples

Defining pairs

PAIR ≡ λxyf.fxyFIRST ≡ λ p. p TSECOND ≡ λ p. p F

First (Pair (a b)) ≡ (λp.p T) (PAIR a b)≡ (PAIR a b) T≡ (λxyf.fxy a b) T≡ (λf. f a b) T≡ T a b≡ a

Tuples

Recursion

A self-regenerating function...

Y ≡ λy.(λx.y(xx))(λx.y(xx))

YR ≡ λy.(λx.y(xx))(λx.y(xx))R≡ (λx.R(xx))(λx.R(xx))≡ R((λx.R(xx))(λx.R(xx)))≡ R(YR)

When Y is applied on an arbitrary function R, R will call itself on(YR). Subsequent equivalent forms appear when R getsexpanded according to its definition. It is R’s responsibility tobreak the recursive loop at some moment by providing acondition for stopping.

Tuples

Recursion - Example

fact = λ f. λ n.if n = 0 then 1 else n * (f (n-1))(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

Outline

1 λ-calculus

4 Food for Tought

Readings

1 Mandatory reading: Chapter 8: pages107-119

2 Optional reading: A Tutorial Introduction to theLambda Calculus, Raul Rojas, FU Berlin,Germany, WS-97/98

Lambda Calculuscs-gw.utcluj.ro/~adrian/pf/pf_13.pdf · 2012-01-10 · λ-calculus Operational...

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