[PDF] - -calculus Motiv ation: primitiv es of and pro cedural PDF Document

SLIDE 1

calculus

and com binatory logic Jan Ma luszy

nski

E++ 1o

r

ro

m

210 Phone: 1483 Mail: janma@ida.liu.se

Motiv

ation

Computation

as rewriting

Lam

b da calculus

Com

binatory logic

Relating

them 1 Motiv ation: primitiv es

f

pro cedural programming Lam b da calculus:

A

formalism that captures the mec hanism

f

{ making expressions to pro cedures, { pro cedure calls

A

general mo del

f

computation.

Reference

concepts for studying pro cedural languages. Pla y ed imp

rtan

t role in dev elopmen t

f

programming languages, ALGOL, LISP ,... Com binatory logic:

A

related formalism

Imp
rtan

t concept

f

c

mbinator
Inuenced

implemen tation tec hniques Computation seen as Rewriting 2 An Example: Simplication

f

P

lynomials

Rules: + x ! x

x

! x(y + z ) ! xy + xz (x + y )z ! xz + y z Applied to an expression: (0 + a)(a + b) ! a(a + b) ! a 2 + ab An alternativ e reduction: (0 + a)(a + b) ! (0 + a)a + (0 + a)b ! (0

a

+ a 2 ) + (0 + a)b ! (0 + a 2 ) + (0 + a)b ! a 2 + (0 + a)b ! a 2 + (0

b

+ ab) ! a 2 + (0 + ab) ! a 2 + ab 3 Simplication

f

P

lynomials:

a Summary W e ha v e:

a

domain: the p

lynomials
a

binary rewrite relation

n

the domain: sp ecied b y rules Desirable prop erties:

an

y sequence

f

reductions is nite (termination).

for

eac h expression the

utcome

is unique. 4

SLIDE 2 Abstract Reduction Systems An ars is a pair (A; !), where

A

is a set,

!

is a binary relation

n

A: a r ewrite r elation. Notation

a

! b: "b is a r e duct

f

a"

!

!: the reexiv e and transitiv e closure

f

!

:

the in v erse

f

!

:

the in v erse

f

! !

=:

(! ! [ )

.

5 T uring Mac hines : : : : : : 1 1 Read/write head 6 S tap e alphab et; Q states;

:

Q

S

) Q

S
fleft

; right g. s 1 : : : s i1 q s i : : : s n represen ts a conguration with

the

tap e s 1 : : : s i1 s i : : : s n

the

state q

the

p

sition
f

the head at s i c

mputation

, r ewriting: If

(q

; s i ) = (q ; t; right ), then s 1 : : : s i1 q s i : : : s n ` s 1 : : : s i1 t q s i+1 : : : s n 6 The con v ertibilit y relation a and b are c

nvertible

(a = b) if a can b e rewritten in to b using the ! relation forw ards

r

bac kw ards a nite n um b er

f

times. a b q q q q q q q q q q : : :

@

@ R @ @ R @ @ R

@

@ R @ @ R

7

Normalizing ars Giv en an ars (A; !):

a

is a normal form : there is no b suc h that a ! b.

b

is a normal form

f

a: a ! ! b and b is a normal form.

!

is we akly normalizing if ev ery a 2 A has a normal form.

!

is str

ngly

normalizing (or terminating , if there is no innite sequence a ! a 1 ! a 2 ! : : : a a 1 b a 2 b 1 a 3 b 2 . . .

@

@ R

@

@ R

@

@ R

@

@ R 8

SLIDE 3 Conuen t ars Giv en an ars (A; !):

!

is Chur ch-R

sser

(CR) (or c

nuent

) if for all a; b; c 2 A suc h that a ! ! b and a ! ! c, there is a d 2 A suc h that b ! ! d and c ! ! d.

!

is we akly Chur ch-R

sser

(WCR) (or we akly c

nuent

) if for all a; b; c 2 A suc h that a ! b and a ! c, there is a d 2 A suc h that b ! ! d and c ! ! d. a b c d CR

@

@ @ @ R @ @ @ @ R @ @ @ @ R @ @ R

a

b c d W CR

@

@ @ @ R @ @ @ @ R @ @ R

a

b c d

?
6

9 Conuen t ars's sp ecify functions Giv en a conuen t ars (A; !):

ev

ery a 2 A has at most

ne

normal form

!

denes a partial function f

n

A: f (a) = 8 > > < > > : b; if b 2 NF (A) and a ! ! b undened else

if

! is w eakly normalizing, then f is a total function.

if

! is strongly normalizing, then f can b e computed with an y reduction strategy . 10 The ars b elo w is W CR but not CR. It is also non-terminating. a b c d

?
6

Newman's lemma: If ! is w eakly conuen t and terminating, then ! is conuen t. 11 Lam b da calculus Lam b da calculus is an ars

Domain:

lamb da terms

Rewrite

relation: r e duction rules 12

SLIDE 4 Lam b da terms In tuition

Making

an expression to a function, e.g. fun twice x = 2*x;

Using

the function, e.g. twice 2; F

rmal

denition:

a

v ariable (x; y ; z ; :::) is a lam b da term,

if

M is a lam b da term, then (x:M ) is a lam b da term (functional abstr action)

if

M and N are lam b da terms then (M N ) is a lam b da term (functional applic ation) 13 Notational con v en tions for lam b da terms

Left-asso

ciativit y

f

application ((xy )z ) ) xy z

Skipping

m ultiple s (x:(y :(M ))) ) xy :M Example: write xy :xy (xy ) instead

f

(x:(y :((xy )(xy )))) 14 In tuition

f

rewriting

x:M

is a function with parameter x: x is b

und

in M , non-b

und

v ariables are fr e e.

(x:M

)N is a \call": r eplac e x in M b y N . (x:x)y ! y (x:xx)(x:x)y ! (x:x)(x:x)y ! (x:x)y ! y But ho w to handle: (xy :xy )y Replacemen t m ust not bind free v ariables. 15 The replacemen t

p

eration [x 7! N ]M (i) [x 7! N ]x is N , (ii) [x 7! N ]y is y for an y v ariable y dieren t from x, (iii) [x 7! N ](P Q) is ([x 7! N ]P [x 7! N ]Q), (iv) [x 7! N ]x:P is x:P , (v) [x 7! N ]y :P is y :[x 7! N ]P if y is dieren t from x and has no free

ccurrence

in N ,

r

if x has no free

ccurrence

in P , (vi) [x 7! N ]y :P is z [x 7! N ]([y 7! z ]P ) if y is dieren t from x and has a free

c-

currence in N , and x has a free

ccurrence

in P . z is a v ariable that do es not

ccur

free in N nor in P . (xy :xy )y ! [x 7! y ](y :xy ) = z [x 7! y ]([y 7! z ]y :xy ) = z [x 7! y ](z :xz ) = (z :y z ) 16

SLIDE 5 ( ) If y is not free in M , then x:M !

y

:[x 7! y ]M . ( ) (x:M )N !

[x

7! N ]M

In

tuition

f

!

:

renaming; x:xy !

z

:z y

Restriction
n

!

:

\fresh" v ariables are to b e used: x:xy 6!

y

:y y

!

!

is

symmetric, e.g.: xy :xy ! !

z

w :z w z w :z w ! !

xy

:xy Hence the

rule

is called

c
nversion

rule . If s ! !

t

then s and t are called

c
ngruent.

17 The

reduction

rule

In

tuition

f

!

:

function call.

Computation

b y

reduction

ma y

r

ma y not terminate, e.g. (y :y y )(y :y y ) ! (y :y y )(y :y y )

Ch

urc h-Rosser theorem: if P ! !

M

and P ! !

N

then there exist

congruen

t terms T 1 and T 2 suc h that M ! !

T

1 and N ! !

T

2 .

Hence,

ev ery lam b da term has a unique (up to

con

v ersion) normal form,

r

no normal form at all. 18 Some rewrite strategies

The

normal-or der r e duction (lazy ev aluation): reduce the left-most redex,e.g.: (x:xx)((y :y )(y :y )) ! ((y :y )(y :y ))((y :y )(y :y )) ! (y :y )((y :y )(y :y )) ! ((y :y )(y :y )) ! y :y

The

applic ative-or der r e duction (call b y v alue): reduce the left-most innermost redex, e.g.: (x:xx)((y :y )(y :y )) ! (x:xx)(y :y ) ! ((y :y )(y :y )) ! y :y 19 The rewrite strategies and termination

If

the normal form exists, the normal-order reduction will compute it.

The

applicativ e

rder

reduction ma y not terminate ev en if there is a normal form, e.g.: (xz :z )((y :y y )(y :y y )) ! (xz :z )((y :y y )(y :y y )) using the applicativ e-order reduction. But: (xz :z )((y :y y )(y :y y )) ! (z :z ) using the normal-order reduction. 20

SLIDE 6 Represen ting n um b ers as lam b da terms Chur ch numer als: Represen t a natural n um b er n b y xy :x n y where x n y denotes n applications

f

x to y . Th us:

xy

:y abbreviated as 0, denotes 0.

xy

:xy abbreviated as 1, denotes 1.

xy

:x(xy ) abbreviated as 2, denotes 2.

etc.

... Notice:

Eac

h Ch urc h n umeral is normal form.

It

can b e seen as a function

n

lam b da terms: giv en lam b da terms M and N : nM N ! ! M n N 21 The p

w

er

f

lam b da calculus

:

an m-ary function

n

N at

is

lamb da-dene d b y a lam b da term F : i for arbitrary natural n um b ers n 1 ; :::; n m F n 1 ::: n m ! ! n i (n 1 ; :::; n m ) = n.

is

lamb da-denable if it is lam b da-dened b y some lam b da term. The class

f

lam b da-denable functions coin- cides with the class

f
T

uring-computable functions 22 Discussion Lam b da terms dening functions are complex Example

Lam

b da term s dening successor: uxy :x(uxy ) sn= (uxy :x(uxy ))(z v :z n v ) ! xy :x((z v :z n v )xy ) ! xy :x((v :x n v )y ) ! xy :x(x n y ) =n+1

Addition:

nxy ! ! x n y , x + y = s x (s y (0)), hence xy :xs(y s 0) denes addition. Its extended form is: xy :x(uxy :x(uxy )(y (uxy :x(uxy ))(xy :y )). 23 Expressing xp

in

ts

Fixp
in

ts

f

functions can b e represen ted b y lam b da terms: { Existence

f

xp

in

ts: F

r

ev ery term M there exists P s.t.: M P =

P

{ Ho w to represen t them: There exists Y s. that for ev ery M M (Y M ) =

Y

M e.g. tak e x:((y :x(y y ))(y :x(y y ))) as Y , then: Y M = x:(y :x(y y ))(y :x(y y ))M ! (y :M (y y ))(y :M (y y )) ! M ((y :M (y y ))(y :M (y y ))) = M (Y M ) 24

SLIDE 7 Com binatory logic

CL

is y et another rewrite system

Equiv

alen t to lam b da calculus

Do

es not use b

und

v ariables

Used

in implemen tation

f

functional languages

In

tuition: to represen t comp

sition
f

functions 25 Com binatory terms (i) A v ariable is a com binatory term, (ii) The basic com binators K and S are com- binatory terms, (iii) If X and Y are com binatory terms, then the applic ation (X Y ) is a com binatory term with subterms X and Y . e.g.: ((S K )K ), (((S K )K )x) Notation: the application assumed left-asso ciativ e, e.g. S S (K (S K K )) denotes ((S S )(K ((S K )K ))) In tuition:

C-terms

represen t (partial) functions

n

C-terms computed b y rewriting.

The

v alue

f

F for A, is the normal form

f

the term (F A).

K

allo ws for creation

f

constan t functions.

S

describ es a sp ecic w a y

f

comp

sition
f

functions. 26 Com binatory reduction

K

X Y ! X

S

X Y Z ! X Z (Y Z ) S (K S )K xy z ! (K S )x(K x)y z ! S (K x)y z ! (K x)z (y z ) ! x(y z )

Com

binatory logic is conuen t.

If

a C-term X has a normal form then reduction

f

X under normal-order strategy terminates. 27 Some natural com binators A c

mbinator

is a v ariable-free C-term. The iden tit y com binator I : S K K S K K y ! K y (K y ) ! y The comp

sition

com binator B : S (K S )K F

r

an y C-terms X ; Y ; Z : B X Y Z = S (K S )K X Y Z ! (K S )X (K X )Y Z ! S (K X )Y Z ! (K X )Z (Y Z ) ! X (Y Z ) 28

SLIDE 8 Sim ulation

f

the functional abstraction in CL Giv en x and C-term M construct a C-term

c

x:M s.t.: ( c x:M )N ! ! C L [x 7! N ]M Dene:

c

x:x = S K K

c

x:P = K P if x do es not app ear in P ,

c

x:P Q = S ( c x:P )( c x:Q) if the previous cases do not apply .

c

x:K x = S ( c x:K )( c x:x) = S (K K )(S K K ): No w, for an y term P , S (K K )(S K K )P ! (K K P )(S K K P ) ! ! K P 29 Compiling lam b da terms to C-terms In ten tion: Sim ulate lam b da calculus in CL The transformation H :

x

H = x for ev ery v ariable x,

(P

Q) H = P H Q H for an y lam b da terms P and Q,

(x:P

) H =

c

x:(P ) H (xy :y ) H =

c

x:(y :y ) H =

c

x:( c y :y ) =

c

x:S K K = K (S K K ) (xy :y )z u !

(y

:y )u !

u

K (S K K )z u ! C L S K K u ! ! C L u 30