On the complexity of beta-reduction 110 - Unibo

On the

complexity Andrea Dipartimento

P.zza di Porta

Abstract We

that

the

reduction

ponential

complexity

technique

in

the

from

of

for

number

this

sure for what plexity”

5, Bologna,

the

of

Lamping’s

as “the

Lamping’s

a new mea-

intrinsic

sified

com-

graph

in two

1. the

of A-terms.

rules

nodes

Twenty

years

redez

famikj

sharing

the

provide

of A-term

a lower

1990,

resentation,

and

step).

However,

plexity

relation

the

reduced tradict

along

family

neither in view

of fact,

the

exponential, with

family

to the propose

more

“intrinsic

comof

fact

does not

(as a mat-

the

that

~opyright

is

by

permission

of

and

ACM,

[GAL92a,

each other

just

Inc.

copy

brackets

required

for

requires

interaction

of this

between

appealing

proposed

in

the

As94,

since

literature

As95]

the oracle perform

a

can be seen as

oracle.

particularly

GAL92b,

an “oracle”

rule

set of rules

the

all after

differ

from

is implemented exactly

the

(in

same

set

we shall

prove

that

exponential

considering

the

Lamping’s

technique

its

abstract

algorithm,

work

required

by the

in

extra

or-

acle. For this nique

reason,

without

we shall

introduce

mentioning

the

implementation

Initial

2.1 Initially, A-term

To

~-

group

reductions).

already

without

(square

set of rules

all of them

paper,

be

nodes

set of rules.

in the way

the sense that

this

here possible

of the

Lamping’s solution

techto

the

oracle.

of

we shall

measure.

the

Lamping

effective

bound

Permission to make digital/hard copies of all or part of ttda material for peraoml or classroom use is granted without fee provided that the copies are not made or dtatributed for profit or commercial advantage, the copYright notice, the title of the publication and its date appear, and notice is given

translations

for

of the first

second

looks

call

are merely

implementation

distinction

In this

control

correct

the

and

responsible

techlenght

lower

of J-terms,

is

better

implementation that

its

algorithm

the

different

can

con-

nor

it works

a reasonable

complexity

families

algorithm,

where

we claim

is not

can be

of redex

Lamping’s

traditional

This

clas-

abstraction

are

(we shall

wh~ch

the first

of fan-nodes;

of abstract

implementation

complexity”

a different

atomical

the lenght

precisely,

be naturally

algorithm);

application

an effective

rep-

any

technique

of the

where

contrary,

reduction

and

This

optimality

examples

the

graphical

a single

to the number

of an actual

to

On

graph

abstract involving

discriminate

pair

L&Ty ’s sense

establish

Lamping’s

are also the examples

respect

niques).

in

not

computation.

the

to

reduction. that

respect

the

in

his algorithm

we prove with

relevance

the

between

paper,

exponential

be reduced did

More

complexity

that

duplication

croissants),

correct

As

reduction

a complex

optimal

had a unique

Lamping

corresponding

In this

ter

redexes

AL93]).

the

rules

and

implementation.

discovered was

2. some

of

of optimal

intrinsic

any possible

that

could

[Le80,

notion

of the family

to the

[Lam90]

is, all sharable

idea

also

length

bound

technique

the

intuitive

(see

in

Lamping

introduced

the

the

reduction,

reduction

[Le78]

J-calculus

consequence,

would

(that

L&Jy

formalize

to

in

a main

In

ago,

can

application,

(fan),

and

of rules

rules

groups:

involving

sharing

Introduction

rewriting

main

reduction

1

reduction

technique

optimal

reductions.

we propose

be considered

graph

can be ex-

L6vy ’s family

consideration,

could

Lamping’s

A-calculus

Italy

unibo.it

2

prove

Starting

Asperti di Matematica

S. Donato

asperti@cs.

graph

of beta-reduction

orheruk%

to republish, to post on servers or to redistribute to lists, requires apecitic permission andlor fee. POPL ’96, St. Petersburg FLA USA @ 1996 ACM @89791 _769-3195\Ol ..$3 .5(3

tree main

in

the

optimal

is essentially

(like

2. we shall nected

graph

reduction

represented

in ordinary

differences,

1. we shall

110

translation

graph

by its

reduction).

technique, abstract There

a

syntax are

two

however:

introduce suppose to their

an explicit that

node

variables

respective

binders.

are

for

sharing;

explicitly

con-

For

instance,

the

representation two=

Jz.Ay.

The

All

variables

(we shall

the connection of a same edge

node

in the

to

respective

Since

fans, its

fan)

the

binders

on the left

multiple

by

where

where

to express

connection

shared

(ports)

initial

6),

of a same variable.

their

this

(Cl, J and

sites

is used

a J towards

graph

tinguished other

are

leaving

(two

z).

it fun)

to the body).

is the

=

occurrences

represent

variable

single

call

connected

always

l.(l)

M

and6=Az.(m

different

are

Figure

A-term

(we shall

between

in

the

(z(zg))

triangle

sharing

graph

of

we shall

have

variables.

a

So, each

has exactly

it will

of

occurrences

three

dis-

be connected

with

ports. (1)

(3)

(2)

k 2.2

I

Reduction

We

shall

now

optimal simplified

term

comes not ing’s

@.

algorithm

As

unresolved.

in:

however,

necessary algorithm,

ideas

of

by

showing

would

we shall

see,

This

is exactly

the

complexity

since

to prove

main

technique

of the

(two

remain

the

reduction

version

sample will

illustrate

graph

the

we shall

issue

the oracle

of the

this

A

a

on our

a crucial

nature

discuss

how

work

where

exponential

not

Lamping’s

oracle

is

of Lamp-

complex

topic

here. Lamping’s

algorithm

rewriting

rules.

we can

usually

consists

At

of

a set

a given

stage

several

reducible

have

of

of the

local

~

graph

(6)

(5)

(4)

computation,

I

configurations

in

!“ /

the

graph.

In

ply

is made

ter

that

this

the

choice

non-deterministically.

much,

Interaction

since

Net

fies a one-step confluence

case,

in

the

but

This

graph

Lafont’s

diamond also

sense

rule

does

rewriting (that

if a term

not

ap-

/1

matis an

and

satis-

it

implies

has

to

system

[Laf90],

property

that,

of next

not

,, ,,, ;,

only

a normal

.. . .

$

/“ ..

~,,

‘/ . . . . . .Y’

f ,,,’

all

normalizing

particular, example (and

derivations

we shall of reduction

sometimes

The

most

ously

redex

is fired

in the

this

local

link

graph shared

in

graph

rewriting the

Figure

~-expression,

duplication

of 6.

1.(2).

rules

In graph

Y

t ,,?’ %.. .

J

term

the

application) function. and

in Figure /3-redex

we must

the

N.

(7)

More-

Figure

1: Graph

reduction

becomes do not

by the

the play

to the@ completely

2.

in

(two

next

eventually graph

(9)

(8)

the

Since

corresponding

be expressed

Since

N

!!?

re-

before

of the M

is obvi-

iV amounts

application

M

In ordinary

; :b: :, .,

criterion

rewriting

above

rule

outermost

In in our

M [N/z].

to the

by the

reduction can

rule

z for a term

representing

reduction

graph firing

+

the variable

body

of graph

role

rule, By

(the

(instantiated)

portions

N)

returned

length).

next

A

~, $,,

convenience).

a variable

connect value

same

the

to a didactical

of the graph (ku.itf

substituting the

any

graphical

important

to explicitly over, the

for

the

choose

according

/3-reduction:

duction,

have

usually

7

.. .. .

form,

6),

redex

we get involves

proceede

reduction,

the a

to the this

Figure

du-

111

2: the @-rule

of (two

6)

plication

would

on the

shared

timal

piece

graph

lazy

way,

body.

from

binder

works

as an

picted

upside-down

fan(-in) body

the

since

the

function

reduction,

it

still

step

point

op-

shall

the in

its

able

against

the

must

the

ure

of the

functionl.

does

not

Since

play

any

the

fans

is to be “paired”

role

have

duplicate 5.(2).

interaction 3. Let

can

between

us now

be formally

expressed

in the

analysis

in

fans

nothing

of the shared

according

we have

#b

.’.

cd

cd

cd

in Figure

of our

example.

d

(2)

Figure interaction

in front case,

By

applying

the fan-a

in Figure and

1.(3).

by

their

1.(5).

We

no fan-a cation

interaction

Now,

firing have

two

more

interactions, process,

particular,

lead

have

,@redexes

we must

following

in

been in

the

graph,

4: not

forbidden, though

optimal

in the

optimal

semantically

carefully

clear:

since

the

some

redex,

its

cution

of the

The

only

fans

inside

shared

).

of the

The

would

possible

the

dotted

Figure rule

such

of two After

application (al-

should

be

could

be involved a double

is between

from

unshakable

and

there

is no

their

intutive

a fan-out,

ular,

keep

in

with

some

fan-in

ing is

its

definition

formal

lar,

is solved

definition obscure the

obvious

the

graph, The

a crucial by of of

the the the

intuitive and

naive

distinction

semantics

a fan-out

the

effect.

determined

precise

that in

sharing

technique,

just

mind

operational

in

case, both

long

to the

now

(locally)

ure

two

is always

the

point

of

“oracle”.

oracle, natnre idea

different;

supposed

delimiting

way

abstract

is quite

its

correct the

optimal

Obviously,

that of

is the

of labeling

to

scope

and

pairing

algorithm

a fan-in

we very abstract fans

paired

to

and In not

the

of fans

paired

they

l.(7).

both

that

obvious

fans,

application.

in Figure

process”, So, the

the

the

are paired:

“duplication

completed.

for

graph

existence

has

rule,

according

be~ been

in

this

to Fig-

problem meet

of deciding

each

other

is established)

a double

in Figure

application that

rule

is the

is the

technique

1.(9),

which

(that

crucial (solved

of this

to apply

question point

is in normal

oracle).

we get

form

two theii

of the optimal

by the rule,

when

of how

the

graph

w.r. t. Lamping’s

algorithm.

fans

give

provide

rules.

redexes

sharing

the

reduction

order

complex

does

be

ween

graph

should

partic-

annihilat-

bet

in

in

of)

of optimal

implies

5.(l).

fans

By

of view

we get

pairs

same

The

pairing

crucial

between

rule,

interaction

already

(class

this

implementation 1 Although

the point

case, is to annihilate

exe-

the

is another

6: fan-@

a configuration

firing

In this

technique

This

is correct

since

imply

interaction region.

this

In

redex.

other

In this

Intuitively,

is strictly

intuition

application

duplication

application.

rule:

dupli-

here.

rule

imlementation

correct

of the following

and

in the

rewriting

duplication

the

Figure

x-w Figure

function-port apply

rule

created,

graph

proceede

be very

graph

of the

we can

5: fan-annihilation

we get the graph

to the

so we must

but

the

/?-redexes

we are

no

rule,

they in Fig-

a fan-out

sb

[1)

3: Fan-A

Since

other,

to the rule

l.(6)),

shared

6.

each

are vari-

of the

of duplicating

to do with

(see Figure

1. (’7).

1.(5)

in

e

Figure

Figure

is a residual

process

we

a differ-

in Figure

is a residual

fan-out

As

in

as a local

a fan and a }, as described

proceede

two

each other,

Now

technique.

be handled

interactions the

in the

ab

this

similar that

of 6, while

two

reduction must

the fan-in

of two,

the

Ax .M

the

variable

de-

body

form

fan

usually

graph

interaction

“paire&:

(fan-out,

operator

optimal

in particular

not

edge leadthis

way

Note

duplicated,

In a sense,

of the see, this

ent

a more

sharing

been

fan on the

variable.

), that

sharing

has

another

%nsharing)’

contrary,

A but

binder

to the

of the

the

global

proceedes

external

the

to introduce

the

On

technique

duplicating

However,

as a umque,

of graph.

reduction

we are forced ing

be performed

a

3

complexity

the would

Before

discussing

the

stract”

algorithm,

we should

complexity

of

Larnping’s

“ah:

particuwork

(see

liminary

[Lam89]).

112

assumptions.

First

start of

by fixing all,

a typical

a few

pre-

feature

of optimal could

techniques

become

A reasonable is that mal

way

obviously

term

P =

M “cost”

the

size

convincing, its

in Lamping’s redexes),

in

would

Our

of the

duplicated, proportional does

we

and to

not

also

seem

duplicated

prerequisite

chains,

(whenever

belong

assumption

~-I

turned

to distinguished

work

will

calculus.

handling

The

that

is still

Lamping’s

be to

reason,

of garbage

techniques lar,

fully

which

anticipated

instance

P to a closed

had

Figure

to

7: two’:

a representation

of Church’s

integer

fam-

to be done

casez.

second

rect

for

a

some

seem

(its

nor-

prefer,

us to avoid

M

nodes

eventually

you

reasoning

duplicating

terminology),

So, we just

in any

gets

work

whose

Consider

this

application

if

allows

would

However,

“extra”

If we apply

M

@redex

since

A and

ilies!

form,

of the of M.

also

(y z z).

this

(or,

that

computation.

to A-terms

examples.

Az.Ay.

in normal

the

account

analysis

hypothesis

work

on in the

constant

degene~ate

term

into

the

atomic

This

of anticipating

later

to take

is an

variable).

all

only

of restricting

form

the

is that

usefull

consider here,

collection

a subject

approach

only

is that

in optimal

cor-

reduction

of investigation

does not seem

terms

the

(in

particu-

to be completely

satisfactory). We shall

now

provide

our

assumptions,

out

considering

exponential

the

Let

a few

us start

sible ing

of the

obtained

(h.(z(z

y))

(n m)

of two

integer

mn,

of)

church

the

in Figure After

these

subterm

2 steps

of fans After ure

where

dence.

Firing

uration

in

2 The

idea

in

evidence

predictable all

copies

not

fired

ently)

of

of

useless

the

=

the

the

The

the

reduction steps with

Jy. (z(z fan

church

is shown

the

term

y))

is shared

in Fig-

“ a“.

process

effacing

by

Now,

requires:

the application; for

(3)

(2)

are ,d-redex.

marked

This

(1)

application

2 steps

all residuals

“ a“. left we

only

fans chains

computation,

inside

rules

a

which

“virtual”

could

by

put final

in

Fig-

in

evi-

config-

we executed

problems

generated we

the up,

some

graph

been

obtain

those

gives

the

has

Summing

redexes of

with

~-redex

redex,

considering

prerequisite

along

two’

consider

is (a representaion

3 steps

8.(5).

disposition

that

reduction

we are

this

new

in shar-

l-term

m gives

of the

a new

Figure

two’

7 is a postwo

result

the A and

with

two’

by con-

n and

duplicated.

7 steps

8.(4),

put

copies

the fans;

marked

Recall

sub term

is fully

these

Let

we are left

the

Figure

Integer

us now

first

for duplicating

for duplicating

In

w)).

four.

two

two

//&~

Since

proceede

the

integers

reductions

of the

with-

is already

reductions.

reducing

expected

integer

where

case.

to itself.

church

8. The

8.(3),

means

~w.(z

so the

oracle)

we shall

Church

by

of two’

(i.e.

terms.

representation

application

this

auxiliary

a simple

kc.~y.

of the

complex,

satisfying

reduction

of family

with

graphs,

ure

cost

number

is quite

of a A-term

“abstract”

extra

in the

the example sidering

an example

whose

are due

needed to

redex: the

perform

3

duplication some

the

(4)

(5)

to

Figure

un-

again,

if are

(appar-

work.

113

8: the

reduction

of (two’

two’)

two

/3-reductions

(actually,

interactions.

Note

has the

shape

Let

same

us now

be clear, graph

the church

Figure

of length

out

of the

Let

us now

outermost

same

fans

example.

As should

2n can be represented we have

marked

plication

with

exactly

effacing

by the

a sequence

ure

2 steps

2n steps fans.

After

of

“ a“ is now

requires:

application;

9: a representation

n and

7 fan-

configuration

one.

previous

integer

and

the final

initial

the

9, where

reductions), that

of the

generalize

in Figure

fan-in

family

moreover

these

This

duplicated.

duplicating

for duplicating

gives

a total

reductions,

the

fans;

n+2

This

du-

A and

the

steps

for

of 4 + 3n operations.

we get the

11. A new fl-redex

has been

configuration created.

in Fig-

13y firing

this

of 2n

a corresponding

sequence

of fan-

length.

apply

this

~-redexes

term we get

to itself.

By

the

in Figure

term

firing

the 10.

two As

Figure

redex

11: the

we obtain

has the

same

can iterate 3* (2n)

our

we shall

quences Summing

case of two,

the

reduction

portion

of (2n 2n)

of graph

inside

given the

two

4n (this

process),

is decremented up,

the

the

this

require same

sequences

is doubled

at every

of the

4 +

/3-redex,

innermost length

we

duplication

a new

of the

graph

10, and

shape of fans

iteration

outermost

se-

by one.

total

number

of fan-interactions

by.

({4+3n)+(4+3*(2n)

114

length

while

now

have

get a graph

the

the

will

we shall

where

length

12. Now, in Figure

of the graph

Then,

and by its firing, 12 but

graph

In particular,

of Figure have

10: the

part

of (2n 2n)

in Figure

of the

reasoning.

operations.

of the

Figure

reduction

the graph shape

of the innermost

in the

fully

for

)+(4+3

*(4n))+.

. .+(4+3

*(2n-’n)))

is

the

term

(g n)

is n-1

f(rl)=9+3*n+~bi i=o Finally,

let

For each teractions in the

us consider

application

the

for duplicating

reduction

Moreover,

bn final

operations

for

1 + 4 + 3 * bi in-

bi + 3 * b,+l

(recall

one single

5 * (n – 1] operations

of fan-interactions.

we have

a,, plus

of (a~ ai)

we have

external

number

of (6 ai),

that

operation

creating

all

of fan-effacement

operations

b~+l

=

bi * 2b~).

internal

to two’,

copies

of 6, and

when

we apply

the

parameters.

Summing

up,

the

n > 0) is given

number

by the

c(n)

of fan-interactions

n-1

c(n)=

(for

formula n

10*n–4+4*~bi+

3*~bi+bn i=O

i=l

n—1

= 10*n+7* Figure

12: the

reduction

~bi+4*b.

>4

*b.

2=1

of (2n 2n) Note

now

2~::01

that

b’ = b..

So,

n-1 =

4n+3n*~2i

=4n+3n*(2”–1)=

n*(3*2n+l)

i=o In contrast, main

we have

loop,

plus

n + 2 family Our

executed

two

at the

just

very

problem

on the

above

mentioned

in which

mal

form

is an atomic

constant).

The

term

we consider

is:

be a term

two’

kz.Ag,

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port

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optimal

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provide

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part

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reduction

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sophisticated

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oracle

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number

have

found

sion

of its

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of the

so far

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the

issues

is to could

A.

Asperti,

Paths

in

munications

of

[DR95]

in

understand

if and

how

Computer

Sci-

To appear

in Theoret-

Danos,

C.

A-calculus:

wzthoui

Laneve,

three

L.

years

understandings.

Reg-

of com-

Proceedings

1994.

of optimal

Symposium guages

we

Reversible GOI

the

big

on

G.

Logic

M. Abadi,

lambda

and

92).

and

ir-

J-machines.

on

of

M.

boxes. Logic

L&ry.

The geom-

Proc.

of the

Programming

19th Lan-

1992.

Gonthier,

without

J.J.

reduction.

Principles

(POPL

[GAL92b]

explo-

Lamping’s

Regnier.

G. Gonthier,

etry

linear

rules,

view,

L.

Computations:

1995.

posium of

V.

the

Danos,

[GAL92a]

of Lamping’s not

V.

Draft.

complexity

of abstract

point

for

II: Re-

As

is now

of an exponential

implentative

Laboratory

Paris.

reversible

complexity. the

Systems Technical

Science.

of LICS’94.

(see [As95]

it is clearly

so far

to

complex-

of research.

respect),

of applications

of

different

in BOHM

crucial

devoted

Interaction

reductions.

of Bologna.

Computer

nier.

re-

opitmal

University

[ADLR94]

to

complexity

a subject

in BOHM,

no evidence

from

problem rithm

is one

Although,

the

The

proposed

this

to

of family

very

examples

from

complexity

understand

it is still

can see by the few

the terms

is actually

techniques and

important

“oracle”.

algorithm

related

looks

to in

we should

a discussion),

you

bound

the

problems it

algorithm

we called of the

of all,

upper

Secondly,

of what

this

First

“abstract”

ductions. ity

open

issue

1993.

C. Laneve. of

UBLCS-93-12,

ence,

Conclusions

There

Special

June

implementations the

5

Science,

Montreal.

Abadi,

Proc.

in

J.J.

of the

Computer

L4vy.

7th

Linear

Annual

Science

Sym-

(LICS

‘92).

1992.

algo-

be compiled.

[Laf90]

Y.

Lafont.

Interaction

Symposium guages

on

Nets.

Principles

(POPL

90).

of

Proc.

of the

Programming

San Francisco.

17th Lan-

1990.

Acknowledgements [Lam89] We

would

teresting

like

to

thank

discussions

Cosimo

on the

Laneve

subject

for

of this

many

J. Lamping.

calculus

in-

mal

Linear

Reductions.

Vol. [As95]

on

Asperti.

Issue

devoted

Conf.

Categories

pp.3-22.

Asperti.

J-calculus

Logic,

Cornonads,

Fundamental to

22, n.1,

A.

J. Lamping.

calculus

References A.

60!E

=

Rewriting

(RTA’95),

Special

Computer

[Le78]

Science,

@timzzinO Proc.

Techniques

Kaiserlautern,

San

of

and

Germany.

OPtimal the

J.J.Levy.

[Le80]

A.

Asperti,

Bologna

Opt imal

cal Report Science, Journal

C.

Applications,

J. To

J.

A.

Hzgher-order

UBLCS-95-9, University

Machine.

Laboratory

of Bologna.

of Functional

Naletto. for

To

[LM92]

Techni-

in

A. and

Asperti, Labels

in

C. the

X.

release RIA

the

lambda

Symposium

Languages

(POPL

et opttmales

de doctorat

duns

d’&at,

Univer-

m

lambda-

1978.

Optzmal J.P.

optimal 17th

reductions

Seldin

Curry,

and

Essays

Calculus

and

Press.

J.R.

on

the

Hindley,

editors,

Combmatory

Formalism,

pages

Logic, 159-191.

1980.

Leroy, 0.5.

M. Mauny.

The

Documentaiton

Technical

Report.

Carol

and September

Light

user’s

system,

manual.

IN-

1992.

Programming. [Ya94]

[AL93]

lambda Report.

The

Computer

appear

VII.

In

H.B.

for

of the

correctes

Th&se

L&y.

Lambda

1995.

Giovannetti,

optimal Internal

1990.

Reductions

de Paris

calculus.

Sixth

algordhm Proc.

Francisco.

Academic [AGN95]

for

PARC

of Programming

le lambda-calcv,l. sit~

1:

An

reductions.

Principles

90).

Opii-

1995.

implementations.

on

and

In formaticae, in

algorithm Xerox

1989.

paper.

[Lam90]

[As94]

An

reductions.

Laneve.

Paths,

J-calculus.

To

Computations appear

in

The

Manual.

Theo-

118

Yale Yale

Haskell

Group.

University.

The October

Yale Haskel/ 1994.

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