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The N e x t 7 0 0 Programming Languages

P. J . Landin

U n i v a c D i v is i o n o f Sp e r r y R an d C o r p . N e w Y o r k N e w Y o r k

. . .

t oday . . . 1,700 special programming languages used to 'com-

municate ' in over 700 application areas. --Computer S o f t w a r e Issues,

an American Mathematical Association Prospectus, July 1965.

A fam i ly o f un implemented computing languages is de-

sc r ibed that is in tended to span d i f f erences o f app l icat ion are a

by a un i f ied f ram ew ork . Th is f ram ew ork d ic ta tes the rules

ck out the uses of user-coined names, and the conventions

ab out cha racte riz ing functional relationships. W ithin 'lhis f ra me -

work ' lhe design o f a spec if ic langu age sp l its into two inde-

pend ent par ts . On e is ' lhe cho ice o f w r i t ten app earan ces o f

programs (or more genera l ly , the i r physica l representat ion ) .

The o:her is the choice of the ab strac t ent it ies (such as numbers,

charac ter-strings, l ists of them, funct ional relat ions amo ng

them) that can be re ferred to in the language.

The system is biase d tow ard s expressions rathe r than

statements. I t inc ludes a nonproc edura l (p urely funct ional)

subsystem fhat a ims to e xpa nd the c lass o f users ' needs that

can be met by a single print - instruct ion, without sacri f ic ing the

important p ropert ies that make convent ional r ight -hand -s ide

expressions easy to construct and understand.

1. I n t r o d u c t i o n

Most programming languages are partly a way of

expressing things in terms of other things and partly a

basic set of given things. The Isw~M (If you See What I

Mean) system is a byprodu ct of an attem pt to disentangle

these two aspects in some current languages.

This attempt has led the author to think that many

linguistic idiosyneracies are concerned with the former

rather than the latter, whereas aptitude for a particular

class of tasks is essentially determined by the latter rather

than the former. The conclusion follows that many

language characteristics are irrelevant to the alleged

problem orientation.

IswI~ is an attempt at a general purpose system for

describing things in terms of other things, that can be

problem-oriented by appropriate choice of primitives .

So it is not a language so much as a family of languages,

of which each member is the result of choosing a set of

primitives. The possibilities concerning this set and what

is needed to specify such a set are discussed below.

Isw~M is not alone in being a family, even after mere

syntactic variations have been discounted (see Section 4).

In practice, this is true of most languages that achieve

more than one implementation, and if the dialects are well

disciplined, they might with luck be characterized as

Presented at an ACM Programming Languages and Pragmatics

Conference, San Dimes, California, August 1965.

1Throe is no more use or mentiol~ of X n this paper--eognoseenti

will nevertheless sense an undercurrent. A not inappropriate title

would have been Church without lambda,

differences in the set of things provided by the library or

operating system. Perhaps had ALGOL 60 been launched

as a family instead of proclaimed as a language, it would

have fielded some of the less relevant criticisms of its

deficiencies.

At first sight the facilities p rovide d in IswI~,~ will app ear

comparatively meager. This appearance will be especially

misleading to someone who has not appreciated how much

of current manuals are devoted to the explanation of

common (i.e., problem-orientation independent) logical

structure rather than problem-oriented specialties. For

example, in almost every language a user can coin names,

obeying certain rules about the contexts in which the

name is used and their relation to the textual segments

that introduce, define, declare, or otherwise constrain its

use. These rules vary considerably from one language to

another, and frequently even within a single language

there may be different conventions for different classes of

names, with near-analogies that come irritatingly close to

being exact. (Note that restrictions on what names can

be coined also vary, but these are trivial differences. When

they have any logical significance it is likely to be perni-

cious, by leading to puns such as ALaOL'S integer labels.)

So rules about user-coined names is an area in which

we might expect to see the history of computer applica-

tions give ground to their logic. Another such area is in

specifying functional relations. In fact these two areas are

closely related since any use of a user-coined name im-

plicitly involves a functional relation; e.g., compare

x ( x - k a ) f ( b + 2 c )

w h e r e x = b -4- 2 c w h e r e

f x ) = x x + a )

ISW~M is thus part. pro gram ming lan guage and par t pro-

gram for research. A possible first step in the research

program is 1700 doctoral theses called A Correspondence

betwe en x and Chu rch 's X-notation. ''~

2 . T h e w h e r e - N o t a t i o n

In ord inary mathem atica l communication, these uses

of

' w h e r e '

require no explanation. Nor do the following:

f ( b - l - 2 c ) ---I- ( 2 b - - c )

w h e r e f ( x ) = x ( x - t - a )

f (b A -- 2 c) - - f ( 2 b - c )

w h e r e f ( x ) = x ( x + a )

a n d b =

u / u + l )

a n d c =

v / v - t - 1 )

g ( f

w h e r e

f ( x ) = a x 2 - ]- b x - I- c ,

u / ( u - 4 - 1 ) ,

v / ( v + l ) )

w h e r e g ( f , p , q ) = f ( p - k 2 q , 2 p - - q )

Volume 9 / Number 3 / March, 1966 C o m m u n i c a t i o n s o f t h e ACM 157

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A p h r a s e o f t h e f o r m ' w h e r e d e f i n i t i o n ' w i l l b e c a l l e d a

w h e r e - c l a u s e . A n e x p r e ss i o n o f t h e f o r m 'e x p r e s s io n

w h e r e - c l a u s e ' i s a w h e r e - e x p r e s s i o n . I t s t w o p r i n c ip a l

c o m p o n e n t s a re c a ll e d, re s p e c t i v e ly , i t s m a i n c l a u s e

a n d it s s u p p o r t i n g d e f i n i ti o n . T o p u t ' w h e r e ' in t o a

p r o g r a m m i n g l a n g u a g e t h e f o l l o w in g q u e s ti o n s n e e d

a n s w e r s .

L i n g u i s t i c S t r u c t u r e .

W h a t s t r u c t u r e s o f e x p r e s s io n

c a n a p p r o p r i a t e l y b e q u a l i f ie d b y a w h e r e - c l a u s e , e .g . ,

c o n d i t i o n a l e x p r e s s i o n s , o p e r a n d - l i s t i n g s , s t a t e m e n t s ,

d e c l a r a t i o n s , w h e r e - e x p r e s s i o n s ?

L i k e w i s e , w h a t s t r u c t u r e s o f e x p r e s s i o n c a n a p p r o -

p r i a t e l y b e w r i t t e n a s t h e r i g h t - h a n d s i d e

(rhs)

of a

s u p p o r t i n g d e f i n i t i o n ? W h a t

contexts

a r e a p p r o p r i a t e f o r a

w h e r e - e x p r e s s i o n , e . g . , a s a n a r m o f a c o n d i t i o n a l e x -

p r e s si o n , a n o p e r a t o r , t h e m a i n - c l a u s e o f a w h e r e - e x -

p r e s s i o n , t h e l e f t - h a n d s i d e

(lhs)

o f a s u p p o r t i n g d e f i n i t io n ,

t h e r h s o f a s u p p o r t i n g d e f i n i t i o n ?

S y n t a x .

H a v i n g a n s w e r e d t h e a b o v e q u e s t i o n s , w h a t

a r e t h e r u l e s f o r w r i t i n g t h e a c c e p t a b l e c o n f i g u r a t i o n s

u n a m b i g u o u s l y ? E . g . , w h e r e a r e b r a c k e t s o p t i o n a l o r

o b l i g a t o r y ? o r o t h e r p u n c t u a t i o n ? o r l i n e b r e a k s ? o r i n -

d e n t a t i o n ? N o t e t h e s e p a r a t i o n o f d e c is io n s a b o u t s t r u c -

t u r e f r o m d e c i s i o n s a b o u t s y n t a x . ( T h i s i s n o t a d e n i a l

t h a t l a n g u a g e d e s i g n e r s m i g h t i t e r a t e , l i k e h a r d w a r e

d e s i g n e r s w h o d i s t i n g u i s h l e v e l s o f h a r d w a r e d e s i g n . )

S e m a n t i c C o n s t r a i n ts o n L i n g u i s t i c S t r u c t u r e .

I n t h e

a b o v e e x a m p l e s e a c h m a i n c l a u s e w a s a n u m e r i c a l ex-

p r e s s i o n ; i . e . , g i v e n a p p r o p r i a t e m e a n i n g s f o r t h e v a r i o u s

i d e n ti f ie r s i n it , it d e n o t e d a n u m b e r . W h a t o t h e r k i n d s o f

m e a n i n g a r e a p p r o p r i a t e f o r a m a i n c la u s e , e . g ., a r r a y s ,

f u n c t i o n s , s t r u c t u r e d e s c r i p t i o n s , p r i n t - f o r m a t s ?

L i k e w i se w h a t k i n d s o f m e a n i n g a r e a p p r o p r i a t e f o r

rhs ' s

o f s u p p o r t i n g d e f i n i t io n s ? N o t i c e t h e r e i s n o t a t h i r d

q u e s t i o n a n a l o g o u s t o t h e t h i r d q u e s t i o n a b o v e u n d e r

l ingu is t ic s t ruc ture .

T h i s i s b e c a u s e a w h e r e - e x p r e s s i o n

m u s t m e a n t h e s a m e k i n d o f t h i n g a s it s m a i n c l a us e a n d

h e n c e r a is e s n o n e w q u e s t i o n c o n c e r n i n g w h a t c o n t e x t s

a r e m e a n i n g f u l . N o t i c e a l s o t h a t t h e q u e s t i o n s a b o u t

m e a n i n g a r e a l m o s t e n t i r e l y i n d e p e n d e n t o f th o s e a b o u t

s t r u c t u r e . T h e y d e p e n d o n c l a s s i f y i n g e x p r e s s i o n s i n t w o

w a y s t h a t r u n a c r o s s e a c h o t h e r .

Outcome.

W h a t i s t h e o u t c o m e o f t h e m o r e r e c o n d i t e

s t r u c t u r a l c o n f i g u r a t i o n s a m o n g t h o s e d e e m e d a d m i s s i b l e ,

e .g . m i x e d n e s t s o f w h e r e - e x p r e s s i o n s , f u n c t i o n d e f i n i ti o n s ,

c o n d i t i o n a l e x p r e s s i o n s , e t c . ?

E x p e r i m e n t a l p r o g r a m m i n g h a s l e d t h e a u t h o r t o t h i n k

t h a t t h e r e i s n o c o n f i g u r a t io n , h o w e v e r u n p r o m i s i n g i t

m i g h t s e e m w h e n j u d g e d c o l d , t h a t w i ll n o t t u r n u p q u i t e

n a t u r a l l y . F u r t h e r m o r e , s o m e c o n fi g u r a t io n s o f ' w h e r e '

t h a t m i g h t f i r st a p p e a r t o r e f l e c t s o m e w h a t p e d a n t i c d i s -

t i n c t i o n s , i n f a c t p r o v i d e c l o s e m a t c h e s f o r c u r r e n t l a n -

g u a g e f e a t u r e s s u c h a s n a m e / v a l u e a n d o w n (see [2, 3]) .

A l l t h e s e q u e s t i o n s a r e n o t a n s w e r e d i n t h i s p a p e r . T h e

t e c h n i q u e s f o r a n s w e r i n g t h e m a r e o u t l i n e d i n S e c t i o n 4 .

O n e o t h e r i s s u e a r i s e s w h e n ' w h e r e ' i s a d d e d t o a

p r o g r a m m i n g l a n g u a g e - - t y p e s a n d s p e c i f i c a t i o n s . A

m e t h o d o f e x p r e s s i n g t h e s e f u n c t i o n a l l y i s e x p l a i n e d i n

[2 ]. I t a m o u n t s t o u s in g n a m e d t r a n s f e r - fu n c t i o n s i n s t e a d

o f c l a ss n a m e s l ik e i n t e g e r , i .e . , w r i t i n g

w h e r e n = round(n)

i n s t e a d o f t h e s p e c i f i c a t i o n

i n t e g e r n

T h u s t h e u s e o f f u n c t i o n a l n o t a t i o n d o e s n o t j e o p a r d i z e

t h e d e t e r m i n a t i o n o f t y p e f r o m t e x t u a l e v i d e n c e .

3 . P h y s i c a l I S W I M a n d : L o g ic a l I S W I M

L i k e A L G O L 6 0 , I S W I M h a s n o p r e s c r i b e d p h y s i c a l

a p p e a r a n c e . A L G OL C 0 'S d e s i g n e r s s o u g h t t o a v o i d c o m m i t -

m e n t t o a n y p a r t i c u l a r s e t s o f c h a r a c t e r s o r t y p e f a ce s .

A c c o r d i n g l y t h e y d i st i n g u is h b e t w e e n p u b l i c a t i o n h m -

g u a g e , r e f e re n c e l a n g u a g e a n d h a r d w a r e l a n g u a g e s .

O f th e s e t h e r e f e r e n c e l a n g u a g e w a s t h e s t a n d a r d a n d w a s

u s e d i n t h e r e p o r t i ts e lf w h e n e v e r p i e ce s o f A L G O L 6 0

o c c u r r e d . P u b l i c a t i o n a n d h a r d w a r e l a n g u a g e s a r e t r a n s -

l i te r a t i o n s o f t h e r e f e r e n c e l a n g u a g e , v a r y i n g a c c o r d i n g t o

t h e i n d i v i d u a l t a s t e , n e e d s a n d p h y s i c a l c o n s t r a i n t s o n

a v a i l a b le t y p e f a c e s a n d c h a r a c t e rs .

S u c h v a r i a t i o n s a r e d i f f e r e n t p h y s i c a l r e p r e s e n t a t i o n s

o f a s in g l e a b s t r a c t i o n , w h o s e m o s t f a i t h f u l p h y s i c a l

r e p r e s e n t a t i o n i s t h e r e f e r e n c e l a n g u a g e . I n d e s c r ib i n g

I s w I ~ w e d i st i n g u is h a n a b s t r a c t l a n g u a g e c a ll e d l o g i c a l

ISWIM, w h o s e t e x t s a r e m a d e u p o f t e x t u a l e l e m e n t s ,

c h a r a c t e r i z e d w i t h o u t c o m m i t m e n t t o a p a r t i c u l a r p h y s i c a l

r e p r e s e n t a t i o n . T h e r e i s a p h y s i c a l r e p r e s e n t a t i o n s u i t a b le

f o r t h e m e d i u m o f t h i s r e p o r t , a n d u s e d f o r p r e s e n t in g

e a c h p i e c e o f I s w I ~ 1 t h a t o c c u r s i n t h i s r e p o r t . S o t h i s

p h y s i c a l r e p r e s e n t a t i o n c o r r e s p o n d s t o r e f e r e n c e A LG O L

6 0 , a n d i s c a l l e d r e f e r e n c e I SW I M , o r t h e I s w I ~ i

r e f e r e n c e r e p r e s e n t a t i o n , o r t h e I sw I ~ ,r r e f e r e n c e h m -

g u a g e .

T o a v o i d i m p r e c i s i o n o n e s h o u l d n e v e r s p e a k j u s t o f

ISWIM, b u t a l w a y s o f l o g i c a l I s w x M o r o f s u c h -

a n d - s u c h p h y s i c a l I SW l M . H o w e v e r , i n l o o s e s p e e c h ,

w h e r e t h e p r e c i s e i n t e n t i o n i s c l e a r o r u n i m p o r t a n t , w e

r e f e r t o I S W lM w i t h o u t q u M i fi e a ti o n . W e a i m a t a m o r e

f o r m a l r e l a t i o n b e t w e e n p h y s i c a l a n d l o g i c a l l a n g u a g e s

t h a n w a s t h e c a s e i n t h e A L G O L C 0. T h i s i s n e c e s s a r y s i n c e

w e w i s h t o s y s t e m a t i z e a n d m e c h a n i z e t h e u s e o f d i f fe r e n t

p h y s i c a l r e p r e s e n t a t io n s .

4 . F o u r L e v e l s o f A b s t r a c t i o n

T h e

p h y s i c a l ~ l o g i c a l

t e r m i n o l o g y i s o f t e n u s e d t o

d i s t in g u i s h f e a t u r e s t h a t a r e a f o r t u i t o u s c o n s e q u e n c e o f

p h y s i c a l c o n d i t i o n s f r o m f e a t u r e s t h a t a r e i n s o m e s e n s e

m o r e e s s e n t i a l . T h i s i d e a i s c a r r i e d f u r t h e r b y m a k i n g a

s i m i la r d i s t i n c t io n a m o n g th e m o r e e s s e n t i a l f e a t u r e s.

I n f a c t I S W l M i s p r e s e n t e d h e r e a s a f o u r - l e v e l c o n c e p t

c o m p r i s i n g t h e f o l l o w i n g :

( 1) p h y s i c a l I s w l M ' S, o f w h i c h o n e i s t h e r e f e r e n c e

l a n g u a g e a n d o t h e r s a r e v a r i o u s p u b l i c a t i o n a n d h a r d w a r e

l a n g u a g e s ( n o t d e s c r i b e d h e r e ).

1 5 8 C o m m u n i c a t i o n s o f t h e A C M V o l t, m e 9 / N u m b e r 3 / M a r c h , 19 66

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( 2 ) l o g i ca l I S W l M , w h i c h i s u n c o m m i t t e d a s t o c h a r -

a c t e r s e ts a n d t y p e f a ce s , b u t c o m m i t t e d a s to t h e s e q u e n c e

o f t e x t u a l e l em e n t s , a n d t h e g r a m m a t i c a l r u le s f o r g r o u p -

i n g t h e m , e .g ., b y p a r e n t h e s e s , i n d e n t a t i o n a n d p r e c e d e n c e

r e l a t i ons .

( 3 ) a b s t r a c t I s w l s ,, , w h i c h i s u n c o m m i t t e d a s to t h e

g r a m m a t i c a l r u l e s o f s e q u e n c e a n d g r o u p i n g , b u t c o m -

m i t t e d a s to t h e g r a m m a t i c a l c a t e g o ri e s a n d t h e i r n e s t i n g

s t r u c t u r e . T h u s a b s t r a c t I sw is ,~ i s a t r e e l a n g u a g e o f

whi ch l og i ca l I sw l M i s one l i nea r i za t i on .

( 4 ) a p p l i c a t i v e e x p r e s s i o n s ( A E s ) , w h i c h c o n s t i t u t e

a n o t h e r t r e e l a n g u a g e , s t r u c t u r a l l y m o r e a u s t e r e t h a n

a b s t r a c t I SW l M , a n d p r o v i d i n g c e r t a i n b a s i c g r a m m a t i c a l

c a t e g o r i e s in t e r m s o f w h i c h a l l o f I s w 1 ~ ' s m o r e n u m e r o u s

c a t e g o r i e s c a n b e e x p r e s s e d .

T h e s e t o f a c c e p t a b l e t e x t s o f :t p h y s i c a l I SW l M i s

s p e ci fi ed b y t h e r e l a ti o n s b e t w e e n 1 a n d 2 , a n d b e t w e e n

2 a n d 3 . T h e o u t c o m e o f e a c h t e x t i s s p e c i fi e d b y t h e s e

r e l a t io n s , t o g e t h e r w i t h a f r a m e o f r e f e r e n c e , i .e . , a r u l e

t h a t a s s o c i a t e s a m e a n i n g w i t h e a c h o f a c h o s e n s e t o f

i den t i f i e r s .

T h e s e a r e th e t h i n g s t h a t v a r y f r o m o n e m e m b e r o f o u r

l a n g u a g e f a m i l y to t h e n e x t . T h e s p e c if i ca t io n o f t h e f a m i l y

i s c o m p l e t e d b y t h e r e l a t i o n b e t w e e n a b s t r a c t I s w I ~ a n d

A E s , t o g e t h e r w i th a n a b s t r a c t m a c h i n e t h a t i n t e r p r e t

A E s . T h e s e e l e m e n t s a re t h e s a m e f o r al l m e m b e r s o f t h e

f a m i l y a n d a r e n o t d i s c u s s e d i n t h i s p a p e r ( s e e [ 1 , 2 , 4 ] ) .

T h e r e l a t io n s h i p b e t w e e n p h y s i c a l I S W l M a n d l o g ic a l

IS W IM i s f i x e d b y s a y i n g w h a t p h y s i c a l t e x t s r e p r e s e n t

e a c h l o g ic a l e l e m e n t , a n d a ls o w h a t l a y o u t is p e r m i t t e d i n

s t r i n g i n g t h e m t o g e t h e r . T h e r e l a t i o n s h i p b e t w e e n l o g i c a l

I s w ~ a n d a b s t r a c t I sw l M is f ix e d b y a fo r m a l g r a m m a r

n o t u n l i k e t h e o n e i n t h e A L G O L 6 0 r e p o r t , t o g e t h e r w i t h a

s t a t e m e n t c o n n e c t i n g t h e p h r a s e c a t e g o ri e s w i t h t h e

a b s t r a c t g r a m m a t i c a l c a t e g o r i e s .

T h e s e t w o r e l a t i o n s c o v e r w h a t i s u s u a l l y c a l le d t h e

s y n t a x o r g r a m m a r o f a l an g u a g e. I n t h is p a p e r

s y n t a x i s n o t d i s c u s s ed b e y o n d a f e w g e n e r a l r e m a r k s a n d

a f e w e x a m p l e s w h o s e m e a n i n g s h o u l d b e o b v i o u s .

T h e r e l a ti o n s h i p b e t w e e n a b s t r a c t I s w l s ( a n d A E s i s

f ix e d b y g i v i n g t h e f o r m o f A E e q u i v a l e n t t o e a c h a b s t r a c t

I s w iM g r a m m a t i c a l c a t e g o r y . I t h a p p e n s t h a t t h e s e l a t t e r

i n c l u d e a s u b s e t t h a t e x a c t l y m a t c h e s A E s . H e n c e t h i s

l i n k i n o u r c h a i n o f re h ~ ti o n s i s r o u g h l y a m a p p i n g o f

I SW I M i n t o a n e s s e n t i a l k e r n e l o f I s w IM , o f w h i c h al l t h e

r e s t i s m e r e d e c o r a t i o n .

5 . A b s t r a c t I S W I M

T h e t e x t s o f a b s t r a c t I SW l M a r e c o m p o s i t e i n f o r m a t i o n

s t r u c t u r e s c a l l e d

amessage's.

T h e f o l l o w i n g s t r u c t u r e

d e f i n i t i o n d e f i n e s~ t he c l a s s a messa ge i n t e r m s o f a cl a ss

ca l l ed

identifier.

I t a l s o d e f i n e s s e v e r a l f u n c t i o n s f o r

m a n i p u l a t i n g a messa ge ' s . T h e s e c o m p r i s e t h e p r e d i c a t e s

2 Writin g a struc tur e definition i~volves coining nam es for the

var ious a l ternat ive formats of a m e s s a g e ' s and thei r components .

The only obscure coinage is beet , which abbrevia t es beta-

redex, i .e . , an express ion amenable to rule ( f l ) ; see Section 7'.

demand, simple, infixed,

e t c ; a l s o t h e s e l e c t o r s

body, ra to

leflarm, nee,

e t c ; a ls o ( t a k i n g f o r g r a n t e d c e r t a i n u n

f o r m a l i z e d c o n v e n t i o n s c o n c e r n i n g s t r u c t u r e d e f i n i t i o n

t h e c o n s t r u c t o r s ,

consdemand, conscombination

( e l s e w h e

~ b b r e v i a t e d t o combine), consstandardade], e t c . E x a m p l e

o f r e f e r e n c e I s w I ~ a r e g i v e n a l o n g s id e , a g a i n s t t h e r i g h

m a r g i n .

An amessage is

either a

d e m a n d ,

and has [P r in t

a + 2

a b o dy which is an aexprcss ion,

or else a definition, [Def

x = a + 2

w h e r e r e c

an aexpression (aexp) is

e i ther

s i m p l e ,

and has

[ C A t h 2 3

a b o dy which is an identifier

o r a c o m b i n a t i o n ,

in which case it has

[ s i n ( a + 2

a rator , which is an aexp, or

and a

r a nd ,

whic h is an aexp, a +

or c o n d i t i o n a l , in which case it is

e i ther

two-armed,

and has

[ p - -* a + 2 b ; 2 a -

a c o n d i t i o n , which is an aexp,

and a

t e f t a r m ,

which is an aexp,

and a r i g h t a r m , which is an aexp,

or one-armed, and has [ q -+ 2 a -

a c o n d i t i o n ,

which is an aexp,

and an a r m , which is an aexp,

or a

l i s t i n g ,

and has

[ a+ b , c + d , e +

a b o dy which is an aexp-list ,

or bee t , and has [x( x+l ) wh ere x = a +

a m a i n c l a u s e ,

which is an aexp, or

and a s u p p o r t let x = a + 2b; x( x+ l

which is an adef,

a n d

an adefinition (adef) is

e i ther s tandard, and has [ x = a + 2

a de f inee (nee ) ,

which is an abe,

and a de f i n i e ns (n i e ns ) , which is an aexp,

or

f u n c t i o n f o r m ,

ar id has [ f (x) =z(x +l

a le f thandside ( lhs ) ,

which is an abe- l i s t of length >2,

and a r ighthandside (rhs) , which is an aexp

or

p r o g r a m p o i n t ,

and has [pp f (x ) = x(x+ l

a b o dy which is an adef,

or c i r c u l a r , and has [ t ee f (n )= (n= 0) -+ l ; n f ( n - 1

a b o dy which is an adef,

or

s i m u l t a n e o u s ,

and has

[ x = a + 2 b

a n d

y = 2 a - -

a b o dy , which is an adef-list ,

or bee t , and has [ f (y) =z( x+y

a m a i n c l a u s e , w h e r e x = a + 2

which is an adef,

and a

s u p p o r t ,

which is an adef.

w here an abe i s

e i ther s i m p l e , and has

body, which is an identifier,

or else, is an abv-lislo.

[x , (y , z ) ,

A p r o g r a m - p o i n t d e f i n it i o n i n t r o d u c e s a d e v i a n t k i n

o f f u n c t io n . A p p l y i n g s u c h a f u n c t i o n p r e c i p i t a te s p r

m a t u r e t e r m i n a t io n o f t h e w h e r e - e x p r e s s i o n c o n t a i n i n

i t , a n d c a u s e s i ts r e s u l t t o b e d e l i v e r e d a s t h e v a l u e o f t h

e n t i r e w h e r e - e x p r e s s i o n .

P r o g r a m - p o i n t s a r e I s w l i ' S , n e a r e s t t h i n g to ju m p i n g

A s s i g n m e n t i s c o v e r e d a s a p a r t i c u l a r c a s e o f a n o p e r a t o

F o r b o t h o f th e s e t h e p r e c i s e s p e c i f ic a t i o n is in t e r m s o f t h

u n d e r l y i n g a b s t r a c t m a c h i n e ( s e e [ 2 ] ) .

V olum e 9 / N um ber 3 / M arch , 1966 C om m unic a t i ons o f t he A C M

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6 . R e l a t i o n s h i p t o L I S P

I s w I ~ r c a n b e l o o k e d o n a s a n a t t e m p t t o d e l i v e r L i s P

f r o n I i t s e p o n y m o u s c o m m i t m e n t t o l i s t s , i t s r e p u t a t i o n

f o r h a n d - t o - m o u t h s t o r a g e a ll o c a ti o n , t h e h a r d w a r e

d e p e n d e n t f l a v o r o f it s p e d a g o g y , i t s h e a v y b r a c k e t i n g ,

a n d i t s c o m p r o m i s e s w i t h t r a d i t i o n . T h e s e f i v e p o i n t s a r e

n o w d e a l t w i t h i n t u r n :

( 1) I s w i ~ h a s n o p a r t i c u l a r p r o b l e m o r i e n t a t io n .

E x p e r i m e n t s s o f a r h a v e b e e n m a i n l y i n n u m e r i c a l w o r k

a n d l a n g u a g e p r o c e s s in g w i t h b r i e f e x c u r s io n s i n t o c o m -

m e r c i a l p r o g r a m m i n g a n d e l s ew h e r e . N o b ia s t o w a r d s o r

a w a y f r o m a p a r t i c u l a r f ie ld o f a p p l i c a t i o n h a s e m e r g e d .

( 2 ) O u t s i d e a c e r t a i n s u b s e t ( c o r r e s p o n d i n g c l o s e l y t o

A LG O L ~ 0 w i t h o u t d y n a m i c o w n a r r a y s ) , I s w IM n e e d s

g a r b a g e c o l l e c t i o n . A n e x p e r i m e n t a l p r o t o t y p e i m p l e -

m e n t a t i o n f o l lo w e d c o m m o n A L GO L 6 0 p r a c t ic e . I t u s e d

d y n a m i c s t o r a g e a l lo c a t i o n w i t h t w o s o u rc e s , o n e L I F O

a n d t h e o t h e r g a r b a g e c o l l e c t e d , w i t h t h e i n t e n t i o n t h a t

t h e L I F O s o u r c e s h o u l d t a k e a s b i g a s h a r e a s p o s s i b l e .

H o w e v e r , a s w i t h A L G O L 6 0 , t h e r e i s a l a t e n t p o t e n t i a l

f o r p r e a ll o e a t i n g s t o r a g e i n c e r t a i n f a v o r a b l e a n d c o m -

m o n l y u s e f u l si t u a ti o n s . C o m p a r e d w i t h L I S P t h e c h i e f

a m e l i o r a t i o n o f s t o r a g e a l l o c a t i o n c o m e s o u t o f a m e r e

s y n t a c t i c d i ff e re n c e , n a m e l y , t h e u s e o f

w h e r e ( o l ' l e t '

w h i c h i s e x a c t l y e q u a l i n p o w e r a n d p r o g r a m s t r u c t u r e ) .

T h i s p r o v i d e s a b l o c k - s t r u c t u r e n o t d i s s i m i l a r i n t e x t u a l

a p p e a r a n c e f r o m A L G O L 6 0 'S , t h o u g h i t s l ip s o f f t h e p e n

m o r e e a s i l y , a n d i s i n m a n y r e s p e c t s m o r e g e n e r M .

( 3 ) L i s P h a s s o m e d a r k c o r n e r s , e s p e c i a l l y o u t s i d e

p u r e L IS P , i n w h i c h b o t h t e a c h e r s a n d p r o g r a m m e r s

r e s o r t t o t a l k i n g a b o u t a d d r e s s e s a n d t o d r a w i n g s t o r a g e

d i a g r a m s . T h e a b s t r a c t m a c h i n e u n d e r l y i n g I s w I ~ , r i s

a i m e d a t i l l u m i n a t i n g t h e s e c o r n e rs w i t h a m i n i n m m o f

h a r d w a r e d e p e n d e n ce .

( 4 ) T h e t e x t u a l a p p e a r a n c e o f I s w I ~ l i s n o t l ik e L i s p ' s

S - e x p r e s s i o n s . I t i s n e a r e r t o L I SP 'S M - e x p r e s s i o n s ( w h i c h

c o n s t i t u t e a n i n f o r m a l l a n g u a g e u s e d a s a n i n t e r m e d i a t e

r e s u l t i n h a n d - p r e p a r i n g L I SP p r o g r a m s ) . I s w lA i h a s t h e

f o l l o w i n g a d d i t i o n a l f e a t u r e s :

( a ) A u x i l i a r y d e f i n i t io n s , i n d i c a t e d b y ' l e t ' o r ' w h e r e ' ,

w i t h t w o d e c o r a t i o n s : ' a n d ' f o r s i m u l t a n e o u s d e f i n i ti o n s ,

a n d

' rec '

f o r s e lf - r e f e r e n t i a l d e f i n i t i o n s ( n o t t o b e m i s t a k e n

f o r a w a r n i n g a b o u t r e c u r s i v e ac t iva t ion, w h i c h c a n o f

c o u r s e a ls o a r i se f r o m s e l f - a p p l i c a t i o n , a n d w i t h o u t s e l f -

r e f e r e n c e ) .

( b ) I n f i x e d o p e r a t o r s , i n c o r p o r a t e d s y s t e m a t i c a l l y . A

l o g i c a l IS W I M c a n b e d e f i n e d i n t e r m s o f f o u r u n s p e c i f i e d

p a r a m e t e r s : t h r e e s u b s e t s o f th e c l a ss o f

ident i f ie rs ,

f o r u s e

a s p r e f i x e d , i n fi x e d a n d p o s t f i x e d o p e r a t o r s ; a n d a p r e c -

e d e n c e r e l a t i o n d e f i n e d o v e r t h e u n i o n o f t h e s e s u b se t s .

( c) I n d e n t a t i o n , u s e d t o i n d i c a t e p r o g r a m s t r u c t u r e . A

p h y s i c a l I s w iM c a n b e d e f i n e d i n t e r m s o f a n u n s p e c i fi e d

p a r a m e t e r : a s u b s e t o f p h r a s e c a t e g o r ie s , in s t a n c e s o f

w h i c h a r e r e s t r i c t e d i n l a y o u t b y t h e f o l l o w i n g r u l e c a l le d

t h e o f fs i de r u l e . T h e s o u t h e a s t q u a d r a n t t h a t j u s t c o n -

t a i n s t h e p h r a s e ' s f i r s t s y m b o l n m s t c o n t a i n t h e e n t i r e

p h r a s e , e x c e p t p o s s i b ly f o r b r a c k e t e d s u b s e g m e n t s . T h i s

r u l e h a s t h r e e i m p o r t a n t f e a t u r e s . I t i s b a s e d o n v e r t i c a l

a l i g n m e n t , n o t c h a r a c t e r w i d t h , a n d h e n c e is e q u a l ly

a p p r o p r i a t e in h a n d w r i t t e n , t y p e s e t o r t y p e d t e x t s. I t s

u s e is n o t o b l i g a t o r y , a n d u s e o f it c a n b e m i x e d f r e e l y w i t h

m o r e c o n v e n t i o n M a l t e r n a t i v e s l i k e p u n c t u a t i o n . A l so , i t

i s i n c o r p o r a t e d i n I s w I ~ t i n a s y s t e m a t i c w a y t h a t a d m i t s

o f a l t e r n a t i v e s w i t h o u t c h a n g i n g o t h e r f e a t u r e s o f I s w ~ . r

a n d t h a t c a n b e a p p l i e d t o o t h e r l a n g u a g e s .

( 5 ) T h e m o s t i m p o r t a n t c o n t r i b u t i o n o f L i s P w a s n o t

i n l i s t p r o c e s s i n g o r s t o r a g e a l l o c a t i o n o r i n n o t a t i o n , b u t

i n th e l o gic ~d p r o p e r t i e s l y i n g b e h i n d t h e n o t a t i o n , t t e r e

I s w i ? i m a k e s l i t t l e i m p r o v e m e n t b e c a u s e , e x c e p t f o r a f e w

m i n o r d e t a i ls , L i s p l e f t n o n e t o m a k e . T h e r e a r e t w o

e q u i v a l e n t w a y s o f s t a t i n g t h e s e p r o p e r t i e s .

( a ) L I s P s i m p l i f ie d t h e e q u i v a l e n c e r e l a t i o n s t h a t

d e t e r m i n e t h e e x t e n t t o w h i c h p ie c e s o f p r o g r a m c a n b e

i n t e r c h a n g e d w i t h o u t a f f e c t in g t h e o u t c o m e .

(b)

L IS P b r o u g h t t h e c la s s o f e n t i ti e s t h a t a r e d e n o t e d

b y e x p r e s s i o n s a p r o g r a m m e r c a n w r i t e n e a r e r t o t h o s e

t h a t a r is e i n m o d e l s o f p h y s i c a l s y s t e m s a n d i n m a t h e -

m a t i e M a n d l o g i c a l s y s t e m s .

T h e s e r e m a r k s a r e e x p a n d e d i n S e c t i o n s 7 a n d 8 .

7 . T h e C h a r a c t e r i s t ic E q u i v a l e n c e s o f I S W I M

F o r m o s t p r o g r a m m i n g l a n g u a g e s t h e r e a r e c e r t a i n

s t a t e m e n t s o f t h e k i n d , T h e r e is a s y s t e m a t i c e q u i v a l e n c e

b e t w e e n p i e c e s o f p r o g r a m l ik e t h is , a n d p i e ce s li k e t h a t ,

t h a t

ne a r l y

h o l d b u t n o t q u i t e. F o r i n s t a n c e i n A L G OL 6 0

t h e r e i s a ne a r l y t r u e s u c h s t a t e m e n t c o n c e rn i n g p r o c e d u r e

c a l ls a n d b l o c k s .

A t f i r s t s i g h t i t m i g h t a p p e a r p e d a n t i c t o q u i b b l e a b o u t

s u c h u n t i d i n e s s - - W h a t ' s t h e p o i n t o f h a v i n g t w o d i f fe r e n t

w a y s o f d o i n g t h e s a m e t h i n g a n y w a y ? I s n ' t i t b e t t e r t o

h a v e t w o fa c il it ie s t h a n j u s t o n e ? T h e a u t h o r b e l ie v e s

t h a t e x p r e s s i v e p o w e r s h o u l d b e b y d e s i g n r a t h e r t h a n

a c c i d e n t , a n d t h a t t h e r e i s g r e a t p o i n t i n e q u i v a l e n c e s t h a t

h o l d w i t h o u t e x c e p t i o n . I t i s a p l a t i t u d e t h a t a n y g i v e n

o u t c o m e c a n b e a c h i e v e d b y a w i d e v a r i e t y o f p r o g r a m s .

T h e p r a c t i c a b i l i t y o f a l l k i n d s o f p r o g r a m - p r o c e s s i n g

( o p t i m i z i n g , c h e c k i n g s a t i s f a c t i o n o f g i v e n c o n d i t i o n s ,

c o n s t r u c t i n g a p r o g r a m s a t is f y i n g g i v e n c o n d i t io n s )

d e p e n d s o n t h e r e b e i n g e l e g a n t e q u i v a l e n c e r u l es . F o r

I s w l M t h e r e a r e f o u r g r o u p s 3, c o n c e r n i n g :

( 1) t h e e x t e n t t o w h i c h a s u b e x p r e s s i o n c a n b e r e p l a c e d

b y a n e q u i v a l e n t s u b e x p r e ss i o n w i t h o u t d i s t u r b i n g t h e

e q u i v a l e n c e c la s s o f th e w h o l e e x p r e s si o n . W i t h o u t t h i s

g r o u p t h e o t h e r r u l e s w o u l d b e a p p l i c a b l e o n l y t o c o m p l e t e

e x p r e s s i o n s , n o t t o s u b e x p r e s s i o n s .

( 2 ) u s e r - c o i n e d n a m e s , i .e . , i n d e f i n i t io n s a n d , i n p a r t i c u -

l a r , f u n c t i o n d e f i n i t i o n s .

( 3 ) b u i l t - i n e n t i t i e s i m p l i c i t i n s p e c i a l f o r m s o f e x -

p r e s s i o n . T h e o n l y i i i s t a n e e s o f t h i s i n I sw llv [ a r e c o n d i t i o n a l

e x p r e s s i o n s , l i s ti n g s a n d s e l f - r e f e r e n t i a l d e f i n i t io n s .

( 4) n a m e d e n t i ti e s a d d e d i n a n y s p e ci fi c p r o b l e m -

o r i e n t a t i o n o f I s w I M .

3 To faci l i t a te subsequ ent di scussion each rule i s preceded

by a name, e.g., (~t) , ( , ) , e tc . These are chosen to conform

with precedents in Curry 's Combinatory Logic.

1 6 0 C o m m u n i c a t i o n s o f t h e A C M V o l u m e 9 / N u m b e r 3 / M a r c h , 1 96 6

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G R O U P 1

( tz ) I f L - = L ' t h e n L ( M ) - = L ' ( M )

@ ) I f M ~ M ' t h e n L ( M ) ~ L ( M ' )

@ ' ) I f M ~ M ' t h e n

( L , . . . , M , . . - , N ) ~ ( L , . . . , M ' , . . . N )

( v ) I f L ~ L ' t h e n ( L - - ~ M ; N ) ~ ( L '- - + M , N )

( v ' ) I f M ~ M ' t h e n

( L -- a M ; N ) ~ ( L - a M ' ; N )

( vi ~) I f N - ~ N ' t h e n ( L - - ~ M ; N ) ~ ( L -- ~ M ; N ' )

( v ~ ) I f M -= M ~ t h e n ( L w h e r e x = M )

~- ( L w h e r e x = M t)

T h e s i g n i f i c a n t o m i s s i o n s h e r e a r e t h e m a i n - c l a u s e i n t h e

l a s t c a s e a b o v e , t h e

r h s

o f a f u n c t i o n d e f i n i t i o n

f ( x ) = M

a n d o f a c i r c u l a r d e f i n i t i o n r e e :c = M .

G R o u e 2

( l e t ) l e t x = M ; L --= L w h e r e x = M

( I ' ) f ( x ) = L ~ f = ( g w h e r e g ( x ) = L )

f ( a , b , c ) ( x , y ) = L ~ f ( a , b , c ) = ( g w h e r e g ( x , y ) = L )

a n d s o o n f o r e a c h s h a p e o f le f t - h a n d s i d e

( I ) ( f w h e r e f ( x ) = L ) M ~ L w h e r e x = M

( ~ ' ) ( x = L ) w h e r e y = M ~ x ~ ( L w h e r e y = M )

( D ' ) x = L a n d y = M a n d ~ - . . - a n d z = N

( z , y , , z ) = ( L , M , - , N )

R u l e s ( I ' ) , ( ~ ' ) , ( D ' ) , t o g e t h e r w i t h ( Y ) b e l o w , e n a b l e

a n y d e f i n i t i o n t o b e s t a n d a r d i z e d , i .e . , e x p r e s s e d i n a

l h s / r h s

f o r m , i n w h i c h t h e

l h s

i s p r e c i s e l y t h e d e f i n e e . T h u s

a n o n s t a n d a r d d e f i n i t i o n c a n b e t r a n s f o r m e d s o a s t o b e

a m e n a b l e t o r u le s ( ~ ) a n d ( ~) ( s ee G r o u p 2 ' ) .

G R O U P 2 f

(f l ) L w h e r e x = M

S u b s t ~ i Ch e r e

M

S u b s t L

X

m e a n s r o u g h l y t h e e x p r e s s i o n re s u l t in g

f r o m s u b s t i t u ti n g A f o r B t h r o u g h o u t C . H e r e ' x ' m a y b e

a n y l i s t - s t r u c t u r e o f

d i s t i n c t

i d e n t i f i e r s , p r o v i d e d t h a t

' M ' h a s s t r u c t u r e t h a t f i t s i t .

T h i s r u l e is t h e m o s t i m p o r t a n t , b u t i t h a s o n l y l im i t e d

v a l i d i ty , n a m e l y , w i th i n t h e p u r e l y f u n c t i o n a l s u b s e t o f

IS W lM t h a t r e s u lt s f r o m n o t u s i n g t h e p r o g r a m - p o i n t

f e a t u r e o r a s s i g n m e n t .

I t s i m p o r t a n c e l i es in a v a r i a n t o f a f a m o u s t h e o r e m o f

m a t h e m a t i c a l l o g i c , t h e C h u r c h - R o s s e r t h e o r e m . T h i s

c o n c e r n s th e p o s s i b i li t y o f e l i m i n a ti n g ' w h e r e ' f r o m a n

e x p r e ss i o n b y r e p e a t e d l y a p p l y i n g t h e r u l e s s t a t e d a b o v e ,

i n c l u d i n g c r u c i a l l y ( ~) . T h e t h e o r e m e n s u r e s t h a t i f t h e r e

a r e s e v e r a l w a y s o f d o i n g t h is t h e y a l l r e a c h t h e s a m e

r esu l t .

T h e a u t h o r t h i n k s t h a t t h e f r u i t fu l d e v e l o p m e n t t o

e n c o m p a s s a l l I S W l M w i ll d e p e n d o n e s t a b l i s h i n g s a f e

a r e a s o f a n ISW lA ~ e x p r e s s io n , i n w h i c h i m p e r a t i v e f e a t u r e s

c a n b e d i s r e g a rd e d . T h e u s e f u ln e s s o f t h i s d e v e l o p m e n t

w i ll d e p e n d o n h o w s u c c e s s f u l ly IS W l M ' S n o n i m p e r a t i v e

f e a t u r e s s u p e rs e d e c o n v e n t i o n a l p r o g r a m m i n g .

GROUP 3

( - -~ ) t r u e - -~ M ; N ~ M

( - - / ) f a l s e - ~

M

N ~ N

( -- -~ ) P ~ M ~ P ~ M ; u n d e f i n e d

( u n d e f i n e d ) u n d e f i n e d ~ s e l f a p p l y ( s e l f a p p l y )

w h e r e s e l f a p p l y ( f ) = f ( f )

( Y ) r e c x = L ~ x = ( L w h e r e r e c x = L )

( D ) ( x , - . . , z ) = M ~ - ( x , ' , z ) =

n u l l ( t k w )

h w , . . . , h ( t ~ - l w )

w h e r e w = M

( f o r k > 2 )

(x, (u, v), z) = M =- (x, (u, v), z) =

n u l l ( t a w )

h ( w ) ,

( n u l l ( t2 w ' ) - - ~

h ( w ' ) , h ( t ( w ' ) )

w h e r e w ' =

h ( t ( w ) ) )

h ( t 2 w )

w h e r e

w = M

arid so on for each shape of definee

( n u l l ) n u l l ( n u l l i s l ) - = t r u e

( n u l l I ) n u l l ( L a, . . ' , L k ) ~

f a l s e

w h e r e ( x , ' , z)

= L b , L k ( k > 2 )

( h ) h ( L b , L k ) ~ x

w h e r e ( x , - . . , z )

= L j , . - . , L i t ( k > 2 )

( t ) t ( L 1 , . . . , L ~ ) ~ y , . . . , z

w h e r e ( x , y , - . - , z )

= L 1 , ' , L k (k _ 3 )

( t ' ) t ( t (L l , L2)) =-- nul l is t

w h e r e ( x , y ) = L1, L2

T h e r u l e s a b o u t l i s ti n g s m a y a p p e a r w i l l f u l l y i n d i r e c t .

T h e m o r e n a t u r a l t r a n s f o r m a t i o n s a r e t h o s e e ff e c t ed b y

a p p l y i n g , f o r e x a m p l e , ( D I ') t h e n ( ~) . B u t t h e s e w o u l d

h a v e s u f f e r e d t h e s a m e l i m i t e d v a l i d i t y a s ( ~) . I n t h e i r

a b o v e s l i g h t l y c a u t i o u s f o r m u l a t i o n t h e v a l i d i t y o f ( D I ' ) ,

e t c . is u n r e s t r i c t e d , a n d t h e m o r e p o w e r f u l eq u i v a l e n c e s

t h a t h o l d f o r n o n i m p e r a t i v e e x p r e s s i o n s a r i s e e n t i r e l y

f rom ( /3) .

G R o u P 4

A p r o b l e m - o r i e n t a t i o n o f Is w I~ r c a n b e c h a r a c t e r i z e d

b y a d d i t i o n a l a x i o m s . I n t h e s i m p l e s t c a s e s u c h a n a x i o m

i s a n I s w i M d e f i n i t i o n . T h e r e s u l t i n g m o d i f i c a t i o n i s c a l l e d

a d e f i n i t i o n a l e x t e n s i o n o f t h e o r i g i n a l s y s t e m .

I n m o r e e l a b o r a t e c a s e s a x i o m s m a y m u t u a l l y c o n s t r a i n

a g r o u p o f i d e n t if i e r s ; e. g . t h e f o l l o w i n g r u l e f o r e q u a l i t y

a m o n g i n t e g e r s :

( = ) S u p p o s e L a n d M a r e I S W I M w r i t t e n i n t e g e rs

( i . e . , s t r i n g s o f d i g i t s ) ; t h e n e i t h e r o n e o r t h e o t h e r o f t h e

f o l lo w i n g h o l d s :

L = M ~ - t r u e

L = M ~ f a l s e

a c c o r d i n g a s L a n d 114 d i f f e r a t m o s t i n l e f t h a n d z e r o s, o r

n o t .

A n o t h e r e x a m p l e p r e s e n t e d e v e n l e s s f o r m a l l y i s t h e

s t r u c t u r e d e f i n i t i o n f o r a b s t r a c t I S W l M .

G r o u p 1 a b o v e m a k e s n o p r o v i s i o n f o r s u b s t i t u t i o n s

w i t h i n e x p r e s s i o n s t h a t a r e q u a l i f i e d b y a s u p p o r t i n g

d e f i n i t i o n o r a r e u s e d t o d e f i n e a f u n c t i o n . H o w e v e r s u c h

a s u b s t i t u t i o n i s l e g i t i m i z e d a s l o n g a s i t d o e s n o t i n v o l v e

t h e d e f i n e e s o r v a r i a b l e s , b y e n c a s i n g i t w i t h i n a p p l i c a t i o n s

o f ru l e ( / 3) a n d i t s i n v e r s e ( w i t h a n y o t h e r r u l e s t h a t m i g h t

V o l u m e 9 / N u m b e r 3 M a r c h , 1 96 6

C o m m u n i c a t io n s o f t h e A C M

161

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6/10

b e n e e d e d t o p r o d u c e s o m e t h i n g t h a t i s a m e n a b l e t o ( ~ ) ,

i . e . , a bee t w i t h a

s t a n d a r d

def i n i t i on) .

E q u i v M e n c e ru l e s c a n b e u se d t o p r o v e t h i n g s a b o u t t h e

s y s t e m . F o r e x a m p l e , t h e r e a d e r w i ll r e a d i ly v e r i f y t h a t

t h e e q u i v a l e n c e o f

f (6) wh ere re ef (n ) = (n=0 ) --~ 1; n f ( n - - l )

a n d

6 (f(5) w h e r e re e f (n) = (n=0 ) --~ 1;

n f

( n - - l ) )

c a n b e e s t a b l i s h e d w i t h t h e f o l l o w i n g s t e p s :

( I ' ) , ( Y ) , ( f~ ), ( Y ) ,

(f3), (I) , (= ), ( i3) backwards, (Y) backwards,

(I ') backwards.

i n t h i s s e q u e n c e w e o m i t t h e a u x i l i a r y a p p l i c a t i o n s o f ( ~ ) ,

e t c. t h a t a r e n e e d e d a t M m o s t e v e r y s t e p t o le g i ti m i z e t h e

s u b s t i t u t i o n .

8 . A p p l i c a t i o n a n d D e n o t a t i o n

T h e c o m m o n p l a c e e x p r es s io n s o f a r i t h m e t i c a n d a l g e b r a

h a v e a c e r t a i n s i m p l i c i t y t h a t m o s t c o m m u n i c a t i o n s t o

c o m p u t e r s la c k . I n p a r t i c u l a r , ( a ) e a c h e x p r e s s i o n h a s a

n e s t i n g s u b e x p r e s s i o n s t r u c t u r e , ( b ) e a c h s u b e x p r e s s i o n

d e n o t e s s o m e t h i n g ( u s u a ll y a n u m b e r , t r u t h v M u e o r

n u m e r i c a l f u n c t i o n ) , ( c ) t h e t h i n g a n e x p r e s s i o n d e n o te s ,

i .e . , i t s v a l u e , d e p e n d s o n l y o n t h e v a l u e s of i t s sub-

e x p r es s io n s , n o t o n o t h e r p r o p e r t i e s o f th e m .

I t i s t h e s e p r o p e r t i e s , a n d c r u c i a l l y ( c ) , t h a t e x p l a i n s

w h y s u c h e x p r e s s i o n s a r e e a s i e r t o c o n s t r u c t a n d u n d e r -

s t a n d . T h u s i t is ( c ) t h a t l ie s b e h i n d t h e e v o l u t i o n a r y

t r e n d t o w a r d s b i g g e r r i g h t h a n d s i d e s i n p l a c e o f s t ri n g s

o f s m a l l , e x p l i c i tl y s e q u e n c e d a s s i g n m e n t s a n d j u m p s .

W h e n f a c e d w i t h a n e w n o t a t i o n t h a t b o r r o w s t h e f u n c -

t i o n a l a p p e a r a n c e o f e v e r y d a y a l g e b r a , i t i s (c ) t h a t g i v e s

u s a t e s t f o r w h e t h e r t h e n o t a t i o n is g e n u i n e l y fu n c t i o n a l

o r m e r e l y m a s q u e r a d i n g .

T h e i m p o r t a n t f e a t u r e o f IS W IM 's e q u i v a l e n c e r u l e s is

c h a t t h e y g u a r a n t e e t h e s a m e d e s i r a b l e p r o p e r t i e s t o

IS W lM 'S n o n i m p e r a t i v e s u b se t . W e m a y e q u a t e a b s t r a c t

o b j e c t w i t h e q u i v a l e n c e c l a ss , a n d e q u a t e d e n o t e s

w i t h

i s

a m e m b e r o f . T h e n t h e p r o p e r t ie s (g ) a n d ( v )

e n s u r e s a n o l o g i e s o f ( c ) a b o v e . T h e y s t a t e t h a t t h e v a l u e

o f an o p e r a t o r / o p e r a n d c o m b i n a t i o n d e p e n d s o n l y o n t h e

v a l u e s

o f i ts c o m p o n e n t s u b e x p r e ss i o n s, n o t o n a n y o t h e r

a s p e c t s o f t h e m .

T h u s c o n d i t i o n s ( g ) a n d ( v ) a r e e q u i v a l e n t t o t h e

e x i s te n c e o f a d y a d i c o p e r a t i o n a m o n g t h e a b s t r a c t o b -

j e c t s; w e c a ll t h i s o p e r a t i o n a p p l i c a t i o n .

T h e t e r m i n o lo g y o f a b s t r a c t o b j e c t s , d e n o t i n g a n d

a p p l i c a t i o n is f r e q u e n t ly m o r e c o n v e n ie n t t h a n t h a t o f

e q u i v a l e n c e r e la t io n s . F o r e x a m p l e , i t s u g g e st s a n o t h e r

w a y o f c h a r a c t e r i z i n g e a c h p r o b l e m - o r i e n t a t i o n o f I S W l M .

W e c a n t h i n k o f a se t o f a b s t r a c t o b j e c t s w i t h a p a r t i a l l y

d e f i n e d d y a d i c a p p l i c a t i o n o p e r a t i o n a n d a m o n a d i c

d e s i g n a t i o n o p e r a t i o n t h a t a s s o c ia t e s a p r i m i t i v e

a b s t r a c t o b j e c t w i t h e a c h o f s o m e c h o s e n s e t o f n a m e s ,

c a l le d t h e c o n s t a n t s o f t h e s p e c ia l s y s t e m .

C o n s i d e r f o r e x a m p l e a p r o g r a m m i n g l a n g u a g e t h a t

c o n t a i n s e x p r e s s i o n s s u c h a s

' w i n e '

A n y o n e w i t h a g e n e r o u s o n t o l o g y w i ll a d m i t t h a t

t h i s 6 - c h a r a c t e r e x p r es s i o n d e n o t e s t h e 4 - c h a r a c t e r - s t r in g

w i n e

F o r s u c h a p e r s o n i t s u s e i n t h e l a n g u a g e i s c h a r a c t e r i z e d

b y

• t h e o b j e c t s t h a t i t i s a p p l i c a b le t o , a n d t h e o b j e c t i t

p r o d u c e s i n e a c h c a s e ( e . g . , s t r i n g s m i g h t b e u s e d l i k e

v e c t o r s , w h o s e a p p l i c a t i o n t o a n i n t e g e r p r o d u c e s a n

i t e m o f t h e s t r i n g ) .

• T h e o b j e c t s t h a t i t is a m e n a b l e t o , a n d t h e o b j e c t i t

y i e l d s i n e a c h c a s e ( e . g . , p r e f i x i n g , a p p e n d i n g , s e l e c t i o n ,

e t c . ) .

T h e s c e p t i c n e e d n o t f e e l l e f t o u t . H e j u s t h a s t o t a l k , a

b i t m o r e c l u m s i l y , a b o u t

' w ine '

b e i n g i n t h e e q u i v a l e n c e c l as s t h a t a ls o c o n t a i n s

concatenate ( 'wi ' , 'he ' )

a n d

append (f i f lhle t terof (romanalphabet) ,

( threelet terstemof ( 'winter ' ))

T h e n h e g o e s o n t o s p e a k o f th e e q u i v a l e n c e c l as s o f

e x p r e s s i o n s t h a t c a n s e r v e a s o p e r a n d o r o p e r a t o r t o a n y

o f t h e a b o v e , a n d t h e e q u i v a l e n c e c l as s o f th e r e s u l t i n g

o p e r a t o r / o p e r a n d c o m b i n a t i o n .

9 . N o t e o n T e r m i n o l o g y

I SW I M b r i n g s i n t o s h a r p r e l ie f s o m e o f t h e d i s t i n c t i o n s

t h a t t h e a u t h o r t h i n k s a r e i n t e n d e d b y s u c h a d j e c t i v e s a s

p r o c e d u r a l , n o n p r o e e d u r a l , a l g o r i t h m i c , h e u r i s t i c , i m p e r a -

t i v e , d e c l a r a t i v e , f u n c t i o n a l , d e s c r i p t i v e . H e r e i s a s u g -

g e s t e d c l a s s i f i c a t i o n , a n d o n e n e w w o r d .

F i r s t, n o n e o f t h e s e d i s t i n c ti o n s a r e c o n c e r n e d w i t h t h e

u s e o f p i d g i n E n g l i sh r a t h e r t h a n p i d g i n a l g eb r a . A n y

p i d g i n a l g e b r a c a n b e d r e s s e d u p a s p i d g i n E n g l i s h t o

p l e a s e t h e g e n e r a l s . C o n v e r s e l y , i t i s a s p e c i a l e a s e o f t h e

t h e s is u n d e r l y i n g I SW l M t h a t a n y p i d g i n E n g l i s h t h a t h a s

s o fa r b e e n i m p l e m e n t e d c a n b e s t r i p p e d t o p i d g i n a l g e b r a .

T h e r e i s n e v e r t h e l e s s a n i m p o r t a n t p o s s i b il i ty o f h a v i n g

l a n g u a g e s t h a t a r e h e u r i s ti c o n a c c o u n t o f t h e i r a p p l i c a -

t i v e s t r u c t u r e b e i n g h e u r i st i c .

A n i m p o r t a n t d i s t i n c t i o n i s t h e o n e b e t w e e n i n d i c a t i n g

w h a t b e h a v i o r , s t e p - b y - s te p , y o u w a n t t h e m a c h i n e t o

p e r f o rm , a n d m e r e l y i nd i c at in g w h a t o u t c o m e y o u w a n t .

P u t t h a t w a y , t h e d i s t in c t i o n w il l n o t s t a n d u p t o c l os e

i n v e s t i g a t i o n . I s u g g e s t t h a t t h e c o n d i t i o n s ( a - e ) i n S e c t i o n

8 a r e a n e c e ss a r y p a r t o f m e r e l y i n d i c a t in g w h a t o u t c o m e

y o u w a n t . T h e w o r d d e n o t a t i v e s e em s m o r e a p p r o -

p r i a t e t h a n n o n p r o e e d u r a l , d e c l a r a t i v e o r f u n c t i o n a l . T h e

a n t i t h e s is o f d e n o t a t i v e is

i m p e r a t i v e .

E f f e c t i v e l y

d e n o t a t i v e m e a n s c a n b e m a p p e d i n t o IS W~ M w i t h o u t

u s i n g j u m p i n g o r a s s ig n m e n t , g i v e n a p p r o p r i a t e p ri m i -

t i ves .

1 6 2 C o m m u n i c a t i o n s of th e ACM Voh, me 9 / Nu mb er 3 / M ar ch , 1966

8/21/2019 Landin - The Next 700 Programming Languages (1966)

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I t f o l l o w s t h a t f u n c t i o n a l p r o g r a m m i n g h a s l i t t l e t o d o

w i t h f u n c t i o n a l n o t a t i o n . I t i s a t r i v i a l a n d p o i n t l e s s t a s k

t o r e a r r a n g e s o m e p i e c e o f s y m b o l i s m i n t o p r e f i x e d o p e r a -

t o r s a n d h e a v y b r a c k e t i n g . I t i s a n i n t e l l e c tu a l l y d e m a n d -

i n g a c t i v i t y t o c h a r a c t e r i z e s o m e p h y s i c a l o r l o g i c a l

s y s t e m a s a s e t o f e n t i ti e s a n d f u n c t i o n a l r e l a t i o n s a m o n g

t h e m . H o w e v e r , i t m a y b e le ss d e m a n d i n g a n d m o r e

r e v e a l i n g t h a n c h a r a c t e r i z i n g t h e s y s t e m b y a c o n v e n t i o n a l

p r o g r a m , a n d i t m a y s e r ve t h e s a m e p u r po s e . H a v i n g

f o r m u l a t e d t h e m o d e l , a s p e c if i c d e s i r e d f e a t u r e o f t h e

s y s t e m c a n b e s y s t e m a t i c a l l y e x p r e s s e d i n f u n c t i o n a l

n o t a ti o n . E u t o t h e r n o t a t i o n s m a y b e b e t t e r h u m a n

e n g i n e e r i n g . S o t h e r o le o f f u n c t i o n a l n o t a t i o n i s a s t a n d a r d

b y w h i c h t o d e s c r i b e ot h e r s , a n d a s t a n d b y w h e n t h e y f a il .

T h e p h r a s e d e s c r ib e i n t e r m s o f h a s b e e n u s e d a b o v e

w i t h r e f e r e n c e t o a l g o r i t h m i c m o d e s o f e x p r e s s i o n , i . e . ,

i n t e r c h a n g e a b l y w i t h e x p r e s s i n t e r m s o f . I n t h is s e n s e

3

+ 4 i s a d e s c ri p t i o n of t h e n u m b e r 7 in t e r m s o f t h e

n u m b e r s 3 a n d 4 . T h i s c o n f li c ts w i t h c u r r e n t u s e o f t h e

p h r a s e d e s c r i p t i v e l a n g u a g e s , w h i c h a p p e a r s t o f o ll o w

t h e l o g i c i a n s . F o r e x a m p l e , a l a n g u a g e i s d e s c r i p t i v e i n

w h i c h t h e m a c h i n e i s t o l d

P r i n t t h e x s u c h t h a t x 2 - - x - - 6 = 0 / ~ x > 0

S u c h a c l a s s i fi c a ti o n o f l a n g u a g e s ( a s o p p o s e d t o m e r e l y

e x p r e s s io n s w i t h i n la n g u a g e s ) i s u s e le s s , a n d e v e n h a r m f u l

b y e n c o u r a g i n g s t u p i d l y r e s t r i c t i v e l a n g u a g e d e s ig n , i f i t

e x c l u d e s t h e f o l l o w i n g :

P r i n t s q u a r e ( t h e x s u c h t h a t x ~ - - x - - 6 = 0 A x > 0 )

P r i n t u ( u + l )

w h e r e u = t h e x s u c h t h ~ t x 2 - x - - 6 = 0 A x _> 0 .

P r i n t f ( 1 , - - 1 , 6 )

where f ( a ,

b , c) = the x such that

a x ~ ÷ b x + c = 0 A x > _0

O n t h e o t h e r h a n d i t m i g h t r e a s o n a b l y e x c l u d e

P r i n t

so l e po s i t i v e z e r o o f

( 1 , - - 1 , - - 6 )

w h e r e

s o l e p o s i t i v e z e r o o ]

h a p p e n s t o b e a l i b r a r y f u n c t i o n .

T h e a u t h o r t h e r e f o r e s u g g e s t s t h a t t h e r e i s a u s e f u l

d i s t i n c ti o n t h a t c a n b e m a d e h e r e c o n c e r n i n g

l a n g u a g e s .

C o n s i d e r t h e f u n c t i o n i , w h i c h o p e r a t e s o n a c l a s s ( o r

p r o p e r t y ) h a v i n g a s o le m e m b e r ( o r i n s t a n c e ) , a n d t r a n s -

f o r m s i t i n to i t s s ol e m e m b e r . W e a r e i n t e r e s t e d i n w h e t h e r

o r n o t a l a n g u a g e p e r m i t s r e f e r e n c e t o i , w i t h m o r e o r

l e ss r e s t r i c t e d d o m a i n .

F o r e x a m p l e th e a b o v e p r o g r a m s b e c o m e :

P r i n t i p w h e r e p ( x ) = x 2 - - x - - 6 A x > O )

P r i n t squa r e ( i ( p

w h e r e

p ( x ) = x 2 - - x - - 6 A x > 0 ) )

P r i n t u (u--}-1 )

w h e r e u = i ( p

w h e r e p ( x ) = x ~ - - x - - 6 A x > O )

P r i n t f ( 1 , - - 1 , - - 6 )

w h e r e f ( a , b , c ) =

i ( p

w h e r e

p ( x ) = a x 2 - b b x + c A x > O)

M o r e p r e c i s e ly , t h e d i s t i n c ti o n h i n g e s o n w h e t h e r , w h e n

a p p l i c a t i v e s t r u c t u r e i s i m p u t e d t o t h e la n g u a g e , i t c a n

b e d o n e w i t h o u t r e s o r t i n g t o i , o r to p r i m i t i v e s in t e r m s o f

w h i c h i c a n b e d e f i n e d .

T h i s d i s c u s s i on o f i r e v e a l s t h e p o s s i b i l i ty t h a t p r i m i t i v e s

m i g h t b e s e n s a t i o n a l l y n o n a l g o r i t h m i e . S o t h e a l g o r i t h m i c /

h e u r i s t ic d i s t in c t i o n c u t s a cr o s s t h e d e n o t a t i v e / i m p e r a t i v e

V o l u m e 9 / N u m b e r 3 / M a r c h , 1 9 6 6

( i. e. , n o n p r o e e d u r a l / p r o c e d u r a l ) d i s t in c t i o n . O n t h e o t h e r

h a n d i f l i m i t e d f o r m s o f i c a n b e a l g o r i t h m i z e d , t h e y s t il l

d e s e r v e t h e t e r m d e s c r i p t i v e . S o t h i s f a c t o r i s a l so

i n d e p e n d e n t .

1 0. E l i m i n a t i n g E x p l ie l t S e q u e n e l n g

T h m ' e i s a g a m e s o m e t i m e s p l a y e d w i t h A L G O L 6 0

p r o g r a m s - - r e w r i t i n g t h e m s o a s t o a v o i d u s i n g l a b e l s a n d

g o t o s t a t e m e n t s . I t i s p a r t o f a m o r e e m b r a c i n g g a m e - -

r e d u c i n g t h e e x t e n t t o w h i c h t h e p r o g r a m c o n v e y s i t s

i n f o r m a t i o n b y e x p l i c i t s e q u e n c i n g . R o u g h l y s p e a k i n g t h i s

a m o u n t s t o u s i ng f e w e r a n d l a r g e r s ta t e m e n t s . T h e g a m e ' s

s i g ni f ic a n c e li e s i n t h a t i t f r e q u e n t l y p r o d u c e s a m o r e

t r a n s p a r e n t p r o g r a m - - e a s i e r t o u n d e r s t a n d , d e b u g ,

m o d i f y a n d i n c o r p o r a t e i n t o a l a r g e r p r o g r a m .

T h e a u t h o r d o e s n o t a r g u e t h e e a s e a g a i n s t e x p l i c it

s e q u e n c i n g h e re . I n s t e a d h e t a k e s a s p o i n t o f d e p a r t u r e t h e

o b s e r v a t i o n t h a t t h e u s e r o f a n y p r o g r a m m i n g l a n g u a g e i s

f r e q u e n t l y p r e s e n t e d w i t h a c h o i c e b e t w e e n u s i n g e x p l ic i t

s e q u e n c i n g o r s o m e a l t e r n a t i v e f e a t u r e o f t h e l a n g u a g e .

F u r t h e r m o r e l a n g u a g e s v a r y g r e a t l y i n t h e a l t e r n a t i v e s

t h e y o f f e r . F o r e x a m p l e , o u r g a m e i s g r e a t l y f a c i l i t a t e d b y

A L G OL 6 0 'S c o n d i t i o n a l s t a t e m e n t s a n d c o n d i t i o n a l e x -

p r e s s i o n s . S o t h e q u e s t i o n c o n s i d e r e d h e r e i s : W h a t o t h e r

s u c h f e a t u r e s a r e t h e r e ? T h i s q u e s t i o n i s c o n s i d e r e d b e -

c a u s e , n o t s u r p r i s i n g l y , i t t u r n s o u t t h a t a n e m p h a s i s o n

d e s c r i b in g t h i n g s i n t e r m s o f o t h e r t h i n g s l e a d s t o t h e

s a m e k i n d o f r e q u i r e m e n t s a s a n e m p h a s i s a g a i n s t e x p l i c i t

s e q u e n c i n g .

T h o u g h A ~G O ~ g 0 is c o m p a r a t i v e l y f a v o r a b l e t o t h i s

a c t i v i t y , i t s h a r e s w i t h m o s t o t h e r c u r r e n t l a n g u a g e s

c e r t a i n d e f i ci e n ci e s t h a t s e v e r e l y li m i t h o w f a r t h e g a m e

c a n g o. T h e a u t h o r ' s e x p e r i m e n t s s u g g e s t t h a t t w o o f t h e

m o s t n e e d e d f e a t u r e s a r e :

• T r e a t a l i s t i n g o f e x p r e s s i o n s a s a s p e c i a l e a s e o f t h e

c l a s s o f e x p r e s s i o n s , e s p e c i a l l y i n t h e a r m s o f a c o n d i t i o n a l

e x p r e s s i o n , a n d i n d e f i n i n g a f u n c t i o n .

• T r e a t , a r g u m e n t l i s t s a s a s p e c i a l e a s e o f l is t s . S o a

t r i a d ic f u n c t i o n c a n h a v e i t s a r g u m e n t s s u p p l i e d b y a

c o n d i t i o n a l w h o s e a r m s a r e 3 - l i s t i n g s , o r b y a p p l i c a t i o n o f

a f u n c t i o n t h a t p r o d u c e s a 3 -l i s t. A s i m i l a r s i t u a t i o n a r i s e s

w h e n a 3 - l i s t i n g o c c u r s a s a d e f i n e e . ( E v e n L I s P t r i p s u p

h e r e , o v e r l i s t s o f l e n g t h o n e . )

T o c l a r i fy t h e i r p r a c t i c a l u s e , h e r e a r e s o m e o f t h e

s t e p s b y w h i c h m a n y a c o n v e n t i o n a l A L G O L e 0 o r P L / 1

p r o g r a m c a n b e t r a n s f o r m e d i n t o a n I s w I~ ,r p r o g r a m t h a t

e x p l o i t s I s w I s l ' s n o n i m p e r a t i v e f e a t u r e s .

( 1) R e w r i t e t h e p r o g r a m s o a s t o us e t w o - d i m e n s i o n a l

l a y o u t a n d a r r o w s t o i l l u m i n a t e t h e e x p l i c i t s e q u e n c i n g ,

i . e . , a s a f l o w c h a r t w i t h a l g e b r a i c s t e p s . R e a r r a n g e t h i s t o

a c h i e v e t h e l e a s t c o n f u s i ng n e t w o r k o f a r r o w s .

( 2) A p p l y t h e f o l l o w in g c h a n g e s r e p e a t e d l y w h e r e v e r

t h e y a r e a p p l i c a b l e :

( a ) R e p l a c e a s t ri n g o f i n d e p e n d e n t a s s i g n m e n t s b y o n e

m u l t i p l e a s s i g n m e n t .

( b ) R e p l a c e a n a s s i g n m e n t h a v i n g p u r e l y l o c a l s i gn if i-

c a n c e b y a w h e r e - c l a u s e .

( e) R e p l a c e p r o c e d u r e s b y t y p e - p r o c e d u r e s ( p o s s i b ly

C o m m u n i c a t i o n s o f t b e A C M 163

8/21/2019 Landin - The Next 700 Programming Languages (1966)

8/10

w i t h m u l t i p l e t y p e ) , a n d p r o c e d u r e s t a t e m e n t s b y a s si g n-

m e n t s t a t e m e n t s .

( d) R e p l a c e c o n d i t i o n a l j u m p s b y c o n d i t i o n a l s t a te -

m e n t s h a v i n g b i g g e r a r m s .

( e) R e p l a c e a b r a n c h w h o s e a r m s h a v e a s si g n ee s i n

c o m m o n b y a n a s s ig n m e n t w i t h c o n d i t i o n a l r i g h t - h a n d

side.

( f) R e p l a c e a j o i n b y t w o c a l ls fo r a p r o c e d u r e .

I t s h o u l d b e o b s e r v e d t h a t t r a n s l a t in g i n t o I SW l M d o e s

n o t

force

s u c h r e a r r a n g e m e n t s ; i t m e r e l y f a c i l i t a t e s t h e m .

O n e i n t e r e s t i n g o b s e r v a t i o n i s t h a t t h e m o s t r e c a l c i t r a n t

u s e s o f e x p l i c i t s e q u e n c i n g a p p e a r t o b e a s s o c i a t e d w i t h

s u c c e s s / f a i l u r e s i t u a t i o n s a n d t h e a c t i o n n e e d e d o n f a i l u r e .

S e c t i o n 2 di s c us s e d a d d i n g ' w h e r e ' t o a c o n v e n t i o n a l

p r o g r a m m i n g l a n g u a g e. T h e o r y a n d e x p e r i m e n t b o t h

s u p p o r t t h e o p p o s i t e a p p r o a c h , t h a t t a k e n i n L l s v , o f

a d d i n g i m p e r a t i v e f e a t u r e s t o a b a s i c a l l y n o n i m p e r a t i v e

l a n g u a g e . O n e b i g a d v a n t a g e i s t h a t t h e r e s u l ti n g l a n g u a g e

w i ll h a v e a n o n i m p e r a t i v e s u b s e t.

T h e s p e c i a l c l a i m o f IS W l M i s t h a t i t g r a f t s p r o c e d u r a l

n o t i o n s o n t o a p u r e l y f u n c t i o n a l b a s e w i t h o u t d i s t u r b i n g

m a n y o f t h e d e s i r a b l e p ro p e r t i e s. T h e u n d e r l y i n g i d e a s

h a v e b e e n p r e s e n t e d i n [2 ]. T h i s p a p e r c a n d o n o m o r e t h a n

b e g i n t h e t a s k o f e x p l a i n i n g t h e i r p r a c t i c a l s i g n i f i c a n c e .

11 . C o n c l u s i o n

T h e l a n g u a g e s p e o p l e u s e t o c o m m u n i c a t e w i t h c o m -

p u t e r s d i f f e r i n t h e i r i n t e n d e d a p t i t u d e s , t o w a r d s e i t h e r a

p a r t i c u l a r a p p l i c a t io n a r e a , o r a p a r t i c u l a r p h a s e o f c o m -

p u t e r u s e ( h ig h le v e l p r o g r a m m i n g , p r o g r a m a s s e m b l y ,

j o b s c h e d u l i n g , e t c ) . T h e y a l s o d i f fe r i n p h y s i c a l a p p e a r -

a n c e , a n d m o r e i m p o r t a n t , i n lo g i ca l st r u c t u r e . T h e q u e s -

t i o n a r i se s , d o t h e i d i o s y n c r a c i e s r e f l e c t b a s i c l o g i c a l

p r o p e r t i e s o f t h e s i t u a t i o n s t h a t a r e b e i n g c a t e r e d f o r ?

O r a r e t h e y a c c i d e n t s o f h i s t o r y a n d p e r s o n a l b a c k g r o u n d

t h a t m a y b e o b s c u r i n g f r u i t f u l d e v e l o p m e n t s ? T h i s

q u e s t i o n i s c l ea r l y i m p o r t a n t i f w e a r e t r y i n g t o p r e d i c t o r

i mq_ uence l ang uag e e vo l u t i on .

T o a n s w e r i t w e m u s t t h i n k i n te r m s , n o t o f l a n g u a g e s ,

b u t o f fa m i l ie s o f l a n g u a g e s. T h a t i s t o s a y w e m u s t

s y s t e m a t i z e t h e i r d e s i g n s o t h a t a n e w l a n g u a g e i s a p o i n t

c h o s e n f ro m a w e l l - m a p p e d s p a c e, r a t h e r t h a n a l a b o r i o u s ly

d e v i s e d c o n s t r u c t io n .

T o t h i s e n d t h e a b o v e p a p e r h a s m a r s h a l l e d t h r e e

t e c h n i q u e s o f la n g u a g e d e s i g n : a b s t r a c t s y n t a x , a x i o m a t i z a -

t i o n , a n d a n u n d e r l y i n g a b s t r a c t m a c h i n e .

I t i s a s s u m e d t h a t f u t u r e c a ll s o n la n g u a g e d e v e l o p m e n t

c a n n o t b e f o r s t a l l e d w i t h o u t g e n e r ~ l i z i n g t h e a l t e r n a t i v e s

t o e x p l ic i t s e q u e n c in g . T h e i n n o v a t i o n s o f p r o g r a m -

p o i n t s a n d t h e o f f -s i d e r u l e a r e d i r e c t e d a t t w o o f t h e

p r o b l e m s ( r e s p e c ti v e l y i n t h e a r e a s o f s em a n t i c s a n d

s y n t a x ) t h a t m u s t c o n s e q u e n t l y b e f a ce d .

Acknowledgments. T h e a u t h o r i s g r a te f u l f o r h e lp f u l

d i s c u s s io n s w i t h W . H . B u r g e . W i d e r i n f l u e n c e s o n t h e

i n v e s t i g a t io n o f w h i c h t h i s p a p e r i s o n e o u t c o m e a r e

m e n t i o n e d i n [ 1 ]. O f t h e s e t h e m a i n o n e s a r e t h e p u b l i c a -

t io n s o f C u r r y a n d o f M c C a r t h y .

~ E F E R E N C E S

1. LANDIN, P. J. The mechanical evalua tion of expressions.

Comput. J. 6, 4 (Jan. 1964), 308-320.

2. -- . A correspond ence betwee n ALGOL 60 arid Church 's

Lambda-notat ion . Comm. ACM 8 (1965), 89-101; 158-165.

3. -- . A formal descrip tion of ALGOL 60. In Formal Language

Description Languages for Computer Programming, T. B.

Steel, Jr. (Ed.), Nort h Hollan d, Amst erdam , 1965.

4. -- . An abstract machine for designers of computing lan-

guages. (Summary). IFIP65 Proc., Part II.

D I S C U S S I O N

Naur: Ilegarding indentation, in many ways I am in sympathy

with this, but I believe that if it came about that this notation

were used for very wide communication and also publication, you

would regret it because of the kind of rearr angem ent of manu-

scripts done in printing, for example. You very frequently run

into the problem that you have a wide written line and then

suddenly you go to the Communications of the ACM and radically,

perhaps, you have to compress it. The printer will do this in any

way he likes; he is used to having great freedom here and he will

foui up your notation.

Landin:

I have great experience with this. (L aughter) I thi nk

I am probably the only person who has run through three versions

of the galley proofs for the

Communications of the ACM.

However,

I think that next time I could do better, and I think it is worth

looking into. At any rate, the principle tha t [ have described here

is a good deal bet ter than some that one might think of ; for example

it does riot depend on details of character width, character by

charac ter--i t is just as good handwritten as it is printed. Secondly,

limiting the breadth of the page, I agree with you, needs more

consideration. By the time I got through with the particular

example I am talking about, by getting it printed, I had devised

what I thought was a fairly reasonable method of communicating

the principles that have been used in indentation.

Floyd:

Anot her object ion that 7[ thi nk is quite serious to

indentati on is that while it works on the micr o-scale --that is, one

page is all r ig ht--wh en dealing with an extensive program, t urning

from one page to the next there is no obvious way of indicating

how far indentat ion stretches because there is no printing at all to

indicate how far you have indented. I would like you to keep that

in mind.

Landin:

Yes, I agree. In practice I deal with this by first making

the page breaks in sensible places.

Floyd:

That ' s a l l right as long as you don't have an indented

region which is simply several pages long.

Landin:

Well in that ease the way I did it was to cut down the

number of carryover levels to about four or five from one page to

another. You can at least make it simpler when you are hand-

writing by putting some kind of symbols at the bottom of the page

and top of the continuation.

Floyd:

Even if you regard your indent ation spaces as characters

there still doesn' t seem to be any way--in fact, I am fairly sure

there is no way--of representing the indentation conventions

within a phrase-structure grammar.

Landin:

Yes, but some indentation conventions can be kept

within phrase structure grammars by introducing two terminal

symbols that are grammatically like parentheses, but are textually

like typewriter keys for settling and clearing tabulation positions.

More precisely, the textual representation of the second of these

symbols can be explained as the following sequence of typewriter

actions: 1) line-feed; 2) back-space as far as the right-most tab

position that is still currently active; 3) clear tab position; and

4) do step 2 again.

While this fits some indentation conventions, the olle I propose

is too permissive to be included. For my language I have written

a formal grammar t hat is not phrase str ucture and includes one

departure that meets this problem.

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Leavenworth:

I should like to raise the question of eliminating

explicit jumps, I mean of using recursion as against interation.

Landin:

It seems to me that there are rather a small number of

function s which you could use if you were writing a Lisp progr am

in the places where ordinary programs would use iterations, and

that if you were to use these the processor might do as well as if

you had writte n a loop. For example, ite ra te (m, f, x) might

apply f, m times to x with the result f ~(x). This is the simplest

kind of loop I know and the function iterate provides a purely

functional notation for this rather simple kind of loop. If a lot of

familiar t ypes of loop can be repre sented by a few functions which

could be defined recursively, I think it is sensible to take these as

primitive. Another such function is while (p, f, x) which goes on

applying f to x until the predica te p becomes false.

Strachey:

I must just interp olate here something which is a bit

of adverti sing l suppose. Nearly all the linguistic feature s, such as

where and while and and and recursive, that Peter Landin has

been talking about are incorporated as an integral part of a pro-

gramming language being developed at Cambridge and London

called CPL. In fact the wher e clauses are a very important feature

of this la nguage.

Irons:

I have put together a program which uses some of these

features and which has a standard output which prints the pro-

gram in an ind ented manne r. If it runs off the ri ght end of the page,

it t:rnduces another page to go on the right, and so forth. While

certainly there are some situations that occur when it would be a

bit awkward to make the paper go around the room, I have found

that in practice, by and large it is true that this is a very profit-

able a ay of operating.

Strachey:

I should like to intervene now and try to initiate a

slightly more general discussion on declarative or descriptive

languages and to try to clear up some points about which there is

considerable confusion. I have called the objects I am trying to

discuss DLs because I don't quite know what they are. Here are

some questions concerning ])Ls: (1) What are DLs? (2) What is

their relationship to imperative languages? (3) Why do we need

DLs? (4) How can we use them to program? (5) How can we

implement them? (6) How can we do this efficiently? (7) Should we

mix l)Ls with imperative languages?

It seems to me that what I mean by DLs is not exactly what

other people mean. I inean, roughly, languages which do not

contain assignment statements or jumps. This is, as a matter of

fact, not a very clear distinction because you can always disguise

the assignments and the jumps, for that matter, inside other state-

meat forms which m~ke them look different. The important

characteristic of DLs is that it is possible to produce equivalence

relations, particularly the rule for substitution which Peter

Landin describes as (~) in his paper. That equivalance relati on,

which appears to be essential in ahnost every proof, does not

hold if you allow assignment statements. The great advantage

then of l)Ls is that they give you some hope of proving the equi-

valence of progra m transf ormat ions and to begin to have a calculus

for combining and manipulating them, which at the moment we

haven't got.

I suggest that an answer to the second question is tha t DLs form

a subset of all languages. They are an interesting subset, but one

which is inconvenient to use unless you are used to it. We need

them because at the moment we don't know how to construct

proofs with languages which include imperatives and jumps.

How should we use them to program? I think thi s is a matter of

learning a new programnling technique. I am not convinced that

all problems are amenable to programming in DLs but I am not

convinced that there are any which are not either; I preserve an

open mind on this point. It is perfectly true that in the process of

rewriting programs to avoid labels and jumps, you've gone half

the way towards going into 1)Ls. When you have also avoided

assignment statements, you've gone the rest of the way. With

many problems yeu can, in fact, go the whole way. LisP has no

assignment statements and it is remarkable what you can do with

pure Lisp if you try. If you thin k of it in terms of the i mple menta -

tions that we know about, the result is generally intolerably

ineffi cient --but then th at is where we come to the late r questions.

How do we implement them? There have not been many at-

tempts to implement DLs efficiently, I think. Obviously, it can be

done fairly straightforwardly by an interpret ive method, but this

is very slow. Methods which compile a runable prog ram run into a

lot of very interesting problems. It can be done, because DLs are

a subset of ordinary programming languages; any progr amming

language which has sufficient capabilities can cope with them.

There arc problems, however: we need entities whose value is a

functi on--not the application of a function but a functi on--and

these involve some problems.

Itow to implement efficiently is another very interesting and

difficult problem. It means, I think, recognizing certain subsets

and transforming them from, say, recursions into loops. This can

certainly be done even if they have been written iu terms of

recursions and not, as Peter Landin suggested, in terms of already

transformed functions like iterate or while.

I think the last question, Should DLs be nIixed with impera-

tive languages? , clearly has the answer that they should, be-

cause at the moment we don't know how to do everything in pure

DLs. If you mix declarative and imperative features like this, you

may get an apparently large programming language, but the

important thing is that it should be simple and easy to define a

function. Any language which by mere chance of the way it is writ-

ten makes it ex tremel y difficult to write compositions of function s

and ve ry easy to writ e sequences of command s will, of course, in an

obvious psychological way, hinder people from using descriptive

rather t han imperative features. In the long run, I think the effect

will delay our underst andin g of basic similarities, which underlie

different sorts of programs and different ways of solving problems.

Smith:

As I understand the declarative languages, there has to

be a mixture of imperative and descriptive statements or no com-

putation will take place. If I give you a set of simultaneous equa-

tions, you may say yes? , meaning well, what am I supposed to

do about it, or you may say yes , meaning yes I underst and, but

you don't do anything until I say now find the values of the vari-

ables . In fact, in a well-developed language there is not just one

question th at I can ask but a n umber of questions. So, in effect, the

declarat ive sta tem ent s are like data which you set aside to be u~ed

later after I give you the imperatives, of which I had a choice,

which get the action.

Strachey:

This is a major poin t of confusion. There are two ideas

here and I think we should try to sort them out. If you give a

quadratic equation to a machine and then say print the value of

x , this is not the sort of language that I call a DL. I regard it as

an implicit language--that is, one where you give the machine the

data and then hope that it will be smart enough to solve the prob-

lem for you. It is very different from a language such as LisP,

where you define a function explicitly and have only one impera-

tive. which says evaluate this expression and print the result.

Abrahams:

I've clone a fair amount of programming in LisP,

and there is one situat ion which I feel is sympt omat ic of the times

when you really do want an imperative language. It is a situation

tha t arises if you are planning to do programming in pure Lisp and

you find that your functions accumulate a large number of argu-

ments. This often happens when you have a nmnber of variables

and you are actually going through a process and at each stage of

the process you want to change the state of the world a little bit --

say, to change one of these variables. So you have the choice of

either trying to communicate them all, or trying to do some sort

of essentially imperative action that changes one of them. If you

try to list all of the transitions from state to stat e and incorporate

them into one function, you'll find that this is not really a very

natural kind of function because the natures of the transitions

are too different.

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L a n d i n : I said in iny talk that LisP had not gone quite all the

way and I think that this difficulty is connected with going all the

way. If we write a function definition where the right- hand side is

a listing of expressions such as

F x ) = E 1 , E 2 , E ~

thel~ we can say that this function will produce a three-list as its

result. If llOW we have ~mother functio n

G ( x ,

y, z) = E, on some

occasion we might have an expression such as

G (a 2 , b 2 , c ~ )

and we

often feel that we should be able to write G ( F ( t ) ) , and ano ther

example which should be allowed is

G ( a > b -- ~ E 1 , E 2 , E 3 else E4 , E5 , E6).

l am not quite sure but I think you can get around your problem

by treating every function as if i t were in fact monadic and ha d

a single argum ent which was the l ist struc ture you are tryi ng to

process.

A b r a h a m s :

This is a difficulty in other progr ammi ng languages

too; you can not define a function of an indefinite numbe r of argu-

ments .

N a u r :

I s t i l l don ' t unders tand th is d is t inct ion about an im-

plicit language. Does i t mean that whenever you have such a

language there is a buil t-in feature for solving equations?

A b r a h a m s : I think the point is whether you are concerned with

the problem or are concerned with the method of solution of the

problem.

I n g e r m a n :

I suggest tha t in the si tuatio n where you have speci-

fied ever ything that you want to know, thoug h the exact sequence

in which you evoke the various operations to cause the solution is

left unspecified, then you have som eth ing which is effectively a

descriptive language; if you have exactly the same pieces of in-

format ion , surrounded wi th p romises tha t you wil l do th is and

then th is , then you have an imperat ive language. The s ign i f icant

point is that i t is not all or nothing but there is a scale and while

it is probably pre tty simple to go all the way with imperative s,

the chan ces of being ttble to get all the wa y descr iptive is abou t

zero, but there is a settle and we should recognize this scale.

S m i l h : I think that there is a confusion between implic