ten mini-languages: a study of topical issues in programming languages

8/9/2019 Ten Mini-Languages: a Study of Topical Issues in Programming Languages

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T e n M i n i - L a n g u a g e s :

A S t u d y o f T o p i c a l I s s u e s in P P o g P a m m i n g L a n g u a g e s

HENRY F. LEDGARD*

Computer Science, The John s H op kins University, Baltimore, M aryla nd

The proliferation of programming languages has raised many issues of language

design, definition, and implementation. This paper presents a series of ten

mini-languages, each of which exposes salient features found in existing

programming languages. The value of the mini-languages lies in their brevity of

description and the isolation of important linguistic features: in particular, the

notions of assignment, transfer of control, functions, parameter passing, type

checking, data structures, string manipulation, and input/output. The

mini-languages may serve a variety of uses: notably, as a pedagogical tool for

teaching programming languages, as a subject of study for the design of

programming languages, and as a set of test eases for methods of language

implementation or formal definition.

Key words and phrases : curriculum for computer science, language design,

compiler design, formal definition, syntax, semantics, assignment, transfer of

control, functions, passing of parameters, type checking, string manipulation,

data structures, input/output

CR calegories: 1.3, 1.52, 4.10, 4.20, 4.22, 4.9

GENERALCONSIDERATIONS

Since the birth of I¢ORTRAN, the notion of

higher-level programming languages has led

to a proliferation of special-and general-

purpose programming languages. It seems

wise, at this juncture, to examine critical

issues raised by the myriad of languages and

to try to isolate certain important features

that seem common to many programming

languages. In this endeavor, we hav.e pre-

pared ten mini-languages, each presenting

in capsule form one or two features taken

from existing programming languages. (The

reader may find the work of Strachey [16]

useful as a companion to this paper, as he

considers many issues related to those pre-

sented here.)

The issues treated in this paper are sum-

marized in Table I. Each mini-language is a

* The work reported herein was performed in

large part at the Programming Research Group,

Computing Laboratory, Oxford University, Ox-

ford, England.

complete (although restricted language in

itself, and each is described only informally.

A modest appeal is made to the reader's

knowledge of constructs in existing pro-

gramming languages. None of the mini-

languages are exact subsets of existing

languages, although much of the notation

and semantic material resembles portions of

existing languages. Many important fen-

tures of existing languages are omitted.

These features include parallel computation,

interrupts and events in real-time, file and

storage management, and simulation.

The format for presenting each mini-

language is as follows:

1) a brief introduction to some topic in

programming languages;

2) a description in English of a mini-

language covering the topic;

3) several example programs in the mini-

language; and

4) a discussion of the mini-language and

Computing Surveys, Vol. 3, No. 3, September 1971


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1 1 6 ° H en r y . L edgar d

C O N T E N T S

General Considerations 11~ 118

Mini-Language 1 Simple Assignment, Transfer of

Control, and Block Structure 118-119

Description of Language Elements

Piogram Execution

Discussion

Mini-Language 2: Generalized Assignment and the

Notion of Locations 119-121


Program Execution

Discussion

Mini-Language 3: Generalized Transier of Control

121-123


Program Execution

Discussion

Mini-Language 4: Functions 123-126


Program Execution

Discussion

Mini-Language 5: Passing of Parameters 126-128


Program Execution

Discussion

Mini-Language 6: Static Type Checking 128 132


Program Execution

Discussion

Mini-Language 7: Dynamic Type Checking 132-134


Program Execution

Discussion

Mini-Language 8: Structured Data 134-139


Program Execution

Discussion

Mini-Language 9: String Manipulation 139 142


Program Execution

Discussion

Mini-Language 10: Input/Output 142-146


Program Execution

Discussion

Acknowledgments 146

References 146

Copy right © 1971, Associat ion for Compu ting

Machinery, Inc. General permission to republish,

but not for profit , all or part of this material is

granted, provided that reference is made to this

publication, to its date of issue, and to the fact

that reprint ing privi leges were granted by permis-

sion of the Associat ion for Computing Machinery.

i t s r e l a t i o n t o i s s u e s i n c u r r e n t p r o g ra m -

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

T h e r e a p p e a r t o b e a t l e a s t f o u r m aj or

a r e a s in w h i c h t h e m i n i - l a n g u a g e s m i g ht

p r o v e u s e f u l :

1 ) a s a p e d a g o g i c a l t o o l f o r t e a c h i n g p r o-

g r a m m i n g l a n g u a g e s;

2 ) a s a s u b j e c t o f s t u d y f o r l a n g u a g e d e-

s i g n ;

3 ) a s a s e t o f t e s t c a s e s f o r p r o p o s e d m e t h -

o d s o f l a n g u a g e i m p l e m e n t a t i o n ; a n d

4 ) a s a s e t o f t e s t c a s e s f o r p r o p o s e d m e t h -

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

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

i m m e d i a t e p r o b l e m i s th a t m o s t l a ng u ag es

a r e s o c o m p l e x t h a t a n y a t t e m p t t o is ola te

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

d e a l o f s t u d y . T h e r e i s c l e a r l y a n e e d fo r

s m a l l , d i g e s t i b l e e x a m p l e s t o i ll u m i n a te

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

j e c t o f s t u d y . T h e t e n m i n i - la n g u a g e s a re

p a r t o f a n e f f o r t t o m e e t t h i s n e e d . T h e is-

s u e s t r e a t e d i n t h e m i n i - l a n g u a g e s a r e c om -

m o n

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

e.g.,

P L / I , ALGOL 60, BASIC, C P L , P A L , FOR-

TRAN, MAD, LISP, SNOBOL, AXLE, GEDAN-

KEN, and ISWlM.

T A B L E I . SUMMARY OF TOPICS TRI~]ATED IN

EACH MINI-LANGUAGE

M i n i - T o p i c

language

1 S imple ass i gnme nt ,

t ran sfer of cont rol ,

and block s t ruc-

ture

2 General ized assign-

ment and the no-

t ion of locat ions

3 Genera l ized t ran sfer

of control

Commands

4 F u n c t i o n s

5 P a s s i n g o f p a r a m e -

t e r s

P r o c e d u r e s

6 Stat ic type checkin g

7 Dyn ami c type check-

i ng

Type checking

8 S t ruc tu red da t a

9 S t r i ng manipu l a t i on

10 Inpu t / ou tpu t

Appl ica t ions

Computing Surveys, Vol. 3, No. 3, September 1971


3/32

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

t r i b u t i n g t o a v i a b l e t h e o r y o f l a n g u a g e d e -

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

m i ng l a n g u a g e s. T h i s is i n d e e d u n f o r t u n a t e ,

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

h in g es c r i t ic a l l y o n o u r u s e o f l a n g u a g e s t h a t

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

p r e se n t e d h e r e w a s m o t i v a t e d b y t h r e e b a si c

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

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

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

d i s t i n c t i o n b e t w e e n s y n t a x a n d s e m a n t i c s .

T h e se c r i t e r ia a r e v a g u e a n d i n c o m p l e t e .

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

s tr in g m a n i p u l a t i o n , s t r u c t u r e s , a n d i n p u t /

o u t p u t d e s e r v e s p e ci a l s t u d y . I t is h o p e d

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

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

l an g u ag e d e s ig n .

T h e m i n i - l a n g u a g e s i l l u m i n a t e s e v e r a l

d if fi cu lt ie s o f l a n g u a g e i m p l e m e n t a t i o n . F o r

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

t r an s f e r o f co n t r o l p o s e s t h e d i f f i cu l t i s s u e o f

l i n k i n g i d e n t i f i e r s w i t h p r o p e r v a l u e s ; t h e

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

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

l ea d t o p r o g r a m e r r o r s ; a n d t h e m i n i - la n -

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

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

r e c o g n it i o n o f s t r in g s d e f i n e d b y a g e n e r a t i v e

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

th e b a s i s f o r a co u r s e o n co m p i l e r d e s ig n o r o n

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

tion.

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

i s s u es i n t h e a r ea o f f o r m a l d e f in i t i o n o f

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

f o rm a l d e f i n i t i o n h a v e b e e n a p p l i e d t o g e n -

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

t i o n , a n d i n p u t / o u t p u t . B y l i m i t i n g t h e

v a r i e ty o f s y n t a c t i c a n d s e m a n t i c c o n t e n t

in e a c h m i n i - l a n g u a g e , i t is h o p e d t h a t w e

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

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

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

a re m a n y c r i t e ri a f o r j u d g i n g t h e a c c e p t a -

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

C e r t a i n l y a m o n g t h e s e a r e c o n c i s e n e s s o f

d e f in i t i o n , p e r s p i cu i ty o f d e f in i t i o n , t h e

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

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

t he d e f i n i n g m e c h a n i s m t o a d a p t i t s e lf t o

T e n M i n i - L a n g u a g e s * 1 17

d i f f e r e n t li n g u i s ti c c o n s t r u c t s . I t s e e m s l i k e l y

t h a t a n a t t e m p t t o d ef in e a l l t e n m i n i -

l a n g u a g e s b y a n y o n e o f t h e e x i s ti n g m e t h -

o d s o f f o r m a l d e f i n i t i o n w i l l f a i l t o s a t i s f y

t h e a b o v e c r i t e r i a , a n d t h a t p e r h a p s a w e d -

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

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

t o r y m e t h o d o f f o r m a l d e f in i ti o n .

I n a n e f f o r t t o k e e p t h e d e s c r ip t i o n s o f t h e

m i n i - l a n g u a g e s m i n i m a l , t h e f o l l o w i n g d e f in i -

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

f o r a l l m i n i - l a n g n a g e s :

Pr imi t ive ob jec t s

a r e t h e n a t u r a l n u m b e r s ,

r e p r e s e n t e d b y t h e n u m e r a l s 0 1 2 . . - .

Iden t i f i e r s

c o m p r i s e t h e s y m b o l s A B . • •Z .

L a b e l s c o m p r i s e t h e s y m b o l s L 1 L 2 . . . .

L i s t s

c o n s i s t o f a n i t e m o r a s e r i e s o f t w o o r

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

F o r ex a m p le , 1 , 6 , 5 1 , 9 i s a l is t o f n u m er -

als .

Sequences a r e d e f i n e d a s a n i t e m o r a s e r i e s

o f t w o o r m o r e i t e m s e a c h o n a n e w l i n e .

F o r e x a m p l e ,

A : = I

B : = 4

C : = B

i s a s e q u e n c e o f a s s i g n m e n t c o m m a n d s .

I n a d d i t i o n , a c l e a r d i s t i n c t i o n i s m a d e b e -

t w e e n p r o g r a m s t h a t a r e e r ro n e o u s b e c a u se

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

a r e e r r o n e o u s b e c a u s e t h e y l e a d t o a n u n d e -

f in e d c o n d i t i o n d u r i n g e x e c u t i o n . I n p a r t i c u -

l a r, c e r t a i n c o n t e x t - s e n s it i v e r e q u i r e m e n t s

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

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

s y n t a x o f s e v e r a l o f t h e m i n i - l a n g u a g e s . A

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

i t s s y n t a x i s s a i d t o b e " s y n t a c t i c a l l y i ll e g a l "

a n d m u s t

n o t

b e e x e c u t e d . F u r t h e r m o r e , t h e

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

t o a c o n d i t i o n t h a t is u n d e f i n e d a c c o r d i n g

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

" i n v i o l a t i o n , " a n d e x e c u t i o n m u s t t e r m i -

n a t e a t t h e p o i n t o f v i o l a ti o n . T h e f o ll o w i n g

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

A n y

at tem pt to eva lua te an iden t if i e r or na m ed ex -

press ion whose va lue i s unde f ined i s in v io la -

t ion .

C o m p u t i n g S u r v e y s , V o l . 3 , N o . 3 , S e p t e m b e r 1 9 7 1


4/32

118 •

Henry F. Ledgard

A s i d e f r o m t h e a b o v e c o n v e n t i o n s , t h e d e -

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

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

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

m a y d o s o .

MINI -LANGUAGE 1 :

S IMPLE ASSIGNM ENT, TRANSFER OF

C O N T R O L , A N D BL O C K S TR UC TURE

P r o b a b l y , t h e m o s t c o m m o n f e a t u r e o f e x is t-

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

t i o n o f c o m p u t a t i o n s b y a s e q u e n c e o f c o m -

m a n d s . E a c h c o m m a n d i n d i c a te s s o m e s e ri es

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

t h a t is t o b e p e r f o r m e d o n d a t a . T h e d a t a

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

t h e n a t u r a l n u m b e r s a r e r e p r e s e n t e d h e r e b y

t h e n u m e r a l s 0 , 1, 2 , e t c . ) , o r t h e r e m a y a p -

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

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

j e ct s. T h e c o m m a n d s a r e g e n e r a l l y e x e e u t e d

s e q u e n t i a l l y , u n l e s s a c o m m a n d i s e x e c u t e d

t h a t e x p l i c i tl y a l t e r s t h e s e q u e n t i a l o r d e r .

H o w e v e r , t w o q u e s t i o n s a r i s e . F i r s t , w h a t

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

u s e d in t h e e x e e u t i o n o f a c o m m a n d ? S e e -

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

w h i c h c o m m a n d i s t o b e e x e c u t e d n e x t ?

S u c h q u e s t i o n s a r e e s p e c i a l l y b a s l e t o la n -

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

c o n t r o l , a n d b lo c k s t r u c t u r e . M i n i - l a n g u a g e

1 t r e a t s t h e s e i s s u e s i n s o m e w h a t s i m p l i f ie d

f o r m . M i n i -l a n g u a g e s 2 a n d 3 t r e a t m o r e

g e n e r a l i z e d n o t i o n s of a s s i g n m e n t a n d t r a n s -

f e r o f c o n t r o l .

Description of Lan gua ge Elements

Commands

i n m i n i - l a n g u a g e 1 a r e o f t h r e e

v a r i e t i e s :

1) a s im p l e a s s i g n m e n t c o m m a n d , w h i c h

c o n s i s t s o f a s t r i n g o f t h e f o r m i : = e ,

w h e r e i i s a n i d e n t i f i e r , a n d e is a n u m e r a l

o r a n i d e n t i f i e r ;

2 ) a h o p t o c o m m a n d , w h i c h c o n s i st s o f a

s t r i n g o f t h e f o r m h o p t o l, w h e r e 1 i s a

l a b e l ; a n d

3 ) a b l o c k ( d e f i n e d b e l o w ) .

A c o m m a n d m a y b e p re f ix e d b y a la b el .

A declaration

c o n s i s t s o f a s t r i n g o f t h e

f o r m i d e n l, w h e r e l is a l is t o f i d e n t i f ie r s ,

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

A block

co n s i s t s o f a s t r i n g o f t h e f o r m

beg i n

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

c i s a s e q u e n c e o f c o m m a n d s s u c h t h a t : 1 )

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

d i f f e r e n t ; a n d 2 ) e a c h l a b e l i n a h o p t o c o m -

m a n d i n c m u s t b e i d e n t i c a l t o s o m e l a b e l

p r e f i x in g a c o m m a n d i n c.

A n i d e n t i f i e r g i v e n i n a d e c l a r a t i o n i s sa id

t o h a v e a " s c o p e , " w h i c h c o n s i s t s o f t he

b l o c k i n w h i c h t h e i d e n t i f i e r i s d e c l a r e d a n d

a l l e n c o m p a s s e d b l o c k s t h a t d o n o t c o n t a i n

a n o t h e r d e c l a r a t i o n o f t h e s a m e i d e nt if ie r.

( N o t e t h a t a l a b e l p r e f ix i n g a c o m m a n d i n c

m a y b e s a id t o h a v e a " s c o p e " s im i l a r to th a t

o f a n i d e n t i f ie r . B e c a u s e o f r e q u i r e m e n t (2)

a b o v e , i t i s n o t n e c e s s a r y i n m i n i - l a n g u a g e 1

t o e x t e n d t h e c o n c e p t o f t h e s c o p e o f a l a b e l

t o m o r e d e e p l y n e s t e d b l o c k s . T h i s i s s u e w i l l

b e t r e a t e d i n m i n i - l a n g n a g e 3 . )

Program Execution

E a c h i d e n t i f i e r d e c l a r e d i n a p r o g r a m re -

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

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

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

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

1 ) E a c h c o m m a n d i n a b l o c k i s s e q u e n t ia l l y

e x e c u t e d u n t i l a h o p t o c o m m a n d (d e-

f i n e d b e l o w ) i s e n c o u n t e r e d .

2 ) T h e e x e c u t i o n o f a s i m p l e a s s ig n m e n t

c o m m a n d o f t h e f o r m i : = e p r o c e e d s as

f o l l o w s :

a ) t h e e x p r e s s i o n e i s e v a l u a t e d , a n d i f e

i s an i d en t i f i e r , i t s v a lu e i s t h e o b j ec t

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

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

t h e a s s i g n m e n t c o m m a n d ;

b ) t h e o b j ec t o b t a in ed i n ( a ) i s a s s ig n ed

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

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

m a n d .

3 ) T h e e x e c u t i o n o f a h o p t o c o m m a n d of

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

c o n t i n u e ( s e q u e n t i a l ly ) f r o m t h e c om -

m a n d t h a t i s p r e f i x e d b y t h e l a b el and

t h a t o c c u r s i n t h e s a m e b l o c k ( no t a

m o r e d e e p l y n e s t e d b l o c k ) a s th e

h o p t o c o m m a n d .

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

w h e n t h e e x e c u t i o n o f t h e o u t e r m o s t b lock

i s t e r m i n a t e d ( u n l e s s , a s m e n t i o n e d e a r l i e r ,

a v i o l a t i o n c o n d i t i o n a r i s es f r o m a n a t te m p t



5/32

to e v a l u a t e a n i d e n t i f i e r w h o s e v a l u e i s u n -

specified).

EXAMPLE i :

b e g i n i d e n A , B

A : = i

B : = 2

b e g i n i d e n B , C

B : = A

C : = 7

e n d

A : = 3

B : = 3

e n d

EXAMPLE 2:


A : = i

B : = 2


B : = A

C : = 7

e n d

h o p t o

L 1

A : = 3

L 1 B : = 3

e n d

EXAMPLE 3

( S y n t a c t i c a l l y i ll e g a l) :


A : = I

B : = 2


B : = A

D : = 7

e n d

L 1 A : = 3

L 1 C : = 3

e n d

I n E x a m p l e s 1 a n d 2 , t h e r e a r e f o u r d e -

c la re d i d e n t i f i e r s : A a n d B i n t h e o u t e r

b l o c k , a n d B a n d C i n t h e i n n e r b l o c k . L e t

us r e f e r to the se fou r ide n t i f ie r s as A~, B~,

B2, a n d C 2. I n E x a m p l e 1 t h e f i n a l v a l u e s o f

A1, Bj , B2, a n d C2 wil l be 3 , 3 , 1 , an d 7;

w h e r e a s i n E x a m p l e 2 t h e f i n a l v a l u e s w i l l

b e 1, 3 , 1, a n d 7 . T h e p r o g r a m o f E x a m p l e 3

is s y n t a c t i c a l l y q u i t e i ll e ga l , n o t a b l y i n t h a t

i d en ti f ie r D i s n o t d e c l a r e d , t h e l ab e l L1 p r e -

fix es t w o c o m m a n d s i n a s i n g l e b lo c k , a n d

Ten Mini-La,~guages • 119

i d e n t i f i e r C i n t h e l a s t c o m m a n d i s n o t d e -

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

Discussion

T h e n o t i o n s o f a s s i g n m e n t a n d t r a n s f e r o f

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

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

g r a m m i n g l a n g u a g e s m a k e a s e r i o u s d e -

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

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

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

a b l y a r o s e f r o m t h e f a c t t h a t , i n t h e i r s i m -

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

t r o l a r e v e r y c l os e t o e q u i v a l e n t c o m m a n d s i n

m a c h i n e l a n g u a g e s . F r o m t h e u s e r ' s p o i n t o f

v i e w , m a c h i n e - o r i e n t e d n o t i o n s l ik e s e q u e n c -

i ng , c r e a t i o n o f d u m m y v a r i a b l e s , a n d s t o r -

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

o b j e c t i v e .

I t i s p r o b a b l y f a i r t o s a y t h a t f u t u r e d e -

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

l e s s a l g o r i t h m i c ( i . e . , l e s s m a c h i n e - o r i e n t e d )

a n d m o r e s y n t h e t i c ( i. e. , m o r e g o a l - o r i e n t e d ) .

F o r e x a m p l e , t h e n o t i o n o f b l o c k s t r u c t u r e

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

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

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

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

t i o n s , t h e p a t t e r n m a tch in g r u l e o f S ~ - O B O L,

a n d r e e u r s i v e d e f i n it i o n s ) a l so f r e e t h e u s e r

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

s p e c i f i c a t i o n o f t y p e s , s t r i n g s c a n n i n g a l g o -

r i t h m s , a n d p u s h d o w n s t a c k s .

M I N I - L A N G U A G E 2 :

G E N E R A L I Z E D A S S I G N M E N T A N D

T HE N O T I O N O F L O C A T I O N S

C o n s i d e r t h e f o ll o w i n g t w o a s s i g n m e n t c o m -

m a n d s :

1) ( i f A > B t h e n X e l s e Y ) : = I

2 ) I : = ( i f A > B

t h e n

X e l s e Y )

T h e t w o e x pr e ss io n s " ( i f A > B

t h e n

X e l s e

Y ) " a n d " I " a r e e v a l u a t e d q u i t e d i f f e r e n t l y

i n c o m m a n d s ( 1) a n d ( 2) . L o o s e l y sp e a k i n g ,

i n c o m m a n d (2 ) w e w a n t t o k n o w w h a t

ob-

ject is c u r r e n t l y a s s o c i a te d w i t h o n e o f t h e

i d e n t i fi e r s , X o r Y ; w h e r e a s i n c o m m a n d ( 1)

w e w a n t t o k n o w w h a t location i s c u r r e n t l y

r e s e r v e d f o r o n e o f t h e i d e n t i f i e r s , X o r Y ,

s o t h a t w e c a n u p d a t e i t s c o n t e n t s . T h e d i f -

Computing Surveys, Vol. 3, No. 3. Seotember 1971


6/32


7/32

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

p a r a m e t e r B o f p r o c e d u r e P i s l i n k e d w i t h

the

locat ion

o f A . H e n c e t h e o c c u r r e n c e o f B

on t h e l e f t s id e of " B = B + 2 ; " r e f e r s t o t h e

l o ca ti o n o f A ( w h i c h i s t o b e u p d a t e d ) ,

w h er ea s t h e o c c u r r e n c e o f B o n t h e r i g h t

side d e n o t e s t h e v a l u e s t o r e d i n t h e l o c a t i o n

of A.

M o s t p r o g r a m m i n g l a n g u a g e s p l a c e r e -

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

can d e n o t e l o c a t i o n s . F o r e x a m p l e , t h e c o m -

mands

( i f A > B t h e n X e l s e Y ) : = e

and

F ( X ) : = e

w h e r e e i s an ex p r e s s io n , a r e u s u a l ly p r o -

h ib it ed , a l t h o u g h t h e r e i s n o c o n c e p t u a l d i fS -

c u lt y i n v i e w i n g B o o l e a n e x p r e s s i o n s o r f u n c -

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

" l e f t - m o d e , " i. e ., y i e l d i n g a l o c a t i o n .

F u r t h e r m o r e , o n e g e n e r a l l y t h i n k s o f a r -

ra ys a s f u n c t i o n s f r o m a n n - t u p l e o f i n te g e r s

in to v a l u e s . B u t c o n s i d e r t h e a s s i g n m e n t

c o m m a n d A [ I] : = A [ I ] + I , w h e r e A is a n

a r r a y a n d I i s a n i n t e g e r . H e r e A [ I ] o n t h e

le ft s id e o f t h e c o m m a n d r e a l l y d e n o t e s a

location,

a n d t h e o c c u r r e n c e o f A l l ] o n t h e

r ig h t s i d e d e n o t e s t h e v a l u e s t o r e d i n t h i s

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

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

in teger s in to

l oca t ions .

W i t h t h i s v i e w , t h e

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

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

l e ft -m o d e a n d i t s r i g h t s i d e i n r i g h t - m o d e .

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

face t h e n o t i o n o f lo c a t i o n s i n p r o g r a m m i n g

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

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

s to r ed t h e r e in .

M I N I -L A N G U A G E 3 :

GENERALIZED TRANSFER OF CON TROL

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

t ha t o f t e n s t im u l a t e s c o n t r o v e r s y , e s p e c i a ll y

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

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

g ua ge 3 t r e a t s t h i s is s u e b y a u g m e n t i n g m i n i -

l a ng u a ge 1 t o i n c l u d e t h e n o t i o n s o f a l a b e l

T e n M i n i - L a n g u a g e s •

121

v a r i a b l e a n d o f t r a n s f e r o f co n t r o l b e t w e e n

b lo ck s .

Descript ion of Language Elements

C o m m a n d s i n

m i n i - l a n g u a g e 3 a r e o f t h r e e

v a r i e t i e s :

1 ) a n a s s i g n m e n t c o m m a n d , w h i c h c o n s i s t s

o f a s t ri n g o f e i t h e r t h e f o r m i : = e o r

i := l , w h e r e i i s an i d en t i f i e r , e i s a

n u m e r a l o r a n i d e n t i f i e r , a n d l i s a l a b e l ;

2 ) a g o t o c o m m a n d , w h i c h c o n s i s ts o f a

s t r in g o f t h e f o r m g o t o x , w h e r e x is

a n

i d e n t i f i e r o r a l a b e l ; a n d

3 ) a b l o c k ( d e f i n e d b e l o w ) .

A c o m m a n d m a y b e p r e f ix e d b y a l a b e l.

A dec lara t ion

c o n s i s t s o f a s t r i n g o f t h e

f o r m

i d e n

i l , i 2 , " - , i , , w h e r e i i , i 2 ,

• . - i ~ a r e i d e n t i f i e r s , e a c h o f w h i c h m u s t b e

d i f f e r e n t .

A b lock

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

(1)

b e g i n

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

a n d c is a s e q u e n c e o f c o m m a n d s s u c h t h a t

e a c h l a b e l p re f i x in g a c o m m a n d i n c m u s t b e

d i f f e r e n t . A n i d e n t i f i e r g i v e n a d e c l a r a t i o n o r

a l a b e l p r e fi x i n g a c o m m a n d is sa i d t o b e

" d e c l a r e d , " a n d t o h a v e a " s c o p e " c o n s is ti n g

o f b l o c k ( 1 ) a b o v e , i n w h i c h t h e i d e n t i f i e r o r

l a b e l i s d e c l a r e d , a n d a l l e n c o m p a s s e d b l o c k s

i n c t h a t d o n o t c o n t a i n a n o t h e r d e c l a r a t i o n

o f t h e s a m e i d e n t i f i e r o r l a b e l .

A p r o g r a m

c o n s i s t s o f a b l o c k s u c h t h a t

e a c h o c c u r r e n c e o f a n i d e n t i f i e r o r l a b e l i n a

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

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

The value of a label l

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

g o t o c o m m a n d c o n s is t s o f t w o p a r t s : 1 ) t h e

( u n i q u e ) c o m m a n d c t h a t i s p r e f i x e d b y 1

a n d l ie s w i t h i n t h e s c o p e o f l ; a n d 2 ) t h e

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

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

l oca t ions

i n w h i c h t h e i r c u r r e n t v a l u e s a r e

s t o r e d .

Program Execution

E a c h i d e n t if i e r i n a p r o g r a m r e f e r s t o a n

( i n i t i a l l y u n s p e c i f i e d ) o b j e c t . T h e e x e c u t i o n

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

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

t o t h e f o l l o w i n g ru l e s :

1) T h e e x e c u t i o n o f a b l o c k r e s u l ts i n t h e

a l l o c a t i o n o f

n e w

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

t h a t a r e l o c a l t o t h e b l oc k , fo l l o w e d b y

Co m p u t in g S u rv ey s , Vo l . 3 , No . 3 , S ep t em b er 1 9 71


8/32


9/32

2) T h e n u m e r a l 5 is p r in t e d . T h e c o m -

m a n d " g o t o X " w o u l d b e h a v e a s i f i t

w e re w r i t t e n a s " g o t o L 2 "

3) T h e n u m e r a l 3 is p r i n t e d . A f t e r e x i t f r o m

t h e i n n e r b l o c k t h e f i r s t t i m e , t h e e n -

v i r o n m e n t f o r L 2 w o u l d l i n k B w i t h t i l e

v a lu e 2 . T h e c o m m a n d " g o t o X " w o u ld

r e s e t t h e e n v i r o n m e n t f o r th e i n n e r b l o c k

s o t h a t t h e v a l u e o f B w o u l d b e 2 , a n d

t r a n s f e r o f c o n t r o l w o u l d s u b s e q u e n t l y

u p d a t e t h e v a l u e o f B t o 3 , w h i c h w o u l d

b e p r i n t e d .

A s m e n t i o n e d e a r li e r, t h e n o t i o n o f t r a n s -

f er o f c o n t r o l i s a d i ff i c u lt a n d d e b a t a b l e

i s s u e . R e a d e r s w i s h i n g a m o r e d e t a i l e d d i s -

c u ss io n o f t h i s s u b j e c t a r e r e f e r r e d t o t h e

i ll u m i n a ti n g p a p e r b y J o h n s t o n [6 ].

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

a re a ls o w o r t h p o i n t i n g o u t . D o e s t r a n s f e r o f

c on tro l m a k e p r o g r a m d e b u g g i n g m u c h m o r e

d i f f i c u l t ? S h o u l d t r a n s f e r o f c o n t r o l o u t o f

p r o c e d u r e s o r i n t o m o r e d e e p l y n e s t e d b l o c k s

be a l lo w e d ? W h a t d o e s " g o t o x , " w h e r e x

is a l a b el v a r i a b l e , m e a n w h e n t h e c o m m a n d

is e x e c u t e d d u r i n g r e c u r s i v e p r o c e d u r e c a ll s?

H o w d i ff ic u l t is i t to i m p l e m e n t o r f o r m a l l y

d efin e a g e n e r a l i z e d t r a n s f e r o f c o n t r o l

f a ci li ty ? I n s h o r t , i s t h e n o t i o n o f t r a n s f e r

of c o n t ro l a u s e fu l f e a t u r e o f p r o g r a m m i n g

l a n g u a g e s ?

O n e of t h e c a u s e s f o r t h e d e b a t e o v e r

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

e r ty o f r e f e r e n t i a l t r a n s p a r e n c y [1 3]; t h a t

is, w e c a n n o l o n g e r i s o la t e a n a r b i t r a r y

p o rt io n o f a p r o g r a m a n d c o n c e p t u a l l y r e -

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

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

d e c l a r e

p r o c e d u r e P ( X , Y ) ;

0 ; c 2 ; ' ' '

; C n ;

e n d

w h e r e C l, c 2 , • • • , cn a r e c o m m a n d s w h o s e

e ff ec ts a r e l o c al to t h e p r o c e d u r e b o d y . T h e

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

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

m o n d i v i s o r o f X a n d Y o r a s to r e t o s t o r e

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

a n d Y . I f w e a d d t h e p o s s i b i l it y t h a t o n e o f

th e c o m m a n d s c l , c 2 , • • • , c ~ m a y r e s u l t i n

a t r a n s f e r o f c o n t r o l o u t o f th e p r o c e d u r e

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

T e n M i n i - L a n g u a g e s • 123

" v a l u e , " a s w e m a y l e a v e th e p r o c e d u r e a n d

i ts s u p p o s e d p o i n t o f r e t u r n .

M o r e g e n e r a l l y , t h e u s e o f t r a n s f e r o f c o n -

t ro l m a k e s t h e a r t o f p r o g r a m m i n g m o r e

s e q u e n c e - o r i e n t e d a n d l e s s g o a l - o r i e n t e d .

F u r t h e r m o r e , a s i n m i n i - l a n g u a g e 3 , t h e r e

a r e o t h e r s i d e - e f fe c t s a s s o c i a t e d w i t h a g e n -

e r a l i z e d t r a n s f e r o f c o n t r o l f a c i l i t y : n a m e l y ,

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

a n d t o w h i c h v a l u e s t h e y a r e t o b e l in k e d .

T h e s e p r o b l e m s i n d i c at e t h a t t h e n o t i o n o f

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

c a r e fu l e y e t o w a r d m a k i n g i t s d e f in i ti o n a n d

i m p l e m e n t a t i o n s im p l e r.

M I N I - L A N G U A G E 4 : F U N C T I O N S

O n e o f th e m o s t u s e f u l f e a t u r e s o f m a n y

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

d e fi n e a n d a p p l y f u n ct io n s . M i n i - la n g u a g e 4

i s d e v o t e d e x c l u s i v e l y t o t h e n o t i o n o f a

f u n c t i o n a n d is d e v o i d o f c o m m a n d f e a t u r e s .

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

c l a s s m a t h e m a t i c a l o b j e c t s ; i n p a r t i c u l a r ,

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

b e a f u n c t i o n , a n d f u n c t i o n s w h o s e d o m a i n

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

b e d e f i n e d . M i n i - l a n g u a g e 4 i s b a s e d o n t i l e

a p p l i c a t i v e e x p r e s s i o n s o f P e t e r L a n d i n [ 8] ,

w h i c h , i n t u r n , a r e b a s e d o n t h e X - c a l c u l u s

o f C h u r c h [1] a n d C u r r y [ 3 ] , a n d o n t h e

m a t h e m a t i c a l n o t i o n o f a f u n c t io n .

Descr ipt ion of Language E lements

Pr imi t ive ob jec t s

i n m i n i - l a n g u a g e 4 i n -

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

1 ) a b i n a r y f u n c t i o n , w h i c h m a p s t w o n a t -

u r a l n u m b e r s i n t o t h e n a t u r a l n u m b e r t h a t

is t h e i r n u m e r i c a l s u m ; a n d 2 ) a q u a r t e r n a r y

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

t h e f ir st t w o a r e n a t u r a l n u m b e r s a n d t h e

s e c o n d t w o a r e a r b i t r a r y o b j e c t s - - i n t o t h e

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

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

o b j e c t i f t h e o p p o s i t e is t ru e . T h e s e b i n a r y

a n d q u a r t e r n a r y f u n c ti o n s a r e r e p r e s e n te d

b y t h e s y m b o l s

+

a n d " s e l e c t , " r e -

s p e c t i v e l y .

A s i mp l e e x p r e s s i o n

i s e i t h e r o n e o f t h e

p r i m i t iv e s y m b o l s { +

s e l e c t

0 1 2 . . . } o r

a n i d e n t if ie r . T h e v a l u e o f a p r i m i t i v e s y m b o l

Computing Surveys, Vol. 3, No. 3, September I971


10/32

124 •

H e n r y F . L e d g a r d

i s t h e p r i m i t i v e o b j e c t r e p r e s e n t e d b y t h e

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

o b j e c t c u r r e n t l y l i n k e d w i t h t h e i d e n t i f i e r

( f o r l i n k i n g o f i d e n t i f i e r s t o o b j e c t s , s e e

d e f i n i t i o n a n d e v a l u a t i o n o f " l e t " e x p r e s -

s io n s , b e lo w ) .

A l i s t expre ss ion

i s a l i s t o f ex p r e s s io n s .

T h e v a l u e o f a li s t e x p r e s s i o n e j ,

e 2 , ,

e . , w h e r e t h e e l ( 1 ~ i ~ n ) a r e e x p r e s s i o n s ,

i s t h e l i s t o f o b j e c t s a t , a 2 , • • - , a ~ o b t a i n e d

b y e v a l u a t i n g ( in a n y o r d e r ) e a c h o f t h e

c o m p o n e n t e x p re s si o ns e j ,

e 2 , • • • , en •

A c o m b i n a t i o n

i s a s t r i n g o f t h e f o r m

r ( l ) ,

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

s y m b o l s d - o r

s e l e c t ,

an d l i s a l i s t ex p r e s -

s io n . T h e v a l u e o f a c o m b i n a t i o n i s o b t a i n e d

b y e v a l u a t i n g r a n d l a n d t h e n a p p l y i n g

t h e v a l u e o f r t o t h e v a l u e o f l. T h i s e v a l u a -

t i o n i s w e l l - d e f i n e d ( i . e . , n o t i n v i o l a t i o n )

o n l y i f : t h e v a l u e o f r i s a f u n c t i o n ; t h e

n u m b e r o f t h e c o m p o n e n t s o f l is t e x p re s s i o n

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

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

o r s e l e c t , t h e v a l u e s o f t h e f i r s t t w o a r g u -

m e n t s o f l m u s t b e n a t u r a l n u m b e r s .

T h e f o l lo w i n g a l t e r n a t e n o t a t i o n s m a y b e

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

i n p l a c e o f

(el + e2)

-4- (e l, e2)

a n d

( i f e l > e2 t h e n ea e l s e e4)

i n p l a c e o f

s e l e c t

(el , e2, ca, e4).

A l et expre ss ion i s a s t r i n g o f e i t h e r t h e

f o r m

1)

l e t

i = el

i n

e2

o r

2) l e t f ( x t , . . . , x ~ ) = e l i n e2

w h e r e i i s a n i d e n t i f i e r, f , X l , " . . , x ~ a r e

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

a n d e l a n d e2 a r e e x p r e s s i o n s . T h e v a l u e o f a

l e t e x p r e s s i o n o f f o r m (1 ) is c o m p u t e d b y

e v a l u a t i n g e l , l i n k i n g i w i t h t h e v a l u e o f e ~,

a n d t h e n e v a l u a t i n g e 2 . T h e v a l u e o f a le t

e x p r e s s i o n o f f o r m ( 2) i s c o m p u t e d b y f o r m -

i n g t h e f u n c t i o n m a p p i n g n a r g u m e n t s i n to

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

l i n k i n g t h e i d e n t i f i e rs x l , • • . , x ~ w i t h t h e i r

r e s p e c t i v e a r g u m e n t s , a n d l i n k in g i d en t if ie r s

o t h e r t h a n X l , . . . , xn w i t h t h e i r c u rr e n t

v a l u e s ; l i n k i n g f w i t h t h e f u n c t i o n t h u s

f o r m e d ; a n d t h e n e v a l u a t i n g e 2 .

I f t h e i d e n t i f i e r f i t s e l f a p p e a r s i n el a n d

is n o t l i n k e d b y a n o t h e r l e t e x p r e s s io n t h a t is

a s u b c x p r e s s i o n o f e l , t h e n t h e o c c u r r e n c e of

f i s t a k e n a s p a r t o f a r e c u r s i v e d e f i n i t i o n o f

f . F o r e x a m p l e , t h e l e t e x p r e s s i o n s :

1 ) l e t Y = 3

2)

l e t

F ( X ) = ( X + X )

3 ) l e t G ( N ) = ( i f N > 3 t h e n 1 e l s e G ( 5) )

l i n k Y , F , a n d G w i t h o b j e c t s w h o s e e f f e c -

t i v e v a l u e s a r e , r e s p e c t i v e l y : 1 ) t h e n a t u r a l

n u m b e r t h r e e ; 2 ) th e f u n c t i o n m a p p i n g a

n a t u r a l n u m b e r i n t o i t s d o u b l e ; a n d 3 ) t h e

f u n c t i o n m a p p i n g a n a t u r a l n u m b e r i nt o

t h e c o n s t a n t v a l u e o n e .

A n

e x p r e s s i o n

i s e i t h e r a s im p le ex p r e s s io n ,

a l i s t ex p r e s s io n , a l e t ex p r e s s io n , o r a co m -

b i n a t i o n .

Program Execution

A p r o g r a m i s

a n e x p r e ss i o n . T h e v a l u e o f a

p r o g r a m i s t h e v a l u e o f t h e e x p r e s s i o n . N o t e

t h a t ( a s m e n t i o n e d i n t h e f i r s t s e c t i o n ,

" G e n e r a l C o n s i d e r a t i o n s " ) i f t h e e v a l u a t i o n

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

i d e n t i f i e r t h a t i s n o t l i n k e d t o a v a l u e r - t h e

p r o g r a m is in v i o l a t i o n .

E X A M P L E 1 :

l e t F ( Y ) = ( Y + 3 )

i n

( F ( 1 ) d - F ( 2 ) )

E X A M P L E 2 :

l e t F ( X ) = ( X + X )

i n l e t G ( P , X ) = ( P ( X ) + P ( 1 ) )

i n G ( F , 2 )

E X A M P L E 3 :

l e t

F ( X ) = ( if X > 3

t h e n

X

e l s e ( X + F ( X + l ) )

i n

F ( 2 )

E X A M P L E 4 :

l e t Y = 2

i n l e t F ( X ) = ( X + Y )

i n F

Computing Surveys , Vol. 3 , No. 3 , September 1971


11/32

E X A M P L E 5

le t P ( F , G , N ) = ( i f N > 1 0 0 t h e n 1 e l s e

G(F, G(N+I)))

i n l e t

Q(F, G, N) = (if N > 100 t hen i

e l s e

F(F, G(N+I)))

i n P ( P , Q , 86 )

The values of the example programs above

are: Example 1, the natu ral number 9;

Example 2, the natural number 6; Example

3, the natural number 9; Example 4, the

function mapping some object X (presum-

ably a natural number) into a summation of

X and the natural number 2; and Example

5, the natural number 1.

i s c u s s i o n

The notion of functions is a well-accepted

and useful feature of conventional mathe-

matics, and notions akin to functions (e.g.,

procedures and subroutines) have been duly

incorporated into most programming lan-

guages. However, one must be quite careful

when one tries to model such programming

language constructs like procedures and sub-

routines with functions. Consider the simple

A L G O L 6 0

procedure declaration:

i n t e g e r p r o c e d u r e P ( X ) ; i n t e g e r X ;

P := X := X+X

Mathematically, P is not a function of type

I---~I, from integers into integers. The no-

tion of assignment in programming lan-

guages, in conjunction with function-like

procedures, forces us to consider

l oca t ions

of identifiers as part of our universe of dis-

course [16]. In particular, let I be the set of

integers, L be a set of locations for the

identifiers in a program, and S (the stores

of values for a program) be a set of functions

mapping the locations of identifiers into

integers. Mathemat ical ly, P is a function

L X S --~ I X S. That is, P takes the loca-

tion, 1EL, of its argument and a store,

slES, as inputs, and returns an integer,

iCI, and a new store, s~ES, as outputs. The

integer i depends on the integer value stored

in location l of sl, and the new store, s=ES,

reflects the side-effect. Hence when we in-

voke P in an expression such as

3 + P(Y) + Y

T e n M i n i - L a n g u a g e s • 125

two events occur: the first member, i E I of

( 4

s) E I × S, is added to 3, and the second

member, s2E S (which reflects the changed

value in the location of Y in S), is used to

obtain subsequent values of Y.

Furthermore, consider the Lisp doublets:

DEFINE ((

(F (LAMBDA X) (LAMBDA (Y) X)))

))

F 1 )

Mathematically, the value of F(1) is the

cons tant function mapping any argument

into the integer one. However, the valua-

tion of F(1) in Lisp yields an error. There-

fore, in LisP, functions can only appear in

constructs where they will subsequently be

applied to arguments yielding

va lues

that

are n o t functions.

For many programming languages, there

are two other features of functions that do

not fit their mathematical counterparts.

First, many programming languages are

generous in the kinds of objects that can be

used as arguments to functions, but restrict

the kinds of objects that can be returned as

values. For example, ALGOL 60 procedures

can take numbers, Boolean values, labels,

switches, arrays, and procedures as argu-

ments, but can return only numbers and

Boolean values as results. Such restrictions

inhibit the usefulness of the language.

Secondly, the notion of recursive defini-

tion of functions is a feature of programming

languages that is avoided by many pro-

grammers and language implementers.

Mathematically, recursive definitions are

often a more succinct notation than inter-

active definitions. Consider the following

two definitions of a function TOTAL ,

which takes a one-dimensional array A and

integer N as arguments and returns the sum

of the N elements A[1] through A[N]:

T O T A L : P R O C ( A , N ) ; A R R A Y A , I N T E G E R I , N , Z ;

Z : = A [ N ]

I : =

N ;

L : I F I = 1 T H E N G O T O M ;

I : = - 1;

Z : = Z + A [ I ] ;

G O T O L ;

R E T U R N Z ;

:

E N D T O T A L ;



12/32

126 •

He nry F . L e dg ard

a n d

TOT AL (A, N) = IF N = 1 TH EN A[N]

ELSE A[N] + TOTAL (A, N -- 1);

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

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

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

o f s e q u e n c e o f o p e r a t i o n s .

I t is h o p e d t h a t f u t u r e d e v e l o p m e n t s in

p r o g r a m m i n g l a n g u a g e s w i l l f u l l y r e c o g n i z e

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

m a t h e m a t i c a l p r o p e rt ie s .


P A S S I N G O F P A R A M E T E R S

O n e o f t h e m o r e c o n f u s i n g f e a t u r e s o f p r o -

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

o f p a r a m e t e r s . T h e i n t e n t o f m i n i - l a n g u a g e 5

is t o d is t in g u i s h t h r e e c o m m o n l y u s e d m e t h -

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

" c a l l b y c o p y v a l u e , " " c a l l b y l o c a t i o n , "

a n d " c a l l b y e x p r e s s i o n . " T h e s e t h r e e m e t h -

o d s a re b a s e d o n t h e " c a l l b y v a l u e " o f

A LG O L 6 0 , " c a l l b y r e f e r e n c e " o f C P L , a n d

" c a l l b y n a m e " o f A LG O L 6 0.

Descr ipt ion of Language E lements

Array ide~ti f iers

a r e c o m p r i s e d o f t h e

s y m b o l s A I A 2 . . . . T h e v a l u e o f a n a r r a y

i d e n t i fi e r i s a o n e - d i m e n s i o n a l a r r a y w i t h t e n

( i n i t i a l l y u n s p e c i f i e d ) e l e m e n t s . Procedure

identifiers

c o m p r i s e t h e s y m b o l s P 1 P 2 . . . ,

a n d

parameter identifiers

c o m p r i s e t h e

s y m b o l s a b • • • z.

A parameter specification

i s a s t r i n g o f

o n e o f t h e f o l lo w i n g f o r m s : e x p l, c o p y v a l l,

o r l o c l, w h e r e 1 is a l is t o f p a r a m e t e r i d e n t i -

f iers . A

procedure declaration

i s a s t r i n g o f t h e

f o r m

p r o e p(l) s c

e n d , w h e r e p is a p r o c e -

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

t ie r s, s a s e q u e n c e o f p a r a m e t e r s p e c i f i c a -

t io n s , a n d c is a s e q u e n c e o f a s s i g n m e n t c o m -

m a n d s ( d e fi n ed b e lo w ) s u c h t h a t : e a c h

p a r a m e t e r i d e n t i f i e r i n l i s d i f f e r e n t ; e a c h

p a r a m e t e r i d e n t i f i e r i n s i s d i f f e r e n t a n d

o c c u r s a s o n e o f t h e p a r a m e t e r s in l ; a n d

e a c h i d e n t i f i e r i n c i s a p a r a m e t e r i d e n t i f i e r .

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

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

p r o c P l ( a , b , a )

e x p a , d

l o c

a

D : = I

e n d

A n expression is e i th e r a n u m e r a l o r a

n a m e d e x p r e s s io n . A n a m e d e x p r e s s i o n is

e i t h e r a n i d e n t i f i e r , a p a r a m e t e r i d e n t i f i e r ,

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

s i o n e n c l o s e d i l l s q u a r e b r a c k e t s .

A n

ass ignme~t command

i s a s t r i n g o f t h e

f o r m e l : = e 2 , w h e r e e l i s a n a m e d e x p re s -

s i o n , a n d e2 i s a n e x p r e s s i o n . T h e e x e c u t i o n o f

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

i n g t h e v a l u e o f e2 t o t h e l o c a t i o n d e n o t e d b y

e ~ . ( F o r a m o r e d e t a i l e d d i s c u s s i o n o f a s si g n-

m e n t , s e e m i n i - l a n g u a g e 2 . ) I n p a r t i c u l a r , i f

e~ o r e2 c o n t a i n s a n e x p r e s s i o n o f t h e f o r m

a[e],

w h e r e a i s a n a r r a y i d e n t i f ie r a n d e i s a n

e x p r e s s i o n w h o s e v a l u e is t h e n a t u r a l n u m b e r

n , t h e n

a[e]

d e n o t e s t h e ~ tth e l e m e n t o f t h e

a r r a y a . *

A procedure command

i s a s t r i n g o f th e f o r m

p(l) ,

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

1 i s a l is t o f e x p r e s s i o n s . T h e e v a l u a t i o n o f

t h e p r o c e d u r e c o m m a n d o f th i s f o r m is a s

f o l l o w s :

1) O b t a i n t h e s e q u e n c e o f a s s ig n m e n t c o m -

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

2 ) F o r e a c h p a r a m e t e r i c a l l e d b y " e x p r e s -

s i o n , " r e p l a c e e a c h o c c u r r e n c e o f i in c

b y t h e c o r r e s p o n d i n g e x p r e s s i o n e i n 1 .

3 ) F o r e a c h p a r a m e t e r i c a l l e d b y " c o p y

v a l u e , " p r e fi x t o c t h e c o m m a n d i : = e,

w h e r e e i s t h e c o r r e s p o n d i n g e x p r e s s i o n

for i in l .

4 ) F o r e a c h p a r a m e t e r i c a l l e d b y " l o c a -

t i o n , " o b t a i n t h e c o r r e s p o n d i n g e x p r e s -

s ion e fo r i : i f e i s an i den t i f i e r , l e t L~

d e n o t e i t s l o c a t i o n ; a n d i f e i s a n a r r a y "

e x p r e s s i o n A [ e '] , l e t L ~ d e n o t e t h e l o c a -

t i o n o f t h e r~ th e l e m e n t o f A , w h e r e n is

t h e v a l u e

e'.

T h e n r e p l a ce e a c h o c c u rr e nc e

o f i i n c b y L ~ .

5 ) E x e c u t e t h e s e q u e n c e o f a s s i g n m e n t c o m -

m a n d s c ' , f o r m e d b y t h e a b o v e r u l e s ,

w i t h th e f o l lo w i n g i n t e r p r e t a t i o n f or

* Fo r example, i f the value of the a rray nam e

A1 is the array w hos e ten elem ents are 4,A,A,5,5,

A,1,2,3,4 (where A denotes an unspecif ied ele-

ment) , then the value of Al[7] is 1, and the value

of Ai[AI[7]] is 4.

Computing Surveys, Vo/. 3, No. 3, September 1971


13/32

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

i s e v a l u a t e d o n t h e

r igh t -hand

s i d e o f a n

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

o b j e c t c u r r e n t l y s t o r e d i n t h e l o c a t i o n L ~ ;

a n d i f a n L ~ a p p e a r s o n t h e

le f t -hand

s ide

o f a n a s s i g n m e n t c o m m a n d , t h e a s s ig n -

m e n t o f t h e v a l u e o f t h e r i g h t - h a n d s id e

is m a d e t o t h e l o c a t i o n L ~ .

Program Execution

A pr og r am

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

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

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

m a n d s s u c h t h a t :

1) e a c h d e c l a r e d p r o c e d u r e i d e n t i f i e r in d

i s d i f f e r en t ;

2 ) e a c h p r o c e d u r e i d e n t i f i e r i n c i s d e c l a r e d

in d ;

3) t h e n u m b e r o f e x p r e s s i o n s in t h e e x p r e s -

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

i d e nt ic a l t o t h e n u m b e r o f p a r a m e t e r s i n

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

t i o n ;

4 ) n o p a r a m e t e r i d e n t i f i e r s o c c u r i n

c;

a n d

5 ) n o p r o c e d u r e d e c l a r a t i o n i n d c o n t a i n s a

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

a p p e a r s o n t h e l e f t si de o f a n a s s i g n m e n t

s t a t e m e n t , a n d c o r r e s p o n d s t o a n e xp r e s -

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

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

i s s y n t ac t i c a l l y i l l eg a l .

EXAMPLE

1 :

p r o c P 1

(a, b)

c o p y

v a l

a, b

X : = a

a : ~ - b

b :~ x

e n d

I : = 3

A [ I ] := 6

P1 ( I , A [ I ] )

E X A M P L E 2 :

p r o c P2(a , b )

l o c a t i o n a , b

x : = a

a : = b

b : = x

e n d

I : = 3

A[I ] := 6

P 2 ( I , A [ I ] )


127

E X A M P L E 3 :

p r o c P 3 ( a , b )

e x p a , b

x : = a

a : = b

b : = x

e n d

I : = 3

a [ I ] : = 6

P 3 ( I , h [ I ] )

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

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

m e n t s , a n d w e m u s t s e l e c t o n e of t h e t h r e e

o p t i o n s a b o v e . W h i c h o n e s h a l l w e c h o o s e ?

T h e t h r e e e x a m p l e s a r e i d e n t i c a l e x c e p t

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

d u r e s P 1 ,

1)2,

a n d P 3 a r e d i f f e r e n t .

I n E x a m p l e 1 : t h e v a l u e s o f I a n d A [3 J

w i l l b e s e t t o 3 a n d 6 , r e s p e c t i v e l y ; a n d

t h e p r o c e d u r e c a l l P I ( I , A [ I ] ) w i l l b e e x e -

c u t e d - i . e . , t h e v a l u e s o f a a n d b w i ll b e s e t

t o 3 a n d 6 , r e s p e c t i v e l y , a n d t h e v a l u e s o f

x , a , an d b w i l l b e s e t t o 3 , 6, an d 3 , re s p ec -

t i v e l y . T h e v a l u e s o f I a n d A [ 3] w i ll r e m a i n

u n c h a n g e d , s o w e s h a l l n o t e l e c t P 1 .

I n E x a m p l e 2 : t h e v a l u e s o f I a n d A [3 ]

w i l l b e s e t t o 3 a n d 6 , r e s p e c t i v e l y ; a n d

t h e p r o c e d u r e c a l l P 2 ( I , A [ I ] ) w i l l b e e x e -

c u t e d - i . e . , t h e l o c a t i o n s La o f I a n d L b o f

A [ I ] w i l l b e d e t e r m i n e d , a n d t h e c o m m a n d s

x : = L a , L a : = L b , a n d L b : = x w i ll b e

e x e c u t e d , w h i c h w i l l r e s u l t i n s e t t l i n g t h e

v a l u e s o f I a n d A [3 1 t o 6 a n d 3 , r e s p e c t i v e l y .

T h u s , p r o c e d u r e P 2 o f E x a m p l e 2 y i el d s t h e

d e s i r e d o u t c o m e .

I n E x a m p l e 3 w e h a v e a r a t h e r s u r p r i s i n g

r e s u l t :

1 ) t h e v a lu e s o f I a n d A [ 3] w i l l b e s e t t o 3 an d

6, r e s p e c t i v e l y ; a n d

2 ) t h e p r o c e d u r e c a l l P 3 ( I , A [ I ] ) w i l l b e

e x e c u t e d - - i . e . , t h e c o m m a n d s

X :~-- I

I : = A i I ]

A[I ] := x

w i l l b e e x e c u t e d , w h i c h w i l l r e s u l t i n 3

b e i n g a s s i g n e d t o x , 6 t o I , a n d t h e n 3

t o

A[6] .

T h e r e f o r e , w e e l e c t

o~ dy

p r o c e d u r e P 2 .

Co m p u t in g S u rv ey s , Vo l . 3 , No . 3 , S ep t em b er 1 9 71


14/32

128 •

H e n r y F . L e d g a r d

Discussion

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

i m p o r t a n t f e a t u r e o f p r o g r a m m i n g l a n -

g u a g e s : n a m e l y , t h e d i f fe r e n c e s i n t h r e e

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

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

e n c e , o n e m a y w e l l a s k : w h a t i s t h e " v a l u e "

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

S t r a c h e y [1 6], w e c o n s i d e r t h e n o t i o n o f a

" s t o r e . "

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

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

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

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

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

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

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

and a s tore

i n t o a n e w s t o r e . A p r o c e d u r e

c o m m a n d is v i e w e d a s a n a p p l i c a t io n o f t h e

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

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

f o r e t h e c o m m a n d i s e x e c u t e d • T h e n e w ,

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

o f t h e p r o g r a m i d e n t if i e rs .

I n p a r t i c u l a r , l e t

I D d e n o t e t h e s e t o f i d e n ti f ie r s f o r a p r o g r a m ;

L O C d e n o t e t h e s e t of l o c a t i o n s a s s o c i a t e d

w i t h t h e i d e n t i f i e r s i n I D ;

V A L d e n o t e a s e t o f v a l u e s ( f o r m i n i - l a n-

g u a g e 5 , V A L c o n s i s t s o n l y o f t h e n a t u r a l

n u m b e r s ) ;

E X P d e n o t e a s e t o f e x p re s s io n s ; a n d

S T R d e n o t e a s e t o f s t o r e s, i .e ., f u n c t i o n s

( L O C ~ V A L ) m a p p i n g l o ca t io n s i n to

v a l u e s .

I f s is a s t o r e , v a v a lu e , a n d l a l o ca t i o n , l e t

s[v/l]

d e n o t e t h e s t o r e

s'

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

s e x c e p t t h a t l i s m a p p e d i n t o v.

T h e v a l u e o f p r o c e d u r e P 1 , w h o s e a r g u -

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

t i o n :

X ( v l ~ V A L , v 2 C V A L , s E S T R ) . s

T h a t is, P 1 i s a f u n c t i o n t h a t , w h e n a p p l i e d

to tw o v a lu e s , Vl an d v 2 , an d a s t o r e , s ,

r e t u r n s t h e i d e n t i c a l s t o r e s . H e n c e P 1

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

w h i c h i t i s a p p l i e d .

T h e v a l u e o f t h e p r o c e d u r e P 2 , w h o s e

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

t i o n :

X (/1E L O C , 1 2 E L O C , s E S T R )

• s [ s ( l ~ ) / l l ] [ s ( l l ) / l ~ ]

T h a t is , P 2 is a f u n c t i o n t h a t , w h e n a p p l ie d

t o t w o l o c a t i o n s , l l a n d

12,

an d a s t o r e , s ,

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

e x c e p t t h a t ll - o v 2 , w h e r e

v2 = s(l~),

an d

12 - -~ Vl, w he re Vl = s(/1) , w hi ch is preci sely

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

d u r e c o m m a n d P 2 ( I , A [ I ] ) s w a p s t h e v a l u e s

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

N I L

T h e v a l u e o f t h e p r o c e d u r e P 3 , w h os e

a r g u m e n t s a r e c a l l e d b y e x p r e s s io n is t he

f u n c t i o n :

X (elE E X P , e~E E X P , s E S T R ) •

s[v2/ll][Vj/12']

w h er e v~ i s t h e v a lu e o f e~ u s in g t h e s to r e o f

v a l u e s i n s ; ll i s t h e l o c a t i o n d e n o t e d b y el

u s i n g t h e s t o r e o f v a l u e s i n s ; s' i s t h e s t o re

s[v2/ll]; Vl i s t h e v a l u e o f e l u s i n g t h e s t or e

o f v a l u e s i n s ; a n d 12' i s t h e l o c a t i o n d e n o t e d

b y e2 u s i n g t h e s t o r e o f v a l u e s i n s '. H e r e t h e

l o c a t i o n 12' i s d e p e n d e n t o n t h e s t o r e s ' =

s[v2/ll];

i . e . , an y u s e o f e l ( u s ed i n o b t a in in g

11) i n e2 ( u s ed i n o b t a in in g l ( ) m ay r e s u l t i n a

di f ferent

l o c a t i o n l~ ' f o r e2 f r o m t h a t ex i st -

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


S T A TIC T Y P E C H E C K I N G

T h e p a r t i t i o n i n g o f o b j e c t s i n t o c l a s s e s o r

" t y p e s " i s a p r o m i n e n t f e a t u r e o f m a n y

p r o g r a m m i n g l a n g u a g e s • O f t e n , o p e r a t i o n s

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

T h i s g i v e s r is e t o a b a s i c q u e s t i o n : i s e a c h

o p e r a t i o n i n a p r o g r a m d e f i n e d fo r t h e p a r ti c -

u l a r d a t a e l e m e n t s i t i s g i v e n ? I f t h is q u e s t io n

c a n b e a n s w e r e d

before

e x e c u t i o n o f a p r o -

g r am ( a s , f o r ex am p le , i n F O R TR A N ), i t i s a

c a s e o f s t a t i c t y p e c h e c k i n g • I f t h i s q u e s t i o n

c a n o n l y b e a n s w e r e d

d u r i n g

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

p r o g r a m ( a s, f o r e x a m p l e , i n P A L o r GE DA N-

K E N ) , w e

h a v e a c a s e o f d y n a m i c t y p e c h e c k-

i n g. I n s o m e l a n g u a g e s ( f o r e x a m p l e , A LG OL

6 0 o r SN O B O L), t h i s q u e s t i o n c an b e an s w er ed

Computing Surveys , Vol. 3 , No. 3 , September 1971


15/32

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

d u r i n g e x e c u t i o n .

M i n i - l a n g u a g e 6 t r e a t s t h e c a s e o f s t a t i c

t yp e c h e c k i n g , w i t h t h e i n t e n t t h a t a p r o -

p o s e d p r o g r a m

not

b e e x e c u t e d u n l e s s a l l

t yp e r e q u i r e m e n t s a r e s a t is f ie d . ( M i n i -

l a n g u a g e 7 t r e a t s d y n a m i c t y p e c h e c k i n g . )

Description of Lan gua ge Elements

A t y p e d e s i g n a t i o n

i s e i t h e r : 1 ) t h e s y m b o l

N ,

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

n a t u r al n u m b e r s ; o r 2 ) a s t r i n g o f t h e f o r m

(t~, • • • , t~ --~ t~+~), w h e re th e t~, 1 ~ i ~4 n -~ 1,

a re t y p e d e s i g n a t i o n s , i n w h i c h c a s e i t d e -

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

th e C a r t e s i a n p r o d u c t o f th e c l a s s es d e n o t e d

b y h , " ' " , t~ a n d w h o s e r a n g e i s t h e c l a ss

d e n o t e d b y t ~ + ~ .

P r i m i t i v e o b j e c t s

i n m i n i - l a n g u a g e 6 i n -

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

a u n a r y f u n c t io n , S Q , f o r c o m p u t i n g t h e

s q u a re o f a n a t u r a l n u m b e r , a n d a n i n fi x

b i n ar y f u n c t i o n , + , f o r c o m p u t i n g th e s u m

of t w o n a t u r a l n u m b e r s . T h e n a t u r a l n u m -

b er s h a v e a t y p e N , a n d t h e f u n c t i o n s S Q

a n d - / - h a v e t y p e s ( N - - ~ N ) a n d ( N , N - - ~ N ) .

A d ec la ra t io n

s p e c i f i e s a n

i d en t i f i e r

a s

r e p r e s e n t i n g a c l a s s o f o b j e c t s o f o n l y o n e

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

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

1 ) d e c l t y p e t

2 ) d e c f ( l ) : = e w h e r e I t

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

n a t io n , f a n i d e n t i f ie r , e a n e x p r e s s i o n ( d e -

f i n e d b e l o w ) , a n d

It

i s a l is t o f t y p e d e s i g n a -

t io n s s u c h t h a t e a c h i d e n t i f i e r i n 1 i s d i f f e r e n t ;

e a ch id e n t i f i e r i n e o c c u r s i n l; a n d t h e n u m -

b e r o f t y p e d e s i g n a t i o n s i n

It

i s i d e n t i c a l t o

t h e n u m b e r o f i d e n t i f i e r s i n 1.

A d e c l a r a t i o n o f f o r m ( 1) s p e c if ie s t h a t

e a c h i d e n t i f i e r i n I h a s t h e t y p e t. A d e c l a r a -

t io n o f f o r m ( 2) : a ) a s s i g n s t o f t h e f u n c t i o n

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

t~, • • . , t~ = It ,

i n t o t h e v a l u e o f t h e e x p r e s -

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

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

s pe cif ie s t h a t t h e t y p e o f f i s ( h , " " ,

• N ote tha t th i s requi rem ent i s

not

made in

mini- language 7, where the types of objects as-

signed to identif iers m ay v ary .


129

tn

- - ~ t n + l ) ,

w he re t : , • • • , tn

=

It a r e t h e t y p e

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

a n d tn +l i s t h e t y p e o f t h e r e s u l t e x p r e s s i o n e

( d e f i n e d b e l o w ) , o b t a i n e d b y u s i n g t h e t y p e s

h , " . , t~ f o r t h e c o r r e s p o n d i n g i d e n ti f ie r s

of l in e .

A n

e x p r e s s i o n

c o n s i s t s o f a s t r i n g o f o n e o f

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

1) p

2) i

3) SQ(e)

( p r o v i d e d e i s o f t y p e N )

4) d+e)

( p r o v i d e d d a n d e a r e o f t y p e N )

5)

f ( e l , . . . , en)

(p ro v i de d f i x o f t y p e (h , " " , tn ----~ n+~) ,

a n d e ~, • • • , en a r e o f c o r r e s p o n d i n g t y p e

t 2 , ' ' ' , t n )

6 )

( i f

d > e

t h e n e l

e l s e

e2)

( p r o v i d e d d a n d e a r e o f ty p e N , a n d

e l

a n d e2 a r e o f t h e s a m e t y p e ) ,

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

i d e n t i f ie r s , a n d d , e , e , , . . . , e , a r e e x p r e s -

s i o n s .

E x p r e s s i o n s a r e o f t h e f o l l o w i n g t y p e s :

1 ) a n e x p r e s s i o n o f f o r m ( 1) is o f t h e s a m e

t y p e a s p ;

2 ) a n e x p r e s s i o n o f f o r m (2 ) is o f th e t y p e

d e c l a r e d f o r i ;

3 ) i f e is of t y p e N , t h e n a n e x p r e s s i o n o f

f o r m ( 3) is o f t y p e N ;

4 ) i f d a n d e a r e o f t y p e N , a n e x p r e s s i o n o f

f o r m ( 4) is o f t y p e N ;

5 ) if t h e t y p e s o f e l , . . . , e~

a r e

t l , " ' ' ,

t ~ , a n d i f f i s d e c l a r e d t o b e o f t y p e

( h , • • • tn - -~ t~+ l) , t h e n an ex pr es s io n o f

f o r m ( 5 ) i s o f t y p e tn+ ~ ; a n d

6 ) i f d , e a r e o f t y p e N , a n d e l , e2 a r e o f t h e

s a m e t y p e , t h e n a n e x p r e s s io n o f f o r m (6 )

i s o f t h e s a m e t y p e a s el o r e 2 .

A n

a s s i g n m e n t c o m m a n d

c o n s i s t s o f a

s t r i n g o f t h e f o r m i : = e , w h e r e i i s a n i d e n t i -

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

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

A p r o g r a m

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

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

a s s i g n m e n t c o m m a n d s s u c h t h a t e a c h i d e n -

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

o n l y o n c e i n d , a n d

al l

o f t h e a b o v e r e q u i r e -

m e n t s o n t y p e s a r e s a ti s fi e d .

C o mp u t in g S u rv e y s , Vo l . 3 , No . 3 , S e p te mb e r 1 9 71


16/32

130 • Hen ry F . Ledgard

Prog ram Execution

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

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

f o r a d d i t i o n , t h e s q u a r i n g o p e r a t i o n , a s s i g n -

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

t i o n a l a p p l i c a t i o n .

EXAMPLE 1:

d e c A , B t y p e N

d e e F ( X ) : = 8 Q ( SQ ( X )) w h e r e N

A : = 2

B : = F ( A )

E X A M P L E 2 :

d e c X t y p e N

d e e

A, B t y p e (N----~N)

d e c

F ( X ) : = S Q ( S Q ( X ) )

w h e r e

N

d e e G ( X , Y ) : = X ( Y ) w h e r e

( N - - I N ) , N

A : = S Q

B : = F

X : = (G(A , 1 ) + G( B , 2 ) )

EXAMP LE 3 ( s y n t a c t i c a l l y i l leg a l ) :

d e e A , B . t y p e N

d e e F ( X ) : = ( X + X ) w h e r e N

d e e

G ( X , Y ) : = i X + Y ) w h e r e N , N

d e e H ( X , Y ) : = X ( Y ) w h e r e (N ---~ N), N

A : = H ( F , 1 )

B : = H ( G , l )

EXAMPLE 4 ( s y n t a c t i c a l l y i l leg a l ) :

d e e A t y p e

N

d e c B t y p e ( N ~ N )

d e c F ( X ) : = X w h e r e N

d e e G ( X ) : = X w h e r e ( N - + N )

d e e H ( X , Y ) : = X ( Y ) w h e r e ((N--+N)---~

(N~N)), (N~N)

g : = H ( G , S Q )

A : = B ( 2 )

B : = H ( F , 2 )

E x a m p l e 1 r e s u l t s i n s e t t i n g t h e v a l u e o f

B t o 1 6 . E x a m p l e 2 r e s u l t s i n s e t t i n g t h e

v a l u e o f X t o 1 7 .

Ex am p l e 3 i s i l l eg a l b ecau s e o f t h e s t a t e -

m e n t B : = H ( G , 1 ) , " w h e r e H , a f u n c t i o n

o f t y p e ( (N - -~ N ) , N --~ X) , i s ap p l i ed t o

a r g u m e n t s G , 1 o f t y p e s ( X , N - + X ) , X .

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

d e t e r m i n e t h a t t h e p r o g r a m o f E x a m p l e 3

i s s y n t a c t i c a l l y i l l e g a l :

1 ) A a n d B a r e d e c l a r e d t o b e of t y p e N ;

2 ) F i s dec la red to be o f type (N-- -aN) ;

3 ) G is d e c l a r e d t o b e o f t y p e ( N , N ~ N ) ;

4 ) H i s d e c l a r e d t o b e o f t y p e ( ( N - - i N ) ,

N ---~ N ) ;

5 ) H (F , 1 ), w h e re F , 1 a r e o f t y p es (N --~N ) ,

N an d H is of ty p e ((N---- iN), N --~ N ),

is o f t y p e N ;

6 ) A : = H ( F , 1 ) i s t y p e - w i s e o f t h e f o r m

N : = N , h e n c e i s t y p e - w i s e c o r r e c t;

t h e r e f o r e ,

7 ) H(G , 1 ) , wh e re G , 1 a r e o f t y p es (N ,

N --~N) , N an d H i s o f ty p e ( (N--~N) ,

N -+ N ) , i s i l leg a l b ecau s e t h e t y p e

(N , N - -~ N ) o f G i s n o t i d en t i c a l t o t h e

t y p e ( N -- + N ) o f t h e f i r s t a r g u m e n t o f H .

Ex am p l e 4 i s i l l eg a l b ecau s e o f t i l e s t a t e -

m e n t " B : = H ( F , 2 ) , " w h e r e H , a f u n c ti o n

of ty p e (((N--)N)---~(N--~N)), (N--~N )

( N - - + N ) ) , i s a p p l i e d t o a r g u m e n t s F a n d 2

o f t y p e s ( N - -+ N ) a n d N . N o t e , h o w e v e r ,

t h a t t h e a p p l i c a t i o n o f H t o F a n d 2 is

s em an t i ca l l y we l l -d e f i n ed , i . e . , r e s u l t s i n

a p p l y i n g F t o 2 , w h i c h r e t u r n s t h e v a l u e 2 .

( P r o g r a m s l i k e t h a t o f E x a m p l e 4 w i l l b e

a l l o wed i n m i n i - l an g u ag e 7 . ) N o t e , a l s o ,

t h a t i n a l l s y n t a c t i c a l l y l eg a l p r o g r a m s i n

m i n i - l a n g u a g e 6 , n o c o n d i t i o n l e a d i n g t o a

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

Discussion

I n m i n i - l a n g u a g e 6 , t h e r e w e r e f o u r m a j o r

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

( T h e r e a d e r i s r e f e r r e d t o t h e w o r k o f M o r r is

[1 1] fo r a d i s cu s s i o n o f t y p es a n d t y p e ch eck -

ing . )

Assumpt ion 1

A t y p e i s e i t h e r a n a t o m i c t y p e , i n w h i c h

cas e i t d en o t e s a c l a s s o f p r i m i t i v e o b j ec t s ,

o r it, i s a f u n c t i o n a l t y p e ( h , " - , t ,

t~ + l) , wh e re h , " " , t , ,, t , + l a r e t y p e s , i n

wh i ch ca s e i t d en o t e s a c l a s s o f fu n c t i o n s .

A s s um p t ion 2

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

A s s u m p t i o n s 1 a n d 2 s e e m i n t u i t i v e , b u t

t h e y d o r a i s e s o m e i m p o r t a n t q u e s t i o n s :

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

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

Co m putin g Surveys , V ol . 3 , No. 3 , September 1971


17/32

p o i n t e r s , i n t eg e r s , r a t i o n a l s , B o o l ean s , an d

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

a b o u t l a b e l s , l o c a t i o n s , P L / I s t o r a g e c l a s s

a t t r i b u t e s ( a u t o m a t i c o r s t a t i c ) , a n d P L / I

f il e a t t r i b u t e s ( b u f f e r e d o r u n b u f f e r e d ) ?

Are t h e s e l an g u ag e i s s u es t o b e co n s i d e red a s

p a r t o f type c o n s i d e r a t i o n s ? C o n s i d e r , f o r

e x a m p l e , a n a s s i g n m e n t s t a t e m e n t o f t h e

f o rm e l : = e 2 . A s p a r t o f o u r t y p e c h e c k i n g

w e u s u a l l y w a n t t o e n s u r e t h a t e~ a n d e2

a r e b o t h n u m e r i c o r b o t h B o o l e a n . M i g h t

w e a l so w a n t t o e n s u r e t h a t e l d e n o t e s a

l o ca t i o n an d e2 a v a l u e , s o a s t o i n c l u d e , s ay ,

( if X > Y t h e n I e l se J)

: = ( i f A > B t h e n 1 e l se 2)

b u t ex c l u d e

(i f A > B t h e n 1 e l s e 2)

: = ( i f X > Y

ten mini-languages: a study of topical issues in programming languages

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