The use of the language Automath for syntax and semantics
of programming languages
Citation for published version (APA):
Bruijn, de, N. G. (1975). The use of the language Automath for syntax and semantics of programming languages. Technische Universiteit Eindhoven.
Document status and date: Published: 01/01/1975
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First d r d t of paper presented to the
International jymposium on Proving and Improving Programs 4rc et SF 7 < - ~ $ , July 1-2-3,
1975.
Thc use of the l a n ~ u a ~ e AUTOMATH for svatax and semantics of programming larkguages.
by
N.G.
de Bruijn Department o f Mathematics Eindhoven University of TechnologyEindhoven, The Netherlands Telephone ( 0 ) 4 0
-
472807Abstract.
Mathematical theories can be written in the form of books in the language
AUTOMATH, and the checking of the correctness of the entire text can be automatized. The idea of this paper is to consider syntax and semantics of computer pro-
programing languages as parts of mathematics and to write semantical theorems concerning particular programs in books that already contain logic, mathema- tics, definition of syntax,and axioms for semantics of the programming language. The paper describes ways for doing this, with a programming
language that is a kind of subset of ALGOL 60 without being much less power- ful.
- 2 -
1 . --- T n t r o d u c t i o n . ---
AIJII~%A"L'~.
\L
l,3,4_!)
i s a Language which p e r m i t s u s t o w r i t e v e r y l a r g e p a r t s of mathematics i n such a p r e c i s e way t h a t v e r i f i c a t i o n of t h e cor- r e c t n e s s of t h e mathematical c o n t e n t s can be c a r r i e d o u t by a computer. The AUTOMATH book t h a t i s p r e s e n t e d t o t h e computer h a s t o c o n t a i n every- t h i n g ; l o g i c a l f o u n d a t i o n s , i n f e r e n c e r u l e s , mathematical f o u n d a t i o n s , axioms, d e f i n i t i ~ n s , f o r m u l a s , a b b r e v i a t i o n s , theorems, p r o o f s , and t h e m u t u a l c o n n e c t i o n s between a l l t h e s e t h i n g s .S i n c e t h e b e g i n n i n g of t h e AUTOMATH p r o j e c t i n 1968 a c o n s i d e r a b l e amount of m a t h e m a t i c a l m a t e r i a l h a s been w r i t t e n and checked, and t h e r e - fore i t seems t o b e f e a s i b l e t o s t a r t t h e u s e of AUTOMATH f o r p r a c t i c a l purposes.
I n many a r e a s of mathematics one may f e e l t h a t t h e r e i s n o t much need f o r such e x t r e m e l y p r e c i s e f o r m u l a t i o n s of complete t h e o r i e s . Such a r e a s can have a s t r o n g i n t u i t i v e background, and t h e f e e l of s a f e t y i s s u p p o r t e d by t h e many a p p l i c a t i o n s t h a t can be checked by v a r i o u s e n t i r e l y d i f f e r e n t methods l e a d i n g a l l t o t h e same r e s u l t .
There a r e f i e l d s , however, where t h e need f o r p r e c i s i o n i s v e r y s t r o n g , where i n t u i t i v e s u p p o r t i s weak and e x p e r i m e n t a l e v i d e n c e mis- l e a d i n g . T h i s may happen i n c a s e s of long p r o o f s c o n s i s t i n g of v e r y many e l e m e n t a r y s t e p s , where i t i s s t r o n g l y f e l t t h a t a c h a i n i s a s weak a s i t s weakest l i n k . I f t h e number of s t e p s r u n s i n t o thousands,the need f o r mechanical v e r i f i c a t i o n can be a v e r y n r a c t i c a l o m . Xn such c a s e s i t may be worth w h i l e t o go i n t o t h e t r o u b l e of coding e v e r y d e t a i l i n t o a f i n a l f o r m a l i z a t i o n , removing a l l t r a c e s of i n t u i t i o n and e x p e r i e n c e .
One such f i e l d can be t h e one t h a t i s devoted to p r o v i n g t h a t a com- p u t e r p r o g r a m has t h e s e m a n t i c s we want
i t
t o h a v e , i . e . t h a t program e x e c u t i o n does what we c l a i m i t t o do. I n t u i t i o n and e x p e r i m e n t a l evidenceN.G. d e B r u i j n
- 3 -
i n t h i s f i e l d a r e known t o b e independable. One r e a s o n , b u t n o t t h e only ( m e , ; $ t b i long programs a r e o f t e n produced i n c o o p e r a t i o n between
, e v e r a l p r o g r a m e r s , p i e c e s a r e t a k e n from l i b r a r i e s , e t c . Another m a t t e r i s t h a t program languages themselves need a thorough f o r m a l d e s c r i p t i o n , b o t h f o r s y n t a x and s e m a n t i c s .
Let us t h i n k of complex s i t u a t i o n s where l o g i c , mathematics, s y n t a x an3 s e n ~ a n t i c s a r e :nterwoven, and where i t i s j u s t t h e i n t e r p l a y between t f - e s e romponentr: t h a t r e q n i r e s a t t e n t i o n . Imagine we have a program f o r
seal hirt , rhe s o l u t i o n s o f A n u m b e r - t h e o r e t i c a l problem, and we w i s h t o prove t h a t t h e program does what we claim. A s i t happens, d u r i n g t h e s e a r c h a l l s o r t s of s h o r t c u t s a r e made, p a r t l y on t h e b a s i s of n u m b e r - t h e o r e t i c a l o r c o m b i n a t o r i a l theorems, p a r t l y on t h e b a s i s of s e m a n t i c a l knowledge about p a r t s of t h e program. I n t h e f u t u r e we might even imagine t h a t we a r e working i n an advanced programming language t h a t a l l o w s u s t o e x t e n d t h e s y n t a x and s e m a n t i c s i n t h e c o u r s e of t h e argument,and t o prove
s e m a n t i c a i theorems on t h e s e e x t e n s i o n s t h a t a r e used i n t h e proof of t h e f i n a l s e m a n t i c a l s t a t e m e n t on t h e t o t a l program ( i . e . t h e s t a t e m e n t t h a t e x e c u t i o n of t h e program produces e x a c t l y a l l s o l u t i o n s of t h e number- t h e o r e t i c a l problem). I n a s i t u a t i o n l i k e t h i s t h e s u p e r v i s i o n over t h e whole system seems t o b e i n a d e q u a t e i f we produce i n t h e "human" way,
c o n s i s t i n g of a patchwork of f o r m a l i z e d and i n t u i t i v e p i e c e s , t i e d to- g e t h e r by o u r f e e l i n g of c o n f i d e n c e . U s u a l l y t h e s i t u a t i o n i s l e s s com- p l e x t h a n we j u s t d e s c r i b e d , b u t a t l e a s t some of t h e s e d i f f i c u l t i e s a r e always p r e s e n t . P a r t of t h e d L f f i c u l t i e s a r e n o t s o much s o u r c e s of e r r o r s , b u t y e t produce a k i n d of c o n c e p t u a l u n e a s i n e s s s i n c e we f e e l u n a b l e t o
l a y a c o n n e c t i o n , o t h e r t h a n by i n t u i t i o n , between t h e f o r m a l i z e d mathe- m a t i c a l n o t i o n i n t h e mathematics r e x c and its f o r m a l i z e d c o u n t e r p a r t
- 4 -
I t is j u s t t h i s k i n d of a need f o r an o v e r a l l s u r v e y of a l a r g e f i e l d , t h a t AUTOMATH was d e v i s e d f o r .
I n ~ r d e r t o b e a b l e t o t a l k i n l e s s g e n e r a l t e r m s , we a c t a s i f t h e r e were o n l y one i n t e r e s t i n g program i n t h e w o r l d , i . e . a program f o r f i n d i n g t h e g . c . d . of two i n t e g e r s . We s t a r t from an AUTOMATH book t h a t c o n t a i n s an amount of b a s i c m a t e r i a l . We can extend t h e book by adding f u r t h e r m a t e r i a l ( e v e r y a c c e p t a b l e a d d i t i o n t o an a c c e p t a b l e book produces a new a c c e p t a b l e book). The book c e r t a i n l y has t o c o n t a i n
( i ) L o g i c a l t o o l s , i n f e r e n c e r u l e s .
( i i ) Mathematical f o u n d a t i o n s , i n p a r t i c u l a r p r o p e r t i e s of n a t u r a l numbers and i n t e g e r s .
( i i i ) D e f i n i t i o n of t h e g . c . d . , and p r o o f s of some of i t s p r o p e r t i e s . We now add a c h a p t e r t o t h e book, i n v o l v i n g
( i v ) A d e f i n i t i o n of a p a r t i c u l a r programming language.
( v ) B a s i c assumptions on t h e s e m a n t i c s of programs i n t h a t language. Next we w r i t e i n t h e book
( v i ) A d e s c r i p t i o n of a program f o r t h e computation of t h e g.c.d. ( v i i ) A s e m a n t i c theorem, w i t h p r o o f , about t h a t program ( t h i s theorem
may s t a t e something t o t h e e f f e c t t h a t i f t h e i n p u t i s m,n, t h e n t h e o u t p u t i s t h e g.c.d. of m and n ) .
However complete t h i s may seem t o b e , y e t t h e r e a r e a number of t h i n g s t h a t have t o be t a k e n f o r g r a n t e d . These t h i n g s concern t h e b a s i c assumptions
on t h e system a s a whole, and n o t such d e t a i l s a s d e s c r i b e d i n ( i i i ) , ( v i ) o r ( v i i ) .
2 . What h a s t o be t a k e n f o r g r a n t e d ?
I n t h e f i r s t p l a c e w e have t o b e l i e v e t h e t h i n g s c l e a r l y i n d i c a t e d
as " p r i m i t i v e n o t i o n s " i n t h e
AUTOMATH
book i t s e l f . Some of them r e f e r t o p r i m i t i v e o b j e c t s f o r which no d e f i n i t i o n i s given: we j u s t g i v e them a name and we s a y of what t y p e t h e y a r e . O t h e r sN.G. de Bruijn
-
5-
refer to axioms, whi.ch are things that have the form of a theorem without p;ocf.
The w a y AUTOMATH deals with assertions is what is now called the
"formulas as typesv presentation of logic (the term comes from Howard
[ 7
I ) .
Objects have a type (the type of3
is "natural number", the type of S is "set"); proofs have a type too: the type of a proof a the (unique) assertionit
proves. So corresponding to a "definition" (showing (a) name of the object; (b) description of its construction; (c) type of the object) we have a"theorem" (sho~~ling (a) name of the proof; (b) construction of the proof;
(c) the assertion). Corresponding to the introduction of a primitive object we have an axiom: in both cases (b) is replaced by "PN" which is just a
warning symbol. There is a third kind of lines in our book: the block
opening lines; they may differ from the
PN
lines in the sense that they are narrowing the context: They can be "let x be an object of type At' or "let p be a proof of assertion B". They embody what we call introduction of a local variable or the introduction of a local assumption. They lose their power as soon as we get back to the original wider context.As it is the case with any formal system, there is an unformalized system of interpretation that goes along with it. For the primitives in the formal text we choose interpretations in the outside world, and we have a system of propagation of interpretation that produces an interpretation of the final results in the formal text. We need not bother about inter- pretation of the intermediate parts of the text. (If the final result is a theorem, the thing we care to interpret is the assertion and nct the proof). Interpretational troubles are restricted to primitives and finals, and this is exactly the reason for the practical use of formal systems.
Let us direct our attention to the book that checks the g.c.d. program. We have to do various things. Firs', the formal soundness of the AUTOMATH
t e x t h a s t o be b e l i e v e d . T h i s h a s t o b e based on t h e b e l i e f t h a t d e f i n i t i o n and t1,eory ~f AUTCMATH a r e sound, and t h a t t h e t e x t checking a l g o r i t h m r e f u s e s e v e r y i n p u t t h a t i s n o t a c c o r d i n g t o t h e language d e f i n i t i o n s ( a c t u a l l y t h i s f a c t s s a n example of a q u i t e complex s e m a n t i c a l theorem). And we have t o b e l i e v e t h a t t h e computer language t h a t i s used f o r d e s c r i b i n g t h a t checking a l g o r i t h m i s a d e q u a t e l y implemented on a computer t h a t makes no m i s t a k e s .
Once t h e AUTOMATH t e x t h a s been t a k e n a s a b s o l u t e l y sound, we s t i l l
have t h e i i l t e r p r e t a t i o n s . We have t o i n t e r p r e t t h e n o t i o n s and axioms of
(i)
and ( i i ) , and t o make s u r e t h e y correspond t o i d e a s i n t h e p a r t l y f o r m a l i z e d m a t h e m a t i c a l world we t h i n k we l i v e i n . T h i s h a s t oremain vague s i n c e t h a t m a t h e m a t i c a l world i s a b i t vague. The o n l y t h i n g we can s a y h e r e i s t h a t t h i s correspondence can be t r a c e d by c a r e f u l t h i n k i n g
i n t h e same s t y l e a s mathematicians always have been t h i n k i n g i n t u i t i v e l y about t h e meaning of t h e i r symbolisms. And e x p e r i e n c e r e i n f o r c e s c o n f i d e n c e .
Next we t u r n t o ( i v ) . The program p r i m i t i v e s and c o n s t r u c t s d e s c r i b e d i n t h e AUTOMATH book, w i l l n o t be e x a c t l y t h e same a s t h e programs w e r e a l l y g i v e t o o u r computer. There w i l l be an a l g o r i t h m t h a t s y n t a c t i c a l l y t r a n s l a t e s one i n t o t h e o t h e r , and we have t o b e l i e v e t h a t t h i s i s sound.
The more i m p o r t a n t t h i n g i s ( v ) . A f t e r we have i n t e r p r e t a t e d t h e s e m a n t i c axioms we s e e t h a t t h e y can b e c o n s i d e r e d a s c o n d i t i o n s t h a t have t o b e s a t i s f i e d by t h e computer system we want t o p r e s e n t o u r g.c.d. program t o . To convince o u r s e l v e s t h a t i t does s a t i s f y t h e s e c o n d i t i o n s , i s p a r t l y a s e m a n t i c a l , p a r t l y an e n g i n e e r i n g problem.
The f i n a l i n t e r p r e t a t i o n of ( v i i ) i s v e r y simple. There w i l l n o t b e any d i f f i c u l t y i n r e a d i n g i t s main r e s u l t a s " t h e g.c.d. a l g o r i t h m e v a l u a t e s t h e g.c.d.".
N.G.
de Bruijn3, U s e of types.
a-_:
-
We want to describe syntax and semantics of an ALGOL-like programming l.,lguahe in terms ~f AUTONA'L'B. The first thing we have to decide on is which things are
-
types and which things have types.In
AUl'OMAT'rI there are 3-expressions, 2-expressions, and the singleI-expression
--
type. Every 3-expression has a unique type, and that type is a 2-expression. Every 2-expression has type as its type. We shall use 2-expressions for some fundamental classes of mathematical objects, and 3-expressions for those objects themselves. Also, we use 3-expressions for proofs, a v d thei-r types are the assertions these proofs prove. In orderto easily build assertions, we create as a primitive notion the
2-expression "bbool" and, again as primitive notion, we build, for every 3-expression b of type bool, the 2-expression ~ ~ ~ ~ ( b o d f ) . If we have a 3-expression p of type TRUE(b), the interpretation is that p is a proof for the truth of the proposition b. From now on the interpretation of logic and mathematics is rather straightforward.
In order to express programs, we start with taking 2-expressions R, interpreted as state spaces. If R is a type, we create the primitive notion "program
(R)"
as a 2-expression. The interpretation of a 3-expression ?rhaving type "program (R)" is that ?r is a program acting on the state
space
R.
4 . --.- Syntax.
In order to describe the syntax of a programming language, we shall introduce primitive programs and primitive program constructs. On the basis of these we are able to construct all programs of the language, but it does not seem to be necessary to state an axiom that every object of type program (R) belongs to this constructed set.
we w r i t e a f e w l i n e s of our book:
I
:= PNtype
i f , 5 :=--
boo 1 TKTJE := PN t y p e (2 :=-
type
program := ' IT l l ~ h o r i z o n t a l 1 i n e s i n d i c a t e t h a t b and $2 a r e v a r i a b l e s ( o r block openers1')
,
t h e v e r t i c a l l i n e s i n d i c a t e b l o c k s w i t h a common c o n t e x t ( t h e c o n t e x t of t h e l i n e w i t h TRUE i s b ) . Note t h a t on t h e b a s i s of t h i s , "program (I?)'' can be used f o r e v e r y p o s s i b l e 2-expressionr ,
i n e v e r y p o s s i b l e c o n t e x t . For example we can s a y a t h i n g 1 i k e : " l e t f be a f u n c t i o n t h a t a t t a c h e s a program onR
t o e v e r y i n t e g e r n N (whence n i s a v a r i a b l e t h e program depends on, and n o t a v a r i a b l e t h e program a c t s o n ) , o r ' l e t f b e a f u n c t i o n t h a t a t t a c h e s a program on
R
t o e v e r y program onR
I ' I 2 . We s h a l l c e r t a i n l y need t h e n o t i o n of a C a r t e s i a n p r o d u c t of two s t a t e s p a c e s , and t h e r e f o r e we w r i t eR2
:= .!iYE c a r t p r o d := PN tYPeL e t u s now d i s c u s s how t o i n t r o d u c e p r i m i t i v e programs. We may c r e a t e programs on e v e r y s t a t e s p a c e a s w e l l a s programs on s p e c i a l s t a t e s p a c e s
( i n ALGOL160 t h e f i r s t c a s e does n o t o c c u r ) . For example one might c r e a t e t h e empty program ( t o b e i n t e r p r e t e d a s a program whose e x e c u t i o n does n o t a l t e r t h e s t a t e ) , and one might c r e a t e f a k e programs, l i k e a program of which we g i v e no semantic i n f o r m a t i o n whatsoever.
N.G. d e B r u i j n
A v e r y i m p o r t a n t g r o u p o f p r i m i t i v e programs a r e a s s i g n m e n t s . A c l e a r d c s c r i p t l o n o f them i s by no means easy. The s i m p l e s t c a s e s a r e t h e r e p l a c e - ments x := y where x and y a r e s t a t e s p a c e v a r i a b l e s . It can be a c a s e where
t h e s t a t e s p a c e i s c a r t p r o d ( R , R ) ; t h e n t h e replacement can be i n t r o d u c e d a s a p r i m i t i v e program ( t h e i n t e r p r e t a t i o n i s , i f we u s e t h e terminology of a n a l y t i c geometry, h o r i z o n t a l p r o j e c t i o n o n t o t h e d i a g o n a l , and i t s seman- t i c s has t o b e a r r a n g e d a c c o r d i n g l y . (Needless t o s a y , i t seems u n a t t r a c t i v e
t o w r i t e s e p a r a t e axioms f o r c a r t e s i a n p r o d u c t s of 2 , 3 ,
...
s p a c e s ; i t i s F e t t e r t o i n t r o d u c e o p e r a t i o n s on f i n i t e sequences of s p a c e s , which i s e a s i e r t o do i f we u s e t h e e x t e n s i o n AUT-QE of AUTOMATH).A h a r d e r q u e s t i o n i s what t o d e c i d e about v a l u e a s s i g n m e n t s l i k e x := c , where c i s a v a r i a b l e t h e program depends on, and x := 0 , x := I , e t c . A t f i r s t s i g h t i t seems r e a s o n a b l e t o p o s t u l a t e ( i f x i s a s t a t e s p a c e v a r i a b l e of t h e t y p e i n t e g e r ) t h a t " f o r e v e r y i n t e g e r c t h e s t a t e m e n t "x := c" i s a program. But one should b e aware of t h e f a c t t h a t t h i s i s s a i d i n a mathe- m a t i c a l language t h a t a l l o w s t o s u b s t i t u t e o t h e r e x p r e s s i o n s f o r c , and i n
p a r t i c u l a r t h e s e might be e x p r e s s i o n s a computer cannot e v a l u a t e . One should r e a l i z e t h a t t h e p o s t u l a t e "x := c i s a program f o r e v e r y i n t e g e r c i n t h e r a n g e 0 5 c I 1 " h a s a s t r o n g e r e x p r e s s i v e power t h a n t h e two p o s t u l a t e s
S f
x := 0 i s a program" and "x := 1 i s a program" have t o g e t h e r , u n l e s s we add
e x t r a axioms i n t h e l a t t e r c a s e t h a t c r e a t e t h e same e f f e c t ( l i k e axioms on " d e f i n i t i o n by c a s e s " ; t h e f o r m u l a t i o n of such axioms seems t o r e q u i r e
I t e q u a l i t y of programs" a s
a
p r i m i t i v e ) .It seems t o b e s a f e t o be v e r y r e s t r i c t i v e about v a l u e a s s i g m e n t s . I n t h e c a s e of i n t e g e r s one could j u s t t a k e "x := 0" and x := 1 " ( a s soon a s we have a d d i t i o n a s a p r i m i t i v e program, we can s i m u l a t e a n a s s i g n m e n t l i k e "x := 216" by a program having no o t h e r a s s i g n m e n t s t h a n "x := 0" and
11
-
10-
On t h e o t h e r h a n d , i f one s t u d i e s m a t h e m a t i c a l a l g o r i t h m s i n s t e a d of c m p u i r :- (Ircgrams, t h e r e i s no o b j e c t i o n a g a i n s t a d m i t t i n g "x := c".
in t h e c a s p t h a t x i s a s t a t e s p a c e v a r i a b l e of t h e t y p e "bool",
w e can taice " X := t r u e T t and
"x :=
f a l s e " as a s s i g n m e n t s ( t r u e and f a l s ea r e p r i m i t i v e s of t y p e b o o l , f o r which we h a v e t h e axioms o f c l a s s i c a l l o g i c i n t h e l o g i c o - m a t h e n i a t i c a l p a r t of o u r b o o k ) . Moreover we can add a f a . o t h e r s l i k e : ~ f x , y , z a r e s t a t e s p a c e v a r i a b l e s of t y p e b o o l , r e a l , r e a l , r e s p e c t i v e l y , t h e n "x := y>z", "x := y=zl' e t c . a r e programs. And, i n
d n a l o g y t c t h ~ a d d i l i o n of i n t e g e r s , we c a n u s e p r i m i t i v e s l i k e c o n j u n c t i o n , i m p l i c a t i o n , c t c . , i n o r d e r t o g e t programs t h a t s i m u l a t e t h e more compli- c a t e d b o o l e a n e x p r e s s i o n s .
A s p e c i a l k i n d o f a s s i g m e n t s i s c o n n e c t e d w i t h a r r a y s . F o r some s p e c i a l d a t a t y p e s A we a l l o w t h e u s e of a s s i g n m e n t o p e r a t i o n s o n v a r i a b l e s of t y p e
[n,Z]A ( i . e . t h e t y p e of a l l mappings of t h e s e t of i n t e g e r s Z i n t o A). I f f i s a v a r i a b l - e of t h i s t y p e , i f x i s a v a r i a b l e of t y p e A , and k a v a r i a b l e of
t y p e Z , we have t o p o s t u l a t e t h e p r i m i t i v e programs whose i n t e r p r e t a t i o n
i s f [ k l : =
x
and x := f C k l . The t r e a t m e n t of a r r a y s i s h a r d i f a r r a y bounds a r e v a r i a b l e s , a s i t i s p o s s i b l e i nALGOL'GO.
We s h a l l n o t d i s c u s s t h i s i n t h i s p a p e r ( i t s h o u l d b e remarked t h a t i t s t r e a t m e n t i n C61
i s u n s a t i s f a c - t o r y s i n c e i t n e g l e c t s t h e f a c t t.hat t h e v a r i a b l e s t h a t p r e s e n t e d t h e a r r a y bounds may change i n s i d e t h e b l o c k ; a c o r r e c t way t o d e s c r i b e how s u c h v a r i a b l e s a r e u n t o u c h a b l e m i g h t i n v o l v e t h e i n t r o d u c t i o n of v a r i a b l e s t h a tc a n n o t b e a s s i g n e d t o s y n t a c t i c a l l y ) .
Thus f a r we d i s c u s s e d p r i m i t i v e programs; n e x t we g e t t o p r i m i t i v e program c o n s t r u c t s , i . e . ways t o compose b i g g e r programs f r o m s m a l l e r o n e s . F i r s t t h e c o n c a t e n a t i n n ( T i r s t n l , t h e n n2):
N.G. d e B r u i j n
-
1 1-
A c o n s t r u c t ( n o t e x i s t i n g i n A L G O L ' ~ ~ ) t h a t h a s t h e same s y n t a c t i c s t r u c t u r e a s t h e c o n c a t e n a t i o n , i s t h e d i s j u n c t i o n I r r l o r
IT^"
(where t h eprogrammer l e a v e s i t t o t h e machine which one t o e x e c u t e ) .
A v e r y i m p o r t n n t p r i m i t i v e c o n s t r u c t i s t h e b i n a r y s e l e c t i o n
" i f b tbren
- -
e l s e r2". We want t o r e s t r i c t t h i s t o t h e c a s e t h a t b i s a s t a t e s p a c e v a r i a b l e of t y p e bool ( o t h e r w i s e we would have t o d e s c r i b e i n o u r brok t h a t b i s a s y n t a c t i c a l l y a d m i s s i b l e boolean e x p r e s s i o n , and t h a t seems t o b e h a r d ) . T r u s t i n g t h e s i m u l a t i v e power of a p r i m i t i v e language, we might l i m i t o u r s e l v e s t o t h e c a s e t h a t b does n o t o c c u r i n t h e s t a t es p a c e $2 of and r2, and t h a t b i s t h e v a r i a b l e t h a t c o r r e s p o n d s t o t h e e x t e n s i o n of
R
t oR
x b o o l . I n o u r book t h i s l o o k s l i k e t h i sNext we c o n s i d e r t h e c o n s t r u c t s " p r o j e c t i o n " and " i n j e c t i o n " t h a t t r a n s f o r m programs i n one s t a t e s p a c e i n t o programs i n a n o t h e r s t a t e s p a c e . We s h a l l b e a b i t s u p e r f i c i a l h e r e , s i n c e a c o r r e c t t r e a t m e n t r e q u i r e s
e l a b o r a t e i n d e x i n g of t h e v a r i o u s components.
Imagine T i s a program on
R
x Z , where $2 i s any s t a t e s p a c e and Zs t a n d s f o r t h e s e t of a l l i n t e g e r s . Now we want t o c o n s i d e r t h e " p r o j e c t i o n " on $ 2 , i . e . t h e program on $2 t h a t would b e d e s c r i b e d i n ALGOL'GO a s
-
12-
b e g i n .---
i n t e g e r n ; IT-
end. 1 1 1 OUT. I,, w e i n t r a ~ i a c e t h i s p r i m i t i v e a s f o l l o w s : 71 ' = program (cartprod(62,Z)) p r o j := PN program(62).
rlne n o t i o n of i n j e c t i o n i s a c a s e of a p r o c e d u r e c a l l e d by name. L e t n b e a program an a s t a t e s p a c e 62 x . . . x ilk, and l e tx,
,..
.
,xk b e t h e names1
of t l ~ r v a r i a b l e s f o r t h e s e k components. L e t
4
be some one-to-one mapping of { I,
. .
.
, k } i n t o { l,
. . .
,m}, and l e t A l x . . .xAm s a t i s f yA
4
( i ) = ni) ( i = 1 , .. .
, k ).
Then we want t o i n t r o d u c e t h e p r o g r a m " i n j e c t i o n (IT)" on4
A l x . . . x Am whose i n t e r p r e t a t i o n i s a s f o l l o w s : t a k e y.
,
a s s t a t e Ym s p a c e v a r i a b l e s f o r A x...xA
R e p l a c i n g i n t h e program IT e v e r y i n s t a n t I m' of X - 1 by Y 4 ( i ) g i v e s t h e program " i n j e c t i o n (IT)':4
The c o n s t r u c t s d e s c r i b e d t h u s f a r a r e mappings of programs o r p a i r s of programs o n t o programs. L e t u s c a l l them "lower" c o n s t r u c t s i n c o n t r a s t t o " h i g h e r " c o n s t r u c t s where t h e v a r i a b l e s a r e n o t programs, b u t o b j e c t s on a h i g h e r l e v e l .
The o n l y example we t a k e i s t h i s one: I f Q i s a f u n c t i o n t h a t maps programs o n t o programs, t h e n t h e p r i m i t i v e h i g h e r c o n s t r u c t " r e c u r s ( Q ) " i s t o be i n t e r - p r e t e d a s t h e r e c u r s i v e l y d e f i n e d p r o c e d u r e "n := Q ( I T ) " . I n o u r book we g e t
SZ := t y p e
I
Q := Cn,program(R)lprogramCR)r e c u r s := PN program(@
N.G. d e B r u i j n
srrcill fragment of A T ~ G O L ' ~ ~ . N e v e r t h e l e s s q u i t e a l a r g e p a r t of ALGOL'GO can
b ~ d s i n ~ u l a t e d by t h i s f r a g m e n t , i n p a r t i c u l a r much of t h e p r a c t i c e c o n c e r n i n g pro#:r d u r e s c a l l e d by v a l u e o r by nanie, and f u n c t i o n p r o c e d u r e s . For some d e t a ~ i s of such a s i m u l a t i o n we r e f e r t o C6
1.
5.---
Semantics. As o u r p o i n t of view we s e l e c t r e l a t i o n a l s e m a n t i c s , T h i s means t h a t s e m a n t i c r e s u l t s on a program n ( w i t h s t a t e s p a c e R) a r e g i v e n i n t h e Lorm of a b i n a r y r e l a t i o n on Q. I f we s a y t h a t T s a t i s f i e s t h e r e l a t i o n P t h e n t h e i n t e r p r e t a t i o n i s a s f o l l o w s : f o r e v e r y p a i r w E R , w ' E Rwhich a r e such t h a t w and w ' can be t h e i n i t i a l and f i n a l v a l u e of t h e s t a t e ,
- . - ---
---
P ( u , , L ' ) ; S t r u e ( b u t we do n o t r e q u i r e t h e c o n v e r s e ) . We s h a l l a l s o e x p r e s s
t h i s by s a y i n g t h a t P p r e s e n t s i n f o r m a t i o n on T . We do n o t r e q u i r e t h i s i n -
- --
f o r m a t i o n t o b e a s complete a s p o s s i b l e : w e - j u s t t r y t o work w i t h P ' S which have a s i m p l e form, a s long a s t h e y a r e adequate. One of t h e advan-
t a g e s of t h i s k i n d of s e m a n t i c s o v e r F l o y d ' s i n d u c t i v e a s s e r t i o n method i s t h a t we s t a t e r e s u l t s about t h e program and n o t about t h e program p l u s a number of a s s e r t i o n s s c a t t e r e d over i t . Our r e l a t i o n a l s e m a n t i c s i s s t i l l p r a g m a t i c : j u s t l i k e i n any mathematical p r o o f , we never b o t h e r about i n t e r m e d i a t e s t e p s b e i n g a s s t r o n g a s p o s s i b l e , i t i s o n l y t h e f i n a l r e s u l t t h a t c o u n t s , And r e l a t i o n a l s e m a n t i c s h a s n o t t h e s l i g h t e s t t r o u b l e w i t h n o n - d e t e r m i n i s t i c programs. I t i s p o s s i b l e t o e x t e n d t h e s t a t e s p a c e by improper e l e m e n t s s t a n d i n g f o r n o n - d e t e r m i n a t i o n and a b o r t i o n ( s e e C87,C61). We s h a l l n o t t r y t o d e s c r i b e t h a t h e r e , s i n c e i t i s a b i t awkward i n c o n n e c t i o n w i t h t h e c a r t e - s i a n p r o d u c t s t r u c t u r e of o u r s t a t e s p a c e s . Acompromise t h a t would make t h i n g s o n l y s l i g h t l y h a r d e r , i s t o e x p r e s s s e m a n t i c i n f o r m a t i o n by means
of a p r e d i c a t e Q and a r e l a t i o n R, w i t h t h e f o l l o w i n g i n t e r p r e t a t i o n : i f t h e i n i t i a l s t a t e w s a t i s f i e s Q(w)
,
t h e n t h e program e x e c u t i o n t e r m i n a t e s , and t h e f i n a l s t a t e s a t i s f i e s R(w,wT). (See C61).11.1 our AUTOMATH book we formulate relational semantics by means of a
~i " Z I P \-riqitive w . We take it to be a 3-expression, and the interpretation
i. a proposition. (Any 3-expression that has TRUE(w) as its type is a proof for our semantic assertion). The variables are
R
(a state space), n (a program) andP
(a relation onR).
The text isAs a general axiom on w we can formulate that for all 71, P,,
P
we 2 have (ifR
is fixed we write w(n,P) instead of w(~,.rr,P)):This implies monotonicity (if P
1
-t P2 then w(n,P1) -+ w(.rr,P2)). One might formulate this for infinite conjunctions too, but that seems to be un- necessary.For every primitive program and for every primitive construct we have to formulate a special semantic axiom. At least for the primitive programs and lower primitive constructs these axioms comerather naturally, and there is
no point where it is a problem how to write the axiom in AUTOMATH once we feel what it should express. We give a few examples. First the axiom for the concatenation. It says that if
P
I are relations (state space
R),
if P I presents information on the program .ii and P on n 2 , then the
1 2
1 1
boolean matrix product'1 P1
*
P 2 presents information on the concatenation of l r l and n 2 . (The boolean matrix product isN.G. d e B r u i j n
-
15-
n o t e t h a t t h e b o o l e a n m a t r i x sum d o e s t h e same t h i n g f o r t h e c o n s t r u c t -. " n 2 t 1 i . F o r c h e b i n a r y s e l e c t i o n we h a v e t h e f o l l o w i n g . If w,? a r e u s e d a s c o o r d i r i a t e s i n c a r t p r o d ( R , b o o l ) , i f P and P p r e s e n t i n f o r m a t i o n on n. 1 2 1 and TI t h e n P p r e s e n t s i n f o r m a t i o n on " i f b t h e n IT e l s e IT ( o r i n 2 " 3- -
1 - 2 t h e t e r m i n o l o g y of o u r book: b i n ~ e l e c t ( R , n . ~ , ~ ~ ) ) , where P ( w , B , w r , 8 ' ) i s 3 deCinec! as ( B = ~ ' ) A ( ( P =--
t r u e ) -+ P I ( w , B , w ' , B f ) ) ~ ( ( B = f a l s e ) + P ( u , ? , u ' , B f ) ) . 2 I n o r d e r t o d e s c r i b e t h e s e m a n t i c s of r e c u r s ( Q ) , where Q i s a program- to-program f u n c t i o n , we s t a r t from t h e f a k e program K ( f o r which we assume0
n o t h i n g b u t
IT
,TRUE), where TRUE i s t h e i d e n t i c a l l y t r u e r e l a t i o n ) . We0 2 3 now b u i l d t h e programs Q ( n )
,
Q
( r o ) ( = Q ( Q ( r 0 ) ) ) ,Q
T. . .
L e t 0 P I , P 2 ,...
b e r e l a t i o n s s u c h t h a t ( f o r n = 1 , 2 , ...) P p r e s e n t s i n f o r m a t i o n n on g n ( n 0 ) . L e t P b e s u c h t h a t f o r e v e r y p a i r w , u l t h e r e e x i s t s a n n s u c h t h a t P n ( w , u l ) -+ P ( w , o ' ) . Then P p r e s e n t s i n f o r m a t i o n on r e c u r s ( Q ) . I n t h i s p r a g m a t i c f o r m u l a t i o n we a v o i d t a l k i n g a b o u t t h i n g s l i k e S c o t t ' s minimal f i x p o i n t ( f o r which we r e f e r t o [ 2 1 ) t h u s l e a v i n g t h e way open f o r non- d e t e r m i n i s t i c c a s e s .A f t e r h a v i n g d e s c r i b e d t h e p r i n c i p l e s of s y n t a x and s e m a n t i c s i n o u r AUTOMATH book, we c a n p r o c e e d by p r o v i n g g e n e r a l lemma's, e . g . on t h e
s y n t a x and s e m a n t i c s of t h e " w h i l e do" s t a t e m e n t . From t h e r e on we c a n s t a r t d e s c r i b i n g s p e c i a l programs and t h e i r s e m a n t i c s , b u i l d i n g up a l i b r a r y of r e s u l t s i n t h e same way as a n
AUTOMATH
mathern2tic. book b u i l d s up i t s own i n f e r e n c e r u l e s , m a t h e m a t i c a l f o u n d a t i o n s , lemma's, t h e o r e m s , up t o t h e f i n a l r e s u l t s whose c o r r e c t n e s s we want t o check. It i s a l o n g way t o g o , b u t t h e f a c t t h a t n o t h i n g n e e d s t o b e done t w i c e , p r o v i d e d t h e s e t t i n g h a s b e e n c h o s e n w i t h s u f f i c i e n t g e n e r a l i t y , may h e l p u s t h r o u g h .6 . Epilogue. -
---
--
A' '0 -I:'!'ir can be used i n v a r i o u s ways f o r l o g i c and mathematics ( c f C51,
s e c t i o n 1 4 ) ; w i t h s y n t a x and s e m a n t i c s we have t h e same s i t u a t i o n . The d i f f e -
r e n c e between d i f f e r e n t ways of u s i n g t h e language l i e s mainly i n t h e u s e we make of t h e t y p i n g o p e r a t i o n . One way t o l i v e w i t h i t i s t o r e s t r i c t i t s u s e
t o a minimum, t h u s c l i n c h i n g a s c l o s e a s p o s s i b l e t o s t a n d a r d p r e d i c a t e calculLus l i u s Z e r n i e l r F r a e n l t e l s e t t h e o r y . But we can a l s o u s e t y p i n g i n a more l i b - . r a l f a s h i o n , t o t h e e f f e c t t h a t t h e i n t u i t i v e i d e a t h a t t h i n g s have
t y p e s , i s ~eLlccr;.cl i n t h e language. The p r e s e n t paper h a s been w r i t t e n i n t h i s l i b e r a l s p i r i t . The e f f e c t i s t h a t AUTOMATH'S machinery of d e f i n i t i o n a l e q u a l i t y can be used f r e e l y f o r t h e h a n d l i n g of programs, b u t on t h e o t h e r hand i t cannot be r e f u t e d any more. This means t h a t , whether we want i t o r n o t , d e f i n i t i o n a l l y e q u a l programs have t h e same s e m a n t i c s .
As long a s t h i s concerns s o - c a l l e d d e l t a - r e d u c t i o n ( i . e . t h e e f f e c t of r e p l a c i n g t h i n g s by t h e i r d e f i n i t i o n ) t h i s i s what everybody wants. On t h e o t h e r hand, i f we have t h e
6-
and n - r e d u c t i o n of t h e A-calculus, we h a r d l y know what i t means when d e a l i n g w i t h programs. I n p a r t i c u l a r we have t h e q u e s t i o n why two programs e q u i v a l e n t by6-
and n - r e d u c t i o n , should n e c e s s a - r i l y have t h e same s e m a n t i c s . The answer i s probably t h i s : A s l o n g a s t h e s y n t a c t i c and s e m a n t i c axioms have a p a r t i c u l a r form ( n o t i n v o l v i n gA-expressions) t h e n i t can b e proved t h a t i f two programs ( n o t i n v o l v i n g A-operations on t h e t y p e "program") a r e d e f i n i t i o n a l l y e q u i v a l e n t , t h i s e q u i v a l e n c e can be e s t a b l i s h e d a l r e a d y w i t h o u t A-operations on t h e t y p e
11
program". A proof of t h i s might e s t a b l i s h t h e f a c t t h a t oui l i b e r a l a t t i t u d e i s n o t j u s t e f f e c t i v e b u t a l s o i n n o c e n t .
N.G. de Bruijn
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d e Bruijn. AUTOMATH, a language for mathematics. Notes (preparedby
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Fawcett) of a series of lectures in the SGminaire de Mathgmatiques SupiSrisures, Universitg de Montreal, 1971.5. --.-" N.G. de Bruijn. Set theory with type restrictions. Colloquia Mathematica
----
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---
N.G, de Bruijn. A system for handling syntax and semantics of computer programs in terms of the mathematical language AUTOMATH. Report,Department of Mathematics, Technological University, Eindhoven, 1973.
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