The Automath mathematics checking project
Citation for published version (APA):Bruijn, de, N. G. (1974). The Automath mathematics checking project. In P. Braffort (Ed.), APLASM 73 Symposium dórsay sur la manipulation des symbols et l'utilisation d'APL, Orsay, December 1973 (Vol. 1). Université Paris Sud.
Document status and date: Published: 01/01/1974
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P l l z a The AUTOMATH Mathematics Checking P r o j e c t
by N.G. de B r u i j n .
This l e c t u r e 1 ) w i l l d e s c r i b e t h e AUTOMATH p r o j e c t , b u t w i l l be ~ r e t t y vague about t h e n a t u r e of AUTOMATH language i t s e l f . We r e f e r t o [ 1
I
a n d C 21
f o r d e t a i l s about t h e d e f i n i t i o n of t h e language; h e r e we s h a l l mainly con- c e n t r a t e on m o t i v a t i o n .One s o u r c e of confusion should be t a k e n away a t t h e s t a r t : AUTOMATH i s a mathematical language and n o t a programming language. N e v e r t h e l e s s t h e two k i n d s of languages have much i n common, and can c e r t a i n l y p r o f i t from each
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o t h e r ' s i d e a s .
The AUTOMATH p r o j e c t was conceived i n 1966. The i d e a was t o develop a system of w r i t i n g e n t i r e mathematical t h e o r i e s i n such a p r e c i s e f a s h i o n t h a t v e r i f i c a t i o n of t h e c o r r e c t n e s s can b e c a r r i e d o u t a u t o m a t i c a l l y , y e t keeping, s t e p by s t e p , c o n t a c t w i t h o r d i n a r y mathematical p r e s e n t a t i o n . A s i m i l a r i d e a p o s s i b l y e x i s t e d i n t h e mind of Leibniz b u t d i d n o t develop a t t h a t time s i n c e t h e r e was n e i t h e r i n t e r e s t n o r e x p e r i e n c e i n f o r m a l ' l i n g u i s t i c s .
The i d e a i s t o make a language such t h a t e v e r y t h i n g we w r i t e i n i t i s i n t e r p r e t a b l e a s c o r r e c t mathematics, a s long a s our w r i t i n g is s y n t a c t i - c a l l y c o r r e c t ( i n c l u d i n g c o r r e c t r e f e r e n c e s t o t h i n g s t h a t have been s a i d b e f o r e ) . T h i s may i n c l u d e t h e w r i t i n g of a v a s t mathematical e n c y c l o p a e d i a , t o which everybody ( e i t h e r a human o r a machine) may c o n t r i b k e what h e l i k e s , and any c o n t r i b u t i o n t h a t h a s been accepted s y n t a c t i c a l l y can be s a f e l y used by o t h e r s . The i d e a of a k i n d of f o r m a l i z e d encyclopaedia was a l r e a d y con-
ceived and p a r t l y c a r r i e d out by Peano around 1900, b u t t h a t was s t i l l f a r from what we might c a l l a u t o m a t i c a l l y r e a d a b l e .
The t a s k of checking s y n t a c t i c c o r r e c t n e s s can be l e f t t o a computer. S i n c e t h e checking only concerns t h e q u e s t i o n whether t h e t e x t h a s been
w r i t t e n according t o t h e r u l e s , we have t o admit t h a t t h e t a s k of t h e checking i s as human a s t h e t a s k of t h e w r i t i n g . Yet, t h e i d e a of a computer i s i n t h e background i n o r d e r t o s e t t h e s t a n d a r d s : what a computer cannot do, cannot b e c a l l e d automatic. Moreover, computers have some p r a c t i c a l advantages over
humans. They t a k e a l l d e t a i l s s e r i o u s l y and never g e t bored. The human w r i t e r i s i n c l i n e d t o change d e t a i l s now and t h e n , and t o b e l i e v e t h i s has no conse- quences elsewhere; t h e computer i s m e r c i l e s s i n t h i s r e s p e c t .
x
1 ) The lecture w a s given i n the S y m o s i m A?LAS?I, nrsap, December 1973, 'and t h e t e x t was p r e s e n t e d i n t h e Proceedings of t h a t symposium (ed. P. ~ r a f f o r t ) . ( T h e o t h e r c o n t r i b u t i o n s of members of t h e AUTOMATH group were C61, C121 and C 1 6 i . ) The " ~ ~ i l o ~ u e " and some a d d i t i o n s t o t h e l i s t of r e f e r e n c e s were w r i t t e n i n November 1981.
The speed of t h e computer i s h a r d l y a problem, s i n c e we do n o t e x p e c t i t t o do much more than t h e human w r i t e r can do. The problems t h e r e a r e , concern s t o r a g e o r g a n i z a t i o n i n t o d a y ' s computer systems. The mathematician has a subdivided memory: f a s t and slow p a r t s of t h e b r a i n , t h e s h e e t he i s working on, h i s own r e c e n t n o t e s , t h e books on h i s d e s k , t h e i n s t i t u t e l i b r a r y , and f i n a l l y o t h e r l i b r a r i e s he has t o depend on i f h i s i n s t i t u t e l i b r a r y f a i l s . S i m i l a r l y t h e computer system's memory c o n t a i n s f l i p - f l o p s , c o r e memory, drum, d i s k s , t a p e , e t c . Both i n t h e human and i n t h e machine-case, t h e u s e r h a s problems t o d e c i d e what t o s t o r e where. I n t h e c a s e of t h e computer, i t i s q u i t e p o s s i b l e t h a t t e c h n o l o g i c a l improvements of f a s t memory w i l l p u t an end t o t h e s e s t o r a g e d i f f i c u l t i e s i n t h e f u t u r e .
One of t h e aims of t h e AUTOMATH p r o j e c t was, r i g h t a t t h e s t a r t , t o make something of a u n i v e r s a l n a t u r e . This i s a d i s a d v a n t a g e over systems t h a t t r y t o t a c k l e s m a l l p o r t i o n s of mathematics o n l y , l i k e p r o p o s i t i o n a l l o g i c , pre- d i c a t e l o g i c , e t c . The need f o r u n i v e r s a l i t y had t h e e f f e c t t h a t no claims could be made i n t h e d i r e c t i o n of theorem proving. That s u b j e c t i s s o d i f f i c u l t t h a t i t can have s u c c e s s only i n s i t u a t i o n s where problems and methods belong t o a v e r y l i m i t e d a r e a , and where language and s y n t a c t i c a n a l y s i s have-been t a i l o r e d e x a c t l y t o t h e expected s i t u a t i o n .
AUTOMATH i s a language i n which we can w r i t e books, c o n s i s t i n g of sequences of l i n e s . The s y n t a c t i c c o r r e c t n e s s of a l i n e depends on t h e p r e v i o u s l i n e s . F o r t h e time b e i n g , we a r e mainly i n t e r e s t e d i n books t h a t f o l l o w o r d i n a r y mathema- t i c a l p r e s e n t a t i o n almost l i n e by l i n e , and do n o t e x p r e s s thoughts t h e
human mathematician would n o t have.
We have t o r e a l i z e t h a t n o language can embrace a l l mathematical a c t i v i t y . Language and n o t a t i o n may have an i n f l u e n c e on t h e f o r m a t i o n of i d e a s , b u t t o r e q u i r e t h a t t h e f o r m a t i o n of i d e a s should always t a k e p l a c e i n a r i g i d language would mean k i l l i n g mathematics. I n p a r t i c u l a r , t h e r e i s n o t much chance of p u t t i n g
g e o m e t r i c a l o r p h y s i c a l i n t u i t i o n i n an o p e r a t i o n a l formal framework. On t h e p u r e l i n g u i s t i c s i d e , i t seems hard t o r e p l a c e i l l u m i n a t i n g , n a t u r a l language by
something more f o r m a l . The p s y c h o l o g i c a l f u n c t i o n of mathematical u n d e r s t a n d i n g i s u s u a l l y more ( b u t sometimes l e s s ) t h a n checking c o r r e c t n e s s : i t can be a f e e l i n g of peace of mind t h a t s e e s a mathematical s i t u a t i o n i n harmony w i t h s i t u a t i o n s t h a t have become f a m i l i a r a l r e a d y . P a r t of t h a t k i n d of t h i n k i n g i s supposed t o be subconscious.
Even i f we do n o t r e q u i r e complete f o r m a l i z a t i o n , b u t j u s t r e q u i r e dependable mathematics a t every s t e p , we would k i l l p a r t s of mathematics, a t l e a s t i n t h e s t a g e of e a r l y development. Important p a r t s of mathematics have been explored on t h e b a s i s of some fundamental e r r o r s , o r a t l e a s t v e r y s e r i o u s gaps. Without knowing what b e a t i f u l t h i n g s t h e r e were a t t h e o t h e r s i d e , one would n e v e r have had t h e energy ( o r t h e methodology) t o r e p a i r t h e e r r o r o r t o b r i d g e t h e gap. I n some c a s e s i t - has been v e r y lucky f o r mathematics t h a t one d i d n o t have t h e i n t e l l e c t u a l f a c i l i t i e s t o d i s c o v e r t h a t t h e r e was an e r r o r o r a gap a t a l l , u n t i l a f t e r one had ex- t e n s i v e e x p e r i e n c e w i t h t h e m a t e r i a l beyond.
Let u s t r y t o d e s c r i b e t h e p r o d u c t i o n of completely f o r m a l i z e d mathe- m a t i c s a s a k i n d of assembly l i n e . I f we t h i n k of an AUTOMATH book a s a f i n a l g o a l , we have t h e f o l l o w i n g phases: ( i ) mathematical i d e a s , ( i i ) formal d e f i n i t i o n s and p r o o f s , ( i i i ) v e r y p r e c i s e d e t a i l e d p r e s e n t a t i o n of t h e s e , ( i v ) a book i n a i n t e r m e d i a t e language, (v) an AUTOMATH book.
We have i n s e r t e d ( i v ) s i n c e AUTOMATH i s n o t s o easy t o w r i t e , because of i t s u n i v e r s a l i t y . Most mathematical m a t e r i a l concerns o n l y a s m a l l p a r t of mathematics, w i t h w e l l - e s t a b l i s h e d t r a d i t i o n s about s h o r t n o t a t i o n s and
s h o r t ways of s a y i n g t h i n g s . T h e r e f o r e we have i n c l u d e d books of t y p e ( i v ) , i n what we may c a l l a problem-oriented language.
What k i n d of p e r s o n e l l do we need on t h e assembly l i n e ? I n o r d e r t o produce ( i ) we need t h e Great Mathematician. (Here w e do n o t mean a s p e c i a l
c l a s s of mathematician: every mathematician can be g r e a t now and t h e n ) . I n o r d e r t o g e t from ( i ) t o ( i i ) we need t h e Good Mathematician, who m a s t e r s t h e f i e l d and i t s t e c h n i q u e s .
The phases ( i ) and ( i i ) have, of c o u r s e , n o t h i n g t o do w i t h AUTOMATH; i t i s t h e f i e l d of s t a n d a r d mathematical p r a c t i c e .
I n o r d e r t o p a s s t h e p a r t l y f i n i s h e d product from ( i i ) t o ( i i i ) we need a Competent Mathematician. He s t i l l ' h a s t o know t h e s u b j e c t , a t l e a s t h e
- - .
should be a b l e t o master t h e s h o r t h a n d t r a d i t i o n s i n t h e s u b j e c t .
The t r a n s i t i o n s from ( i i i ) t o ( i v ) , from ( i v ) t o ( v ) , and t h e f i n a l checking of (v)
,
can be l e f t t o cheap labour. Much of t h i s , c e r t a i n l y t h e checking of ( v ) , can be l e f t t o v e r y cheap l a b o u r i n t h e form of a computer.There a r e many t h i n g s t h a t a u n i v e r s a l language l i k e AUTOMATH might achieve. S e v e r a l of t h e s e a r e , by themselves, n o t s u f f i c i e n t as a m o t i v a t i o n f o r t h e AUTOMATH p r o j e c t , b u t t h e i r t o t a l i t y seems important enough f o r going i n t o some e f f o r t . Let us c l a s s i f y t h e o b j e c t i v e s i n t o two groups: checking and u n d e r s t a n d i n g
.
-
-
- When s a y i n g "checking", t h e f i r s t t h i n g t h a t may come i n t o o u r mind i s t h e checking of long t e d i o u s p f o o f s , where t h e c h a i n i s as weak a s i t s weakest l i n k , and where, q u i t e o f t e n , t h e d e p e n d a b i l i t y of t h e proof i s n o t s u p p o r t e d by i n t u i t i o n o r e x p e r i m e n t a l evidence. I n p a r t i c u l a r one might e x p e c t t h i s s i t u a t i o n i n complicated c o m b i n a t o r i a l problems where a l a r g e number of c a s e s and subcases have t o be checked. Under t h i s heading wea l s o f i n d problems concerning t h e semantics of computer programs. The number of elementary s t e p s t o be t a k e n , and t h e amount of a d m i n i s t r a t i o n t o be c a r r i e d o u t , may be s o l a r g e t h a t human methods become v e r y u n r e l i a b l e . It i s i n t h i s f i e l d t h a t w e have t o t h i n k a l s o about t h e problems of t e a m w o r k and of man-machine c o o p e r a t i o n . Both r e q u i r e a v e r y r i g i d cammunication system. It seems t o be worthwhile t o work i n t h i s f i e l d , s i n c e tremendous sums of money a r e s p e n t on computer s o f t w a r e , and i t i s of q u i t e some i n t e r e s t t o know what i s r e l i a b l e and what i s n o t .
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Let u s now look a t o b j e c t i v e s t h a t f a l l under t h e heading "understanding". F i r s t we remark t h a t t h e mere f a c t of having a f i x e d well-defined language f o r mathematics i s an advantage a l l by i t s e l f . I t e n a b l e s u s t o s u b d i v i d e
mathematical d i s c u s s i o n i n t o ( i ) s a y i n g t h i n g s i n t h e l a n g u a g e , ( i i ) d i s c u s s i n g how t h i n g s a r e s a i d i n t h e language, and ( i i i ) connecting t h i n g s s a i d i n t h e
language w i t h t h i n g s i n a n o t h e r w o r l d , l i k e s t a n d a r d mathematics, p h y s i c a l r e a l i t y , e t c . We might r e f e r t o ( i i ) a s t o "metalanguage" and t o ( i i i ) a s " i n t e r p r e t a t i o n " .
Most mathematicians do n o t have a c l e a r i d e a of t h e f o u n d a t i o n of t h e i r awn mathematics.'This may p a r t l y be t h e f a u l t of t h e l o g i c i a n s who, f i n d i n g s o many i n t e r e s t i n g t e c h n i c a l problems i n t h e i r f i e l d , n e g l e c t e d t h e i r o r i g i n a l
m i s s i o n , t o b u i l d a b a s i s f o r o t h e r s . Many mathematicians have a vague i d e a t h a t p r e d i c a t e l o g i c p l u s s e t t h e o r y form a complete b a s i s f o r t h e i r own a c t i v i t y , b u t i f they look i n t o t h e s e f i e l d s they s e e t o t h e i r s u r p r i s e t h a t l o g i c and s e t t h e o r y c o n s i s t of mathematical a c t i v i t y too! I n s t e a d of f i n d i n g a f o u n d a t i o n of t h e mathematical p a t t e r n of aXi0ms-definitions- i n f e r e n c e rules-proofs-theorems, they seem t o f i n d t h e same p a t t e r n
a g a i n , a l l over t h e p l a c e . What i s l a c k i n g ,
i s
a good language. A c t u a l l y , i n AUTOMATH t h e s e t h i n g s become q u i t e c l e a r . The language c o n t a i n s h a r d l y anything t h a t can b e c a l l e d l o g i c , and once we have t h e language and s a y t h i n g s c o r r e c t l y ( i n t h e s y n t a c t i & l s e n s e ) t h e q u e s t i o n of what a r e axioms, i n f e r e n c e r u l e s , d e f i n i t i o n s , assumptions, t h e o r e n s , e t c . i s j u s t m e t a l i n g u a l and i n t e r p r e t a t i o n a l . It h a s n o t t h e s l i g h t e s t i n f l u e n c e on t h e r e s u l t s of an AUTOMATH book whether we c a l l a t h i n g a d e f i n i t i o n o r a theorem o r anything e l s e ; i t i s j u s t c o r r e c t a s i t s t a n d s .Another o b j e c t i v e i n t h e d i r e c t i o n of "understanding" i s a n a l y s i s of complexity. Some t h i n g s a r e more d i f f i c u l t t h a n o t h e r s , and a complete formal p r e s e n t a t i o n i s a b l e t o show t h i s . It i s p o s s i b l e t o c l a s s i f y p i e c e s of
mathematics a s t o t h e i r "depth". The mathematics of t h e 19-th c e n t u r y was c e r t a i n l y d e e p e r t h a n t h a t of t h e 18-th c e n t u r y . I n a somewhat s t y l i z e d way one can say t h i s : i n t h e 18-th c e n t u r y one could t a l k about f u n c t i o n s
one had e x p l i c i t l y c o n s t r u c t e d , b u t one could not s a y "let f b e a f u n c t i o n " , s i n c e t h e word " f u n c t i o n " was m e t a l i n g u i s t i c . I n t h e same s t y l i z e d f a s h i o n one might s a y t h a t 18-th c e n t u r y mathematics can be expressed by means of PAL, which i s t h e sublanguage of AUTOMATH we g e t by l e a v i n g t h e lambda c a l c u l u s
I n t h i s c o n n e c t i o n i t may be remarked t h a t AUTOMATH v i o l a t e s t h e h i s t o - r i c a l o r d e r . Already i n PAL, t h i n g s l i k e "proofs" a r e t r e a t e d i n t h e same way a s t h i n g s l i k e "numbers", whereas even i n t h e second p a r t of t h e 20-th
c e n t u r y most mathematicians f e e l t h a t a "proof" i s a m e t a l i n g u i s t i c n o t i o n a r d t h a t a "number" i s an "object". The i d e a s about what i s an o b j e c t and what i s n o t , a r e u s u a l l y vague. The d i f f e r e n c e between o b j e c t s and non-objects i s a p p a r e n t l y p a r a l l e l t o t h e d i s t i n c t i o n between language and metalanguage; one f e e l s t h a t an o b j e c t i s something we can denote by a symbol. Many people b e l i e v e t h a t it i s b e t t e r t o t a l k about s e t s t h a n about p r e d i c a t e s . Rather
t h a n s a y i n g t h a t x s a t i s f i e s t h e p r e d i c a t e P , t h e y form t h e s e t of a l l t h i n g s s a t i s f y i n g t h a t p r e d i c a t e , and t h e n say t h a t x belongs t o t h a t s e t .
U s u a l l y t h i s i s caused by f e a r f o r p r e d i c a t e s , which a r e n o t b e l i e v e d t o be o b j e c t s .
Coming back t o "understanding": i t has o f t e n been s a i d t h a t mathematics i s t a u g h t by i n t i m i d a t i o n and l e a r n e d by i m i t a t i o n . The o n l y way t o f i n d o u t how much t r u t h i s i n t h i s , i s t o c o d i f y e v e r y t h i n g i n a v e r y r i g i d language.
Under t h e heading "understanding" one may f i n a l l y p u t t h e i n f l u e n c e t h a t every new n o t a t i o n (provided i t has some power) h a s on t h e development
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of mathematics whether one asked f o r such an i n f l u e n c e o r n o t . *
Apart from "checking" and "understanding" t h e r e a r e some advantages i n t h e f a c t t h a t machines can p r o c e s s t h e mathematics we produce. For example, we can imagine we g i v e a book on a n a l y t i c number t h e o r y t o a machine, saying: "I am i n t e r e s t e d i n t h e Prime Number Theorem only. P r i n t e v e r y t h i n g t h a t i s needed f o r t h i s theorem and omit e v e r y t h i n g e l s e " (Some *people s a y t h a t E. Landau was such a machine; he wrote h i s books t h a t way). O r we can say:
" P r i n t Theorem 325, and a l l d e f i n i t i o n s needed f o r r e a d i n g what i t s a y s , s t a r t i n g from s c r a t c h " . I n t h i s c a s e a l l p r o o f s w i l l b e o m i t t e d too.
I n o r d e r t o show a glimpse of how mathematical r e a s o n i n g i s expressed i n an AUTOMATH book, we have t o e x p l a i n a l i t t l e about t h e language. F i r s t we n o t e t h a t books a r e organized i n t o n e s t e d %locks" of l i n e s . The f i r s t
l i n e of a b l o c k has a s p e c i a l form. I t s i n t e r p r e t a t i o n is t h a t we i n t r o d u c e e i t h e r a v a r i a b l e t h a t can be used i n s i d e t h e b l o c k , o r an assumption v a l i d throughout t h e block.
The l i n e s a l l have t h i s form:
"In t h e c o n t e x t A t h e name B i s d e f i n e d a s C and i s of t y p e D
".
Here B i s a new i d e n t i f i e r , n o t used i n p r e v i o u s l i n e s . C and D a r e e x p r e s s i o n s i n terms of o l d i d e n t i f i e r s , w i t h t h e u s e of a few c o n n e c t i v e s l i k e b r a c k e t s , p a r e n t h e s e s , commas, e t c . Some l i n e s l t h e b l o c k openers) have j u s t a b a r (-) i n s t e a d of C ( i n t e r p r e t a t i o n : a v a r i a b l e i s i n t r o d u c e d by g i v i n g i t a name and s a y i n g what t y p e i t h a s ) . I n e a c h l i n e , t h e A i s a s t r i n g of p r e v i o u s l y
i n t r o d u c e d block opening i d e n t i f i e r s . The A-parts of t h e l i n e s s e r v e t o i n d i c a t e t h e b l o c k s t r u c t u r e of t h e book, i n d i c a t i n g f o r each l i n e t o which b l o c k s i t be longs.
Sometimes t h e C i s n o t an e x p r e s s i o n , b u t t h e s p e c i a l symbol: "PN". The l i n e s where t h e C-part i s PN, s e r v e t o i n t r o d u c e p r i m i t i v e n o t i o n s , which a r e n o t
can be used from t h e n on. A PN-line i s n o t a block opener, i t j u s t o c c u r s somewhere i n s i d e a block.
We have t o mention t h e p o s s i b i l i t y t h a t t h e D-part of a l i n e i s n o t an e x p r e s s i o n , b u t j u s t t h e symbol "type1'. Such l i n e s i n t r o d u c e a new t y p e , e i t h e r by d e f i n i t i o n , o r a s a p r i m i t i v e , o r a s a v a r i a b l e .
T h i s d e s c r i b e s v e r y roughly t h e s t r u c t u r e of t h e language
PAL,
mentioned e a r l i e r i n t h i s l e c t u r e . The languages of t h e AUTOMATH f a m i l y a r i s e from PAL i f we add some kind of typed lambda c a l c u l u s . We s h a l l n o t d i s c u s s t h i s h e r e .
L e t u s s a y a few t h i n g s a b o u t ' i n t e r p r e t a t i o n . F i r s t , t h e c o n t e x t
i n d i c a t i o n ( t h e A-part) i s a t h i n g t h a t i s u s u a l l y n o t e x p l i c i t l y s t a t e d i n mathematics. P a r t s of i t can be d e r i v e d from t h i n g s l i k e s u b d i v i s i o n i n t o
c h a p t e r s and s e c t i o n s , o t h a r p a r t s can be t r a c e d by c a r e f u l r e a d i n g of t h e p r e v i o u s t e x t , The B-part h a s t h e u s u a l i n t e r p r e t a t i o n o f - t h e name given t o a new o b j e c t we form o r assume. The i n t e r p r e t a t i o n of t h e C and D p a r t s i s a s d e s c r i b e d by s a y i n g t h a t B i s d e f i n e d by C and i s of t y p e D. L e t us i n t r o - duce t h e symbol
E
f o r t h i s t y p i n g : CE
D
:
I n n a t u r a l language we s a y t h i n g s l i k e " 3 i s a number", b u t s i n c e t h e word "is" i s used f o r many d i f f e r e n t t h i n g s , we p r e f e r "3 E number".Some of t h e t y p e s we s h a l l b e u s i n g , have s e t - l i k e i n t e r p r e t a t i o n s . I n s t e a d of " 3 P number" one might t h i n k of 3 E S , where S i s t h e s e t of a l l numbers, but we should be c a r e f u l n o t t o confuse E and € . I n AUTOMATH, t h e t y p e of a t h i n g C ( i . e . t h e D w i t h C €
D)
i s u n i q u e l y determined, and can be e v a l u a t e d by means of an algorithm. With 3 E S t h i s i s n o t s o s i n c e Scan be any s e t c o n t a i n i n g 3.
Apart from t h e t y p e s w i t h s e t - l i k e i n t e r p r e t a t i o n we can have o t h e r s . The most i m p o r t a n t ones a r e t h e p r o p o s i t i o n a l t y p e s . I n l i n e w i t h t h i s k i n d of i n t e r p r e t a t i o n , t h e D-part corresponds t o a p r o p o s i t i o n , and t h e C-part t o i t s proof. We o p e r a t e on p r o o f s : i f t h e y depend on v a r i a b l e s we can s u b s t i t u t e e x p r e s s i o n s f o r t h e s e v a r i a b l e s , i n t h e same way a s t h i s i s done i n t h e c a s e of o b j e c t s depending on v a r i a b l e s . T h i s h a s t h e e f f e c t t h a t a modified proof (modified by s u b s t i t u t i o n ) i s a c c e p t e d a s a proof f o r t h e c o r r e s p o n d i n g l y modified p r o p o s i t i o n , Note t h a t t h e B-part of t h e l i n e is a name f o r t h e proof C , and n o t f o r t h e p r o p o s i t i o n D. The whole l i n e can be c a l l e d a theorem; l a t e r a p p l i c a t i o n s of t h a t theorem a r e made by means of r e f e r e n c e s t o B . Note t h a t t h e m a j o r i t y of t h e theorem l i n e s w i l l be j u s t s t e p p i n g s t o n e s l e a d i n g f i r l a l l y t o one important theorem l i n e , which
a mathematician would c a l l a theorem; he would n o t b o t h e r t o c a l l t h e o t h e r l i n e s even lemmas.
There a r e a l s o block openers w i t h p r o p o s i t i o n a l i n t e r p r e t a t i o n . These seem t o say: " l e t x be a proof f o r t h e p r o p o s i t i o n D". That i s , t h e s e l i n e s i n t r o d u c e assumptions, v a l i d throughout t h e block. And t h e r e can be l i n e s where t h e C-part i s PN. These s e r v e t o i n t r o d u c e t h e t r u t h of t h e p r o p o s i t i o n D a s an axiom. Thus we have t a k e n c a r e of t h e t h r e e t y p e s of p r o p o s i t i o n a l
l i n e s : theorems, assumptions and axioms. -
We a r e a b l e t o c r e a t e new t y p e s i f we w i s h , and we can a l s o s e l e c t i n t e r -
L
p r e t a t i o n s . For i n s t a n c e , i f we want t o make a mathematical t h e o r y of p l a n e g e o m e t r i c c o n s t r u c t i o n w i t h r u l e r and compass, we need n o t go t o t h e t r o u b l e
of coding c o n s t r u c t i o n s a s s e t s ( a c c o r d i n g t o t h e dogmatic i d e a t h a t e v e r y t h i n g i s a s e t ; f o r c r i t i c i s m s e e [ 3
1
) , b u t we can i n t r o d u c e a t y p e " c o n s t r u c t i o n "d i r e c t l y .
.
.We mention a n o t h e r case. For every s e t R , we i n t r o d u c e a t y p e "program(a)lg. I f we have C E program(R), t h a n t h e i n t e r p r e t a t i o n i s t h a t C i s a program a c t i n g on t h e s t a t e s p a c e Q. By means of PN-lines we i n t r o d u c e p r i m i t i v e programs and p r i m i t i v e ways t o c o n s t r u c t b i g g e r programs from s m a l l e r components. I n o t h e r words, we d e s c r i b e t h e s y n t a x of a programming language i n t h e same book where we have t h e l o g i c and t h e mathematics ( t h e r e is n o e s s e n t i a l d i f f e r e n c e between t h e l a t t e r two). Next we can develop, i n t h e same book, axioms about t h e
s e m a n t i c s of t h e arogramming language p r i m i t i v e s . And, s t i l l i n t h e same book, we can d e r i v e l o g i c a l theorems ( d e r i v e d i n f e r e n c e r u l e s ) , mathematical theorems, s e m a n t i c theorems, s p e c i a l programs, and s e m a n t i c r e s u l t s on t h o s e programs. The v a r i o u s p a r t s can be interwoven. For example, t h e r e can b e a mathematical t r e a t m e n t of t h e g.c.d i n a n u m b e r - t h e o r e t i c a l s e t t i n g , a d e s c r i p t i o n of a computer program f o r f i n d i n g t h e g.c.d., and a proof t h a t t h e e x e c u t i o n of t h e computer program t e r m i n a t e s and produces t h e v a l u e of t h e number-theoretical f u n c t i o n g.c.d. (For e x p l i c i t and e x t e n s i v e p r o p o s a l s f o r s e m a n t i c a l t r e a t m e n t of ALGOL-like languages, s e e
C
41
). It would n o t do any harm t o w r i t e s y n t a x and semantics of two d i f f e r e n t programs i n one book, and t o prove, i n t h a t book, t h a t programP,
i n language Q, h a s t h e same semantic e f f e c t s a s program P i n2
language Q2. P r o o f s of t h i s k i n d can be l o n g , t e d i o u s and y e t i m p o r t a n t , and may be t y p i c a l c a s e s where a u t o m a t i c v e r i f i c a t i o n i s adequate.
When r e l a t i n g a book l i k e t h i s t o t h e o u t s i d e world, t h e r e i s q u i t e an amount of i n t e r p r e t a t i o n . A s long a s we have no f u r t h e r formalism t o handle i n t e r p r e t a t i o n , we have t o "convince" o u r s e l v e s t h a t t h e p r i m i t i v e s (whether
l o g i c a l , mathematical, s y n t a c t i c a l o r s e m a n t i c a l ) e x p r e s s what t h e y a r e
supposed t o mean i n t h e o u t s i d e world. And we have t o "convince" o u r s e l v e s t h a t t h e i n t e r p r e t a t i o n s of t h e ~ r i m i t i v e s g e n e r a t e i n t e r p r e t a t i o n s of f u r t h e r m a t e r i a l , and t h a t i n t e r p r e t a t i o n s of t h e f i n a l r e s u l t s can be o b t a i n e d w i t h o u t b o t h e r i n g about the i n t e r p r e t a t i o n of t h e i n t e r m e d i a t e p i e c e s of t h e book, l y i n g between p r i m i t i v e s and f i n a l r e s u l t s . And we b e l i e v e t h a t t h e f i n a l i n t e r p r e t a t i o n s a r e mathematically c o r r e c t .
-
This s i t u a t i o n w i t h computer language i n t e r p r e t a t i o n i s more complex than w i t h s t a n d a r d mathematics, b u t n o t e s s e n t i a l l y d i f f e r e n t from i t . I n t e r p r e t a t i o n always has t o r e m a h on a r a t h e r i n t u i t i v e b a s i s , a s long a s t h e " o u t s i d e world" has n o t been completely f o r m a l i z e d .
We end t h i s paper w i t h a s h o r t d e s c r i p t i o n of t h e AUTOMATH P r o j e c t Group a t t h e Department of Mathematics of t h e T e c h n o l o g i c a l U n i v e r s i t y , Eindhoven, The Netherlands. The group has been growing slowly s i n c e 1967; e a r l y 1974 i t c o n s i s t s of 4 f u l l - t i m e mathematicians ( t a k e n i n t h e s e n s e t h a t i n c l u d e s b o t h l o g i c i a n s and computer s c i e n t i s t s ) , t h r e e p a r t - t i m e mathematicians ( i n c l u d i n g t h e a u t h o r of t h i s p a p e r , who l e a d s t h e p r o j e c t ) ,
a programmer and a p a r t - t i m e punch-typist. We mention some of t h e t h i n g s t h a t have been done t h u s f a r .
( i ) Language checkers have been produced, and a r e now a v a i l a b l e i n conver- s a t i o n a l mode w i t h i n t h e framework of a time-sharing system. Text can b e f e d l i n e - b y - l i n e i n t o t h e machine, which responds w i t h i n a t most a few seconds. If t h e checker r e f u s e s t o a c c e p t t h e l i n e , i t g i v e s complete d i a g n o s t i c s , which u s u a l l y e n a b l e s t h e man i n charge t o improve t h e t e x t ( p o s s i b l y a f t e r c o n s u l t i n g , over t h e t e l e p h o n e , t h e mathematician who produced t h e t e x t ) . U n t i l September 1973, t h e computer was t h e E l e c t r o l o g i c a X 8 , a f t e r t h a t a Burroughs 6700. I n b o t h c a s e s t h e a v a i l a b l e multiprogranming systems re- q u i r e d t h e u s e of ALGOL 60 a s t h e programming language.
( i i ) T h e o r e t i c a l work on t h e languages of t h e AUTOMATH f a m i l y c e n t e r e d around problems of n o r m a l i z a t i o n , s t r o n g n o r m a l i z a t i o n and t h e Church-Rosser theorem. Almost a l l g o a l s have been achieved. For a d e t a i l e d r e p o r t on one of t h e languages (AUT-SL) we r e f e r t o [ , l l ] . We n o t e t h a t t h e r e i s some o v e r l a p w i t h work of o t h e r s ( [ 8
1
,
C
91
,
C I O
1
) who s t a r t e d t o i n t e r p r e t l b g i c i nterms of t h e typed lambda c a l c u l u s i n d e p e n d e n t l y , roughly a t t h e t i m e of t h e s t a r t of t h e AUTOMATH p r o j e c t .
( i i i ) As a kind of t e s t - c a s e , t h e work was undertaken t o t r a n s l a t e a v e r y meticulous mathematical text i n t o AUTOMATH. The choice f e l l on E.Landau'snGrundlagen d e r Analysis". The t r a n s l a t i o n , c a r r i e d o u t by
L.S. van Benthem J u t t i n g , i s about h a l f completed. It has n o t been t r i e d t o r e a r r a n g e t h e t e x t i n o r d e r t o make t h e t r a n s l a t i o n i n t o AUTOMATH
e a s i e r , b u t Landau's t e x t was followed a s p r e c i s e l y a s p o s s i b l e ( t h u s g e t t i n g a l l d i s a d v a n t a g e s and none of t h e advantages). It i s hoped t h a t t h e e x p e r i e n c e o b t a i n e d w i l l be of g r e a t h e l p i n d e c i d i n g what i n t e r m e d i a t e a u x i l i a r y language should be t a k e n f o r more g e n e r a l use. S e v e r a l p o s s i b i - l i t i e s a r e b e i n g explored a t t h e moment.
The AUTOMATH p r o j e c t depends v e r y s u b s t a n t i a l l y on f i n a n c i a l s u p p o r t by t h e Netherlands o r g a n i z a t i o n f o r t h e advancement of pure r e s e a r c h (Z.W.O.).
-
--Epilogue (added November 1981).
The p r o j e c t s t i l l e x i s t s i n 1981, although f i n a n c i a l s u p p o r t by Z.W.O. e n t i r e l y stopped around 1976. We mention a number of q u i t e e x t e n s i v e sub-projects t h a t were s u c c e s f u l l y completed in t h e meantime.
(i) L. S. van Benthem ~ u t t i A g ' s complete computer-checked t r a n s l a t i o n of Landau's book "Grundlagen d e r Analysis" ( t e x t i n C141, r e p o r t on t h e t r a n s l a t i o n i n C131.
(ii) D.T. van Daalen's e l a b o r a t e t r e a t m e n t C71 of t h e language theory.
(iii) J.T. Udding's s h o r t e r v e r s i o n CIS] of t h e theory of r e a l numbers, a l s o computer-checked.
( i v ) J. Zucker's foundation of a n a l y s i s , w r i t t e n i n h i s own v e r s i o n of t h e language (AUT-PI). H i s language i s explained i n C171. Unfortunately t h e r e has n o t been enough manpower f o r t h e c o n s t r u c t i o n of a checker f o r t h a t language,
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