SEMIPAL 2, an extension of the mathematical nottional language SEMIPAL

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SEMIPAL 2, an extension of the mathematical nottional

language SEMIPAL

Citation for published version (APA):

Bruijn, de, N. G. (1969). SEMIPAL 2, an extension of the mathematical nottional language SEMIPAL. Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1969

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N a t - i t f r , Nr 43 ( I 966-1

969)

1 1 1969.

Technological University Eindhoven Department of Mathematics

I n t e r n a l Report

SEMIPAL 2, an extension of the mathematical notational language SEMIPAL.

by N.G. de Bruijn.

1. SEMIPAL i s a simple language, o r rather a notational system f o r the abbreviation of expressions. It was described i n [21, sec.

4.3 -

4.8.

In

t h i s note we s h a l l extend t h a t language by extending the notion of the i n d i c a t o r s t r i n g s used i n SEMIPAL.

I n SEMIPAL every l i n e represents the d e f i n i t i o n of a new i d e n t i f i e r i n terms of previously introduced ones, a s f a r a s the new i d e n t i f i e r was not introduced a s a variable o r a s a primitive notion. And every l i n e has a context i n d i c a t o r , which i s e i t h e r 0 o r a previous block opener. I f the i n d i c a t o r of a l i n e i s xn, and i f , f o r every j ( I G j 4 n )

,

x

i s the

j -1

i n d i c a t o r of the l i n e whose i d e n t i f i e r i s x and i f 0 i s t h e i n d i c a t o r of

'

the l i n e whose i d e n t i f i e r i s x,

,

then (x,

,

.

. .

,x _n

)

i s called the i n d i c a t o r s t r i n g of the given l i n e .

With t h i s d e f i n i t i o n of the indicator s t r i n g s , the following property holds. I f ( x

,

x _n

)

and (y

,...

,Ym) a r e the i n d i c a t o r s t r i n g s of two

I

d i f f e r e n t l i n e s , then there i s an i n t e g e r k ( 0 G k S min(n,m)

)

such t h a t

x = yl

,...,%

= yk, and such t h a t there i s no f u r t h e r case where an x

1 i

equals a n y I n the extended form of SENIPAL, t o be discussed presently,

j '

this property no longer holds, and the indicator s t r i n g s can no longer be obtained from single indicators,

In SEMIPAL 2, a s we s h a l l c a l l t h i s version, we s h a l l write an i n d i - c a t o r s t r i n g i n f r o n t of every l i n e , and no longer a single indicator,

.

2. In order t o f a c i l i t a t e our discuasion, we number the l i n e s by i n t e g e r s

l , 2 , J , . . . ; the m-th l i n e i s called Am, the i n d i c a t o r s t r i n g of A _{i s pm,}

m

the i d e n t i f i e r p a r t of i s am, the d e f i n i t i o n p a r t of i s

hm.

The order i n the i n d i c a t o r s t r i n g s i s , a s i s usually done i n s t r i n g s , referred t o a s

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l e f t t o r i g h t , with l e f t

<

r i g h t .

The following r u l e s define SENIF'AL 2 uniquely.

( i )

am

i s e i t h e r the symbol

-,

o r the symbol PN, o r an expression (see sec. 3).

(ii) a i s c a l l e d a block opener i f bk =

-.

m

(

i i i ) p _m i s e i t h e r 0 or a l i n e a r l y ordered set of block openers taken from the sequence ( a

,...,

a

).

The order i n pm need not be the same a s

1 m-1 i n ( a

,...,

a

).

1 m-1

(id

If a . occurs i n pm, then p i s a subset of p

.

Moreover, the order

J j m

i n p

.

U { a .

)

i s preserved i n p ( t h e order i n p U {aj

]

i s defined by taldng

J J m j

the e x i s t i n g order i n p and requiring b

<

a f o r a l l b E p

.).

j

'

j J

(v) I f 6 i s an expression, then the p a i r ( m , 6

)

i s a "valid pair"

m m

(see sec.

3).

3-1 The no t i o n "expression" ( o r "parentheses expression"

)

i s defined by recursion: i f p i s an i d e n t i f i e r then p i s an expression; i f p i s an i d e n t i f i e r and i f C

,

. . .

,C a r e expressions, then P(C

,

. .

.

,C

)

i s an

1 k 1 k

expression.

3- 2 We s h a l l a l s o define the notion of "valid p a i r t t by recursion. The l e t t e r s m , j ,n,k w i l l stand f o r integers.

( i ) I f a .

E

%

(whence a i s a block opener), then (m , a .

)

is a v a l i d

J j J

p a i r .

( i i ) Assume t h a t 1 --( j

<

m , and t h a t a . i s not a block opener. Let n

J

a be the number of elements of p l e t k be an i n t e g e r s a t i s f y i n g 0 _,₍k _,₍n.

3'

Let C,

,

.

,Ck be such t h a t (m,C

)

, ,

( m ,

)

a r e v a l i d pairs. And assume

1 k

t h a t the f i r s t n - k e n t r i e s of p occur i n p

.

Under these circumstances

j m

( m , ~ ) i s a l s o a v a l i d p a i r , where

These descriptions a r e t o be modified i n the obvious way i f k = n , o r k = 0, o r both.

4. Examples. The SEMIPAL book i n [2], sec. 4.1 (categories t o be omitted) can be transformed a t once i n t o the following SEMIPAL 2 book:

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n a t r e a l prod n power Y d

The following example i s a line-by-line t r a n s l a t i o n of

a

SEUIPAL book:

--

Remark -

j

1

.

In some canes (see

f;

:= PN)'we define a thing i n terms of what seems t o be

independent variables, i n other eases b i k e q

s-

f(w)) the variables seem t o be l e s s

independent. There does not seem t o be much of a reason to admit the l a t t e r possi

b i l i t y , but

i t

w i l l become s i g n i f i c a n t i f we wish t o a t t a c h a category t o each

variable, l i k e we do i n

PAL.

I n that case, the categary of a variable w m a y de-

pend on the variables occurring i n the indicator s t r i n g of the l i n e where w i s

introduced.

Remark 2.

It

ma;y

help the

reader

t o write the i d e n t i f i e r s (apart from the

block

openers) a s functions of the variables i n the indicator string:

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Remark

3.

Note that the variable w (see above) has

x

i n i t s i n d i c a t o r s t r i n g , and t h a t t h i s has the e f f e c t t h a t

x

has to occur i n every i n d i c a t o r s t r i n g containing w.

5.

Completion.

If ( k , ~ ) i s a v a l i d

mir,

then we define the %ompletion" + ' Z 8 of C.

(cf. [2], see 4.6).

(1) I f

C

i s a single block opener, then

C

I = X

( 2 ) Let CE a .(C1

,

".

.Ck) ( f o r notation see sec. 1

)

,

and l e t bl

,

.

. .

,bn bc

J

-

the f i r s t n-k e n t r i e s of p Then

3"

with obvious modifications i f k = n, o r k = 0, o r both.

6.

Normal form.

Let ( k , ~ ) be a valid pair. We shall give a recursive d e f i n i t i o n of the "normal form" of C. It w i l l become an expression C* that again has t h e property t h a t (k,C*) i s a valid pair.

( i ) I f C i s a single block opener;then 'IC* = C.

(ii) I f C i s not a single block opener, we f i r s t form the_@ompletion.

- -

XI

of C. Let

(possibly with n = 0).

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If 6 . i s an expression, then we first produce the normal f o m 6? of

.J J

6 Let pj =

(u,~..

.

, U

)

be the indicator s t r i n g of t h e j-th l i n e , The ex- n

pression 6* may contain u a t various places, i t may contain u

a t

various

3 1 2

"

places, etc. We now replace i n

W

each of these u l l s by

7 ,

each of the u p ' s J

by

P,

etc.

h his

substituion r e f e r s only to t h e w, I s , u2's,.

. .

that were

2

o r i g i n a l l y present i n 6*, and not t o those u . % tthat e n t e r i n t o t h e formula

J 1

since they occurred i n

y,

3,.

.

).

This s u b s t i t u t i o n procedure leads t o what we c a l l

P.

I f both ( k , ~ )

and

(m,C) are v a l i d , then t h e normal form evaluated with reference t o k is i d e n t i c a l to the one evaluated with reference t o m.

7.

Every SEMIPAL book A turns i n t o a SEMIPAL 2 book AQf we a t t a c h t o every l i n e an i n d i c a t o r s t r i n g obtained from t h e indicators i n t h e way described i n sec. I . The normal form of an expression i n A (normal form a s defined f o r

SEMIPAL

i n [2, sec.

4.71)

w i l l be i d e n t i c a l to the normal form of the same expression a s defined i n SENIPAL 2 by means of sec.

6

above,

8. Let A be a c o r r e c t SEMIPAL 2 book.

An

expression C i s c a l l e d a A-expression i f there i s a k such that (k, C) i s a v a l i d p a i r (with respect t o A ) .

Every A-expression has a uniquely defined normd form. Two A-expressions a r e c a l l e d d e f i n i t i o n a l l y equivalent i f they have the same normal form.

I n order t o t e s t d e f i n i t i o n a l equivalence

i t

i s not always necessary t o evaluate the normal forms, One of the devices t h a t w i l l enable us t o reduce the amount of work is the "reduced i n d i c a t o r s t r i n g " , t o be discussed presently.

9,

Reduced i n d i c a t o r strings,

I f 6 . i s an expression, we define the reduced i n d i c a t o r s t r i n g pw as

J j

the s t r i n g obtained from y by d e l e t i n g a l l variables that do no$ occur i n

j

the normal form of 6

j'

A correct book can sometimes be made incorrect i f we replace a l l p !s

N J

by

p,

Is,

In

the

f i r s t place

we

night have

t o

omit

s u b e q r e s s i o n s whose

J

p o s i t i o n i n an expression r e f e r s t o a variable omitted from t h e corres- ponding i n d i c a t o r s t r i n g . Secondly we should bear i n mind t h a t replacing

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pm by

pi

might endanger the v a l i d i t y of rule ( i v ) of sec. 2.

10. We can extend PAL and AUTOMA.TH (deseribed i n [l

1,

[ 2 ] ) t o

PBL

2 and ATJTOMATH 2 by means of t h e same r u l e s on indicator s t r i n g s as d i m cussed f o r the case of SEMIPAL i n t h i s note.

AUTOMBTH 2 has some obvious advantages over ADTOMATH. It can be e a s i e r t o write, and may require a smaller number of l i n e s , On the other hand, d i s - advantages of AUTOMATH 2 are:

( i ) Writing indicator s t r i n g s (and reduced indicator s t r i n g s ) can give more trouble than writing single indicators.

( i i ) We loose the easy s t r u c t u r e of a book, a s i l l u s t r a t e d by the " t r e e of knowledge".

Even i f we write AUTOMATH instead of AUTOMATH 2.,

i t

may be very useful t o evaluate t h e reduced indicator s t r i n g f o r every l i n e , This can of course be done by an AUTOMATH processor. By means of these reduced indicator s t r i n g s , the processor can economize considerably on the amount of checking

i t

has t o do.

References :

1. N. G. de Brui jn : AUTOMATH, a language f o r mathematics.

Report 68-WSK-05 (1 968) Technological University Eindhoven, The Netherlands.

2, N.G. de Bruijn .: The mathematical language AUTOMATE, i t s usage, and some of i t s extensions.

To be published i n the Proceedings of the Symposium on Atit omatic Demonstration

(IRIA,

Versailles, Decem- ber 1 968).