On the use of bound variables in Automath

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On the use of bound variables in Automath

Citation for published version (APA):

de Bruijn, N. G. (1970). On the use of bound variables in Automath. Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1970

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Technische Hogeschool, Eindhoven. O n d e r a f d e l i n g d e r Wiskunde.

N o t i t i e 9

,

d.d. 26 november 1970.

On t h e u s e of bound v a r i a b l e s i n AUTOMATH. by N.G. d e B r u i j n .

The b a s i c s y n t a x of AUTOMATH was d e s c r i b e d i n a l i t t l e n o t e [ 2 1 . The s y n t a x of i t s e x t e n s i o n AUT-QE was n o t d e s c r i b e d t h e r e , b u t i t can be o b t a i n e d by j u s t r e p l a c i n g t h e r u l e

< c a t e g o r y > ::=

-

t y p e

I

< e x p r e s s i o n > by

< c a t e g o r y > : := < e x p r e s s i o n >

1

t y p e

/

C <dummy v a r i a b l e > , < e x p r e s s i o n >

1

< c a t e g o r y > We s h a l l now e x p l a i n how t o d i s t i n g u i s h f r e e and bound occurences of

dummy v a r i a b l e s , To f a c i l i t a t e o u r d i s c u s s i o n , we d e f i n e

I

< v a r i a b l e >

I

1

(

1

)

1 ,I

C

1

11 I

1

3

and we n o t e t h a t each e x p r e s s i o n can be d e s c r i b e d a s a s t r i n g , where each e n t r y i n t h e s t r i n g i s a copy of one of t h e symbols. We speak of "copy of x" r a t h e r t h a n "x" i n o r d e r t o b e a b l e t o d i s t i n g u i s h between d i f f e r e n t o c c u r e n c e s . We s h a l l a l s o r e f e r t o t h e s e c o p i e s a s o c c u r r e n c e s of t h a t dummy i n t h a t e x p r e s s i o n .

Moreover we s h a l l u s e c a p i t a l Greek l e t t e r s ( l i k e C , A ,

...

) a s

m e t a l i n g u a l symbols. They s t a n d f o r e x p r e s s i o n s we do n o t wish t o w r i t e down themselves.

An occurence of a dummy v a r i a b l e i s c a l l e d b i n d i n g i n t h e e x p r e s s i o n C ,

i f i t immediately f o l l o w s a copy of

[.

To e a c h e x p r e s s i o n C , and t o each b i n d i n g occurence of any dummy, we s h a l l d e f i n e a s e t of occurences which w i l l be c a l l e d bound ( i n C) by t h a t b i n d i n g occurence. T h i s w i l l be done i n such a way t h a t each copy of a d m y i s bound by a t most one b i n d i n g copy of t h a t dummy; i f i t i s n o t b i n d i n g and n o t bound by any b i n d i n g copy, i t w i l l be c a l l e d

-

f r e e i n C.

To s t a r t w i t h an example (where we u s e l e t t e r s s , t f o r dummies, x , y f o r v a r i a b l e s , a , b f o r c o n s t a n t s ) :

C S ,

a ( s , t ) l CCs,tl a ( t , Cs, b ( s , x , y ) l s , C t , y l s ) ) a ( t )

i s an e x p r e s s i o n . There a r e seven occurences of s ( t o b e c a l l e d t h e f i r s t ,

...,

s e v e n t h , r e a d i n g from l e f t t o r i g h t ) . The f i r s t , t h i r d and f o u r t h a r e b i n d i n g . The f i r s t s b i n d s n o t h i n g , t h e t h i r d s b i n d s t h e f i f t h and t h e s e v e n t h , t h e f o u r t h s b i n d s t h e s i x t h . Only t h e second s i s f r e e .

As f a r a s t i s concerned: t h e f i r s t t h r e e a r e f r e e , t h e f o u r t h i s b i n d i n g b u t b i n d s n o t h i n g , t h e f i f t h i s f r e e .

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We shall define the relation between binding and bound dummies by recursion, taking the rule

as a basis. This rule gives

6

ways to make an expression, and for each way we have to describe the relation.

If the expression is ofrhe form<constant> or <variable>, we have nothing to define. If C is of the form <constant> (<expression string>) we note that eacht binding occurrence of a dummy is situated in one of the expressions

C 1 ,

...

,Z this expression string consists of, and we agree that such an

k

occurrence binds in C exactly those copies of dummies it was already binding in the Ci it belonged to.

A similar arrangement is made for the case {<expression>) <expression>. Here we have just two constituents C

1 ' C2, which form {C 1 } C 2 ' A binding copy

of a dummy in C binds just the copies of dummies it did bind already in C 1 , 1

and a binding copy of a dummy in C2 binds just the copies of dummies it did bind already in C

2 '

If the expression is of the form <dummy variable>, we consider that single occurrence as free.

If the expression C is of the form [<dummy variable>

,

<expression>l <expression>, we agree on the

follow in^.

Write it as [x, C 1 l C2. The first x is binding, and we agree that in C it binds all occurrences of x in C that were free in C and no others. For the

2 2

'

rest, a binding dummy in C binds (in

2)

just those occurrences of dummies 1

it did bind already in C and a binding dummy in C binds (in C) just those

1 2

occurrences of dummies it did bind already in C 2 '

The above definition of binding is

-

not the one of the report C11, but there is no objection to change the system of

C11

into the present one. With the present definition it is easier to handle expressions: we are not so often under the

obligation to change names for dummy variables.

References.

1 .

N.G.

de Bruijn, Automath, a language for mathematics. THE-report 68-WSK-05

( 1 9 6 8 ) Technological University, Eindhoven.

2. The syntax of PAL and AUTOMATH, Techn. Univ. Eindhoven, Internal Report, Notitie

32,

9 April 1970.