Description of the language Automath

Description of the language Automath

Citation for published version (APA):

de Bruijn, N. G. (1967). Description of the language Automath. Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1967

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Description of the language kUTOMArPII.

1. The grammar of the language consists of a s e t of rules according t o which

1.1 A book i s a l i n e a r l y ordered f i n i t e s e t of "lines". If we wish, the l i n e s o m be numbered: l i n e 1,

.

..

,

l i n e

N.

The r u l e s of grammar w i l l be r u l e s f o r adding an (N+I )-st l i n e t o a book of N l i n e s (and there

is,

of a

rule

f o r writing a f i r s t l i n e ) . I f these r u l e s are obeyed, we say that

t

( ~ + 1 )-st l i n e i s acceptable.

The rules f o r a c c e p t a b i l i t y of t h e ( ~ + l ) - s t li n e make senae only i f each n (1 9 n

<

N ) the (n+l)-st l i n e i s acceptable with respect t o t h e book consisting of l i n e s l,...,n.

1.2 A l i n e a o n s i s t s of f i v e partsr an indicator, an i d e n t i f i e r ,

a

d e f i n i t i o n ,

a

oaterroq. Occasionally we add a f i f t h part, c a l l e d a

hint.

1.2.1 The h i n t

i s intended

t o assist

the reader in hi8 attempts

t o

oheok

whether

p"r9

,..

tho d e l e . have been properly applied. The hint. w S L be e - r e s a & ~ ~

means

of

a

d i f f e r e n t from those we a r e going t o diacuss next. The h i n t

af

a

oertafn

s i t s r 8 l e only when ahecking t h a t l i n e ;

it

i r s

not t o be consulted when

tes l i n e s . For the time being we shall disregard the h i n t s e n t i m w r

- - O f Z p o s*,

are

The h s s i a symbols

of

which the other parts

The seven separation marks, l i s t e d herer

Arbitrarily m,ny other symbols t o be a a l l e d i d e n t i f i e r s , mutually ( i i i )

d i s t i n c t , and d i s t i n c t from the 11 symbols l i s t e d under ( i ) aad ( i i )

1.2.3 The S d e n t i f i e r p a r t of a l i n e consists i s a single i d e n t i f i e r . It

hae

t o be d i f f e r e n t

fmm

the i d e n t i f i e r part of any previous l i n e . There would be no object- i o n against gystematic use of positive integers i n such a way t h a t t h e number

a

i s the i d e n t i f i e r p a r t of the n-th l i n e . However, i n order t o make books e a s i e r t o read, and e a s i e r t o compare with existing ways t o express mathemtics, one may p r e f e r t o choose more suggestive symbols l i k e words, o r words with numbers added

t o them. Note t h a t

an

i d e n t i f i e r i s t o be considered as

a

single symbol, It has already been s t i p u l a t e d that i d e n t i f i e r s have t o be d i a t i n o t from t h e other basio symbols (see 1.2.2). I n a printed t e x t an i d e n t i f i e r may be represented by ar

string

of l e t t e r s , d i g i t s o r other signs, containing no separation mrsdcs.

1.2.4 The i n d i c a t o r part of a l i n e i s e i t h e r t h e symbol 0 o r t h e i d e n t i f i e r

of

any previous l i n e . The i d e n t i f i e r of the

k-th

l i n e

m y

be used

as

i n d i c a t o r of

a

l a t e r l i n e i f and only i f t h e d e f i n i t i o n

part

of the, k-th l i n e i e t h e eymbol --

.

I n t h e discussion of t h e language (not i n the

l m g m p

i t s e l f ) r e o f t e n c o m i d e r the n i n d i c a t o r strizqf of

a

line. We oan describe

i t ~ ~ s i r s l y .

If

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

i s

0, then t h e indicator s t r i n g

i s

empty. I f t h e indicataa. of the n-th l i n e i e t h e i d e n t i f i e r part

of

the k-th l i n e (whenoe k C

n),

t

the i n d i c a t o r s t r i n g

of

t h e n-th l i n e is t h e i n d i c a t o r of the n-th

line

&

t h e i n d i o a t o r string of the k-th l i n e ,

lkanple 1 If t h e book oonsiats of 8 l i n e s , with i d e n t i f i e r s 1,2,3,4,5,6,7,8,

and i n d i c a t o r s 0,1,1

,O,4,5,5,4,

then t h e i n d i c a t o r a t r i w are

If

a and T a m

strings,

r e w r i t e a c T i f e i t h e r o - 7 o r ? e q d ~ u f o l l a e d by some other

string.

If a

i s any

string (possibly empty),

them

a

bloak

ia

the

r e t

af

a l l lines

whose indioator s t r i n g T r s t i s f i e s a e t

,

I n the above example the blocks a r e ( 2 ) .

( 3 ) ,

( 6 ) ,

(7),

(2,3), (1,2,3),

consecutive l i n e s .

We do not always have t h i s simple block structure. X.g., i f t h e i n d i c a t o r s

We

o d

represent the indicators by means of an oriented rooted t r e e ,

The

points of the t r e e are i d e n t i f i e r s , a p a r t from t h e point

0

t h a t fig me^ apl t h e

root of the tree. We draw a directed path from t h e i d e n t i f i e r b t o a point,@ir

# /I

if

and o n l y

if

p i s the i n d i c a t o r of t h e l i n e whose i d e n t i f i e r ie.3.

1.2.5 The d e f i n i t i o n

mrt

of a l i n e

can

be one of t h e following things:

( i > The symbol

-

( t o be called "bar"

)

( i i ) The symbol

PN

( t o be a a l l e d "primitive notion")

(

iii) An emreasion, This i a a

certain

string of eymbols, aonaiet of eeparation marks and i d e n t i f i e r s , I f an i d e n t i f i e r iaaed SA

an

expression occurring i n the

k-th

l i n e i s not t h e i d e n t u i e part of a previourr l i n e , then

it i s cralled

a "bound

nariable".

Althougfi

it

is not s t r i c t l y neoeasary,

i t i a b e t t e r t o

think

of

a

bound WLritbbl~ aa

sn

i d e n t i f i e r that doe8

m t

oaour

a8

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

of

the

k-th

o r

any

f u r t h e r

line

either.

We a h a l l explain

later how expreemime should

be

b u i l t .

1,2*6 The aategoxy wrt;

of

a

line

can.

be one of t h e following t h b g ~ :

(i

>

The symbol sort.

Li-stic variables. In our d e s c r i p t i o n of t h e language r u l e s w e shall

use greek l e t t e r 8 as l i n g u i s t i c variables. They denote i d e n t i f i e r s , expmssionn o r other parts of a book. They occur i f i general statements about t h e language, but do not appear l i t e r a l l y i n the books, I n o r d e r t o avoid confusion, we ~haI.1 agree t h a t i d e n t i f i e r s ( s e e 1 . 2 . 3 ) a r e composed e n t l r e l y of symbols d i f f e r e n t from greek l e t t e r s , so a s t o minimize reconfu:iion be4,ween t h e contents of t h e

book on t h e one hand, and our discussion about t h e book on t h e o t h e r hand.

The following n o t a t i o n w i l l be used i n t h e sequel. I f j i s a positive! i n t e g e r , w e denote by A t the j L i t h e j u : t h e j ' j : t h e b j : t h e

J

r t h e j-th line!. i n d i c a t o r of

J'

i n d i c a t o r s t r i n g of h

3'

i d e n t i f i e r part of A j* d e f i n i t i o n p a r t of A

2'

category of

X

j'

1 . 3

P r i m i t i v e ATVTOMBTH language. I n order t o f a c i l i t a t e t h e exposition

ra

8hal.l f i r s t d e s c r i b e a s e t of r u l e s f o r a language that uses only

a

part

of

t h o oomplete AUI'OMATH language. A book w r i t t e n i n the p r i m i t i v e

language

( t o

be o a l l e d PAL) w i l l

a l s o be acceptable

i n

t h e complete language

( t o

be

orlled

AL).

In PAL we uee only

#

(

1

as s e p a r a t i o n marks, and we do not

use

bound vaxiables.

PAL

is

an

abbreviated form of a language LONGPAL,. The l a t t e r has eimpler r u l e s , but has t h e p r a c t i c a l disadvantage of very l o n g expressione. We s h a l l

f h s t d e f i n e LONGPAL.

1.4.1 We s h a l l desaribe the rules i n t h e following way. We f i r s t say when a book i s aalled p a r t l y correct, next w e give an addition law, i,e.

a

s e t of rulee

f o r

adding a l i n e t o a p a r t l y correct book, and w e show t h a t t h i e l e a d s again t o a partly correct book. F i n a l l y w e call a book correct i f t h e f i r a t l i n e i s correct, and i f i t can 3e obtfiine?? step-by-step by repeated application

of the addition law.

1.4.2 A partly c o r r e c t book i s a set of l i n e s A ₁

,.

. .

,AN

s a t i s f y i n g the following c m d i t i o n s :

( i ) For 1 j N t h e indicator L e i t h e r is equal t o 0 or t o aokae

j

4.

with 1 6 i < j. I n the latter aase

i t

i s required t h a t

bi

, -

-

,

W e have b1 P - o r PN; Y1 I ~ o r t .

Aa a preparation t o condition ( i i i ) we define the notion PALe~preseion*

''1

Moreover,

expressions

,

then $ (.I+,

.

.

.

,Z

_t

)

i s

a

PAL-exprassion

.

We now phrase t h e l a s t condition f o r a book t o be

partly

aorrect t

( i i i ) I f 1

<

j 6 N, then 6 . i s either -- or

PN

or a

PAL-exprsrrsio~r,

J

Moreover, y i s e i t h e r s o r t o r

a

PAL-erpressi~n,

j

-

1.4.3 We define i n d i e a t o r & r i n g s as i n 1.2.4. Given a partly aorract book A1,...,Ap the t a m "admissible s t r i n g "

will

e i t h e r denote t h a indioator s t r i n g of ane of the l i n e e , o r tha indicator string ,that ca

partly

a o r r e c t

book A,

,..

.

s i g h t have

at

i t a

l a s t

line. Thrst is,

i t

might be tha a t r i n g obtained

by

adding

Pi t o

i t s i n d i c a t o r a t r i n g ai,

for an

i

with

Li

--

.

T h b

string doaa not neceesarily occur a s the i n d i c a t o r string o e

lh.8

in

the book,

*)

This

notion

doe8

not

depend

on

having

any

partioulsr

book,

but

o n l y on

the

presenae of s s e t of d i s t i n c t ~ymbole called i d e n t i f i e r s .

from t h e i d e n t i f i e r s

P I ,

..

.

,

PI[.

Let a be a s t r i n g consistin?; of k - 1 ) 1 - e l n f i i c l n t i f i e r s , taken

from

-

-

-

This operator transf~rms PN i n t o i t s e l f , according t o t h e following rule.

Let hE

PN.

This expression

h

may contain t h e i d e n t i f i e r represented by fl a number of t i n e s . Ne replace t h i s i d e n t i f i e r , wherever i t occurs,

il

by t h e symbol C Ve repeat t h i s procedure with

,

. .

,Pi

.

Next we replaoe

2 k

t h e symbols El

,.. .

,Z

oy t h e expressions they denote. This d e f i n e s k

(Q

o(zl,.

. .

, \ ) ) A . Note that i: l , .

. .

&

k themselves may contain

,

.

.

.

,

i l ik t h a t t h e simple order : " r e p l a c e by Z ₁ everywhere, $ _i2 by everg- where

...."

would be q u i t e confusing.

1.4.5 Admissible t r i p l e s . Let t h e p a r t l y c o r r e c t book

3,.

.

.

,

$

be given, We

rrhall

d e f i n e the notion "admissible triple'!, The admissible t r i p l e s

a l l have

t h e form ( u , A , ~ ) h e r e U E S ~ , A E P ~ ,

r

-

-

sort' o r ~ E P ~ .

F ~ ~ T - L -

b - + . h k

Su,

G

We d e f i n e t h e s e t of admissible t r i p 1 es recursively,

I f oE

SN,

%?

i i s such t h a t f3 i s one of t h e e n t r i e s of u

,

then (u,fli,yi) i s an admissible t r i p l e .

This

f3

5

i s j u s t

P,

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

J J

strin J

I f i t i s not, and o.

J

mF$

l1 2 ik

,

then

p*

J =

3

(f3 il g * . * , P

ik

).

h hat

is,

t h e expression t3Sr i s formed by w r i t i n g t h e i d e n t i f i e r

J

denoted by

pj,

an opening p r e n t h e s i s , t h e i d e n t i f i e r

denoted

bJ

f3

,

a comma, etc.), il

Now we require r if 1

<

j 4 N, a E S N ,

ani

o c

u

t h e n

3

(a,f3* Y

)

i s

an

admissible t r i p l e . j9

3

Moreover,

Then a l s o

Let k > 0, and l e t

( p i

,

.

.

.

,Pi

) = CJ E

P

Assume that

1 k

N*

u ( 5 ,Cl ,PI

),

.

.

.

,

(T

,

4,111-) are admissible t r i p l e s . assume that

J

the following trip1 e i s admissible

Examples. Consider the book A

,,...

, A l 4 of 1.4.

Take

o E x,y,u. Thes t h e

f o l l m i n g a r e examplea of acceptable t r i p l e s r (0, x

,

e l t ) ,

1 . 4 6 Addition l a w .

AN+, s h o u l d be

Let h,

,

,

.

.

,

%

be a p a r t l y c o r r e c t book. The i n d i c a t o r of e i t h e r 0 o r a /3 w i t h 1 a; j 4 N, b r

-

5

3

Thie d e f i n e s the

i n d i c a t o r string uN+l

.

Now we admit two cases f o r b _{N+1 and}

_YW

_'

-

-

o r PN, s i o n

with

t h e

t r i p l e .

Finally, t h e i d e n t i f i e r

pN+l

should be d i f f e r e n t from

P1

,

.

. .

,

PN,

and d i f f e r e n t from t h e o t h e r b a s i c symbols ( s e e 1.2.2).

partly

aorrsat

(see 1.42) and if, f o r j

-

2,..,,N, t h e j-th l i n e ha8 been added t o

the

bmk

hl,

..,,4-,

acoording t o t h e a d d i t i o n law (see

1.4.6).

(

ii.1

( i i i )

A t t h e j-th l i n e b . and y

.

( i f they a r e not

-,

PN o r s o r t ) J J

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

P,

,

.

. .

,p

j-l (and of course parenthes.26 and cn7nrna

'

s )

.

If

pi

occurs i n an expression, then y i s

-

o r PN. i

I f (0,n.P) i s an admissibl e t r i p l e f o r t h e book A,

,..

.

,AN,

and if l? s o r t , t h e n (o,17,

-1

i s a l s o an admissible t r i p l e f o r t h a t book.

1.5 Description of PAL. Having described LONGPAL completely,

it

i s q u i t e eaey t o say I r h a t B L is, The d i f f e r e n c e l i e s only i n t h e f a c t t h a t t h e PAL-expreesion a r e abbreviated n o t a t i o n s f o r the LOI?GPALexpressions. Actually

a

book w r i t t e n i n PAL can be t r a n s l a t e d i n t o a book i n LONGPAL by t h e simple procedure of r e p l a c i n g e v e r y b and every y . ( i f they a r e not

--,

PN

o r s o r t ) by t h e

j J

LONGPAL-express ions they a r e abbreviations for. On t h e other hand, every book w r i t t e n i n LONGPAL i s a l s o a c o r r e c t book i n PAL. It may be p o s s i b l e t o a b b r e v i a t e some of i t s expressions, but i t i s by no means

an o b l i g a t i o n

1.5.1 Our present d e s c r i p t i o n of PAL i s given by means of LONGPAL. However, part of t h e p r a c t i c a l value of PAL l i e s i n t h e f a c t t h a t i t i s p o s a i b l e t o m ~ n i p u l a t e with t h e abbreviated expressions themselves, m t h e r then

translating

i n t o LONCPAL a t mry stage.

1.5.2 The expmasione occurring i n PAL a r e s t i l l of t h e form deeoribed

in

11.4.2, and a book w r i t t e n in PAL s t i l l s a t i s f i e s ( i ) of 1.4.7, but

it

doer n o t neoeaearily e a t i s f y ( i i ) of 1.4.

Let A,,..

,AN

be a book m i t t e n in LONGPAL, and l e t o be

an

admiasiblo

rtring

( m e 1.4.3). We shall d e f i n e (by r e c u r s i v e d e f i n i t i o n )

an

o p e n t o r

To

that mps

a

o e r t a i n subset of pN i n t o PN ( a s i n 1.4.4, t h i s pN i n the a e t

of

a l l P A L - ~ I ~ ~ B E ~ O ~ ) , T ~ ( E ) rill be oalled t h e normal

forn

of 2

,

and

Z

i13

died

an

a b b m i a t i o n f o r a ( ~ ) .

(

ii)

Let 1

c

j 4 N, l e t 6 P -,and assume that

p i

i s one o f t h e

j

e n t r i e s of a

.

Then

P

. is a P-4L-expression, and w e d e f i n e J

Let 1 S j G N, l e t T be a str~rg such t h a t both T c a .

and

J

s c a. Let k be t h e length of T

,

an7 l e t h

+

lc be t h e leng-th of a Let the s t r i n g o c o n s i s t of

p i

,...,pi

,

pi

,...,$

.

j* j ₁

h

h+l

lh4.k

If k

>

0, l e t El,.

.

.

, E b expressions f o r which To(Z1

),

. . .

,T=(%)

k

have already been defined. Let now Z

be

t h e

expression

P

( 2

,..

.

,C]

j 1 Then w e d e f i n e

T

)

Q

(P

'j

i~

,***,Pi

k

T~(Z~)~.**,T,(~~)))

i f b . i s d i f f e r e n t from

-

and

PN,

and J

(pf

as

defined

in

1 . 4 5 ( i i ) ) i f

L

-

FN.

j

I n t h e case that k = 0, we define

if b

.

is

d i f f e r e n t f r o m

-

and PI?, and J

I - 1 1

-

, I 'i

.

-

-.

.*

1 , t b ,, / I :;' ,

Correct PAL book, A book i s c a l l e d correct \fS''PAL i f it can be obtained from a LONGPBL book A,

,

.

.

.

,

%

in the following way : If 1

s

j

'

n, and if 6

3

is

not

-

o r

W, then replaoe b by any expression b ! which i s such t h a t T ( b t )

-

j J

_%

_j

9

and

s i m i l h l y replace y

,

i f

i t

i s n o t

sort,

by some y! J with T ( y ' )

-

y

.

I o t i o

7

j

3

t h a t and y! an, not uniquely defined, and t h a t i n p a r t i c u l a r T ( 8

)

=

J

₃

j

%'

T ( y j )

-

Yj.

9

G

Examples. We r e f a r t o 1.4 f o r the PAL version of a LONPAL book. ( l i n e s A',O,..., A' 14

).

AL an6 LQNGAL. These languages have the same s t r u o t u r e

as

PAL

aad

LOWPAL.

They a m richer, since they admit expressions t h a t do not e x i s t

in

PAL

e r

L O W -

PAL. Every PAL-book i s a l s o an AL-book, but not t h e o t h e r w s y

m a .

The

mlatld

between AL and LONGBL i s s i m i l a r t o the r e l a t i o n between PAL

and

LOISCPBL.

Bgsin,

any LONGAL-book i s

a1

s o

an

&-book. And e w q . AL-book

oan

be considered t o u i . 4 from a LONGAL-book by abbreviation of the expreesione given

as

d e f i n i t l o n e .nb

aategoriee.

1.6.1 I n 1.4.1 r e defined the notion of PAL-expnasion. The d e f i n i t i o n of

AL-

expremion w i l l be similar. A s i n 1.4.1, the d e f i n i t i o n w i l l be ohossn r a t h e r l i b e r a l l y , and i t i. not automatic t h a t every A l - e r p n s a i o n can a c t u a l l y oaour

in

an AL-book.

Again we have a s e t of d i s t i n c t symbols oalled i d e n t i f i e r s . Moreover

no

now have an i n f i n i t e s e t K of symbols which

are

mutually different,

aad

diffora

from the i d e n t i f i e r e . The element8 of K a r e c a l l e d indetermiaater. We s h a l l uaro t h e word l e t t e r t o denote something that i s e i t h e r

an

indetexminatb o r

an

i d e n t i f br. The eret of all l e t t e r s i s denoted by S.

\'I

1.6,2 ~ h ~ ~ ~ ~ t ' i o n of "AL-expnssiontt i s defined by

'

-.

( i ) If

$3

i s a l e t t e r , then

P

i s an AL-expression.

( i i

>

I f k 3 1, if

p

i s an i d e n t i f i e r , and i f XI

,

.

.

.

,

i;x

a r e

a-

expressions, then

,

,

.

.

,

,

i s an R1.-expression.

( i i i ) If

Z,

and 1: a r e AL-expressions, t h e n

(z,

}.Y2 i s an AL- expression.

( i d If L1 and C2 a r e AL-expressions, and i f A i s an indeterminate,

1.6.3 F r e e variables.

Uz

of S t o be c a l l e d recursively.

To every Al-expression C we s h a l l a s s i g n a subset t h e s e t of f r e e v a r i a b l e s of

C

.

We d e f i n e

it

an i d e n t i f i e r , then U f3 is empty. an indeterminate, then UA

-

{A).

( i i i ) I f k 3 1 , i f i s an i d e n t i f i e r , and X I ,

...,\

a r e hl- expressions, then t h e s e t of f r e e v a r i a b l e s of $(z,

,.

.

.

, X k

)

i s t h e union of those of Z,

,

.

.

.

,

zk*

(

i v ) I f C 1 and X2 a r e AL-eqressions, t h e s e t of f r e e variables of

{x,

}z2

i s t h e union of t h o s e of Z1 and Z2.

(4

Lf

X 1 and G 2 a r e ALeexprassions, and ff A € S, t h e n t h o ref o f f r e e v a r i a b l e s of [A,

$

@

i s defined as

1

A.4

We shall d e s c r i b e LONGAL i n j u s t t h e same r a y ae o u t l i n e d i n 1.4.1 f o r LONGPAL.

1

,6.5

A

LONGAL-book

is

called

partly

correct

if oonditions (i),

(ii),

(iii)

of

1.4.2 hold, provided that ( i i i ) is phrased w i t h &-erpressions instead of PAL-expre~sions.

1i6.6 The i n d i c a t o r s t r i n g s &d admissible s t r i n g s a r e defined as i n 1.2.4

1.6.7 S u b s t i t u t i o n operator. Let a be a f i n i t e s t r i n g of k d i f f e r e n t l e t t e r s ( k

>

1

)

Let

5 ,

.

.

.

,

be AL-eqressions. We s h a l l define an operator Q

( 5

,.

.

.

,

s)

th8t

u

maps a c e r t a i n claes of AL-expressions i n t o AL-expressions.

o or

t h e

special

case t h a t

u

contains no i n d e t s m i n a t es and that Z.,

,

.

.

.

,

%

a r e PAL-expreeeiolllr, t h e e f f e c t of the operator on PAL-expressions i s t h e one described i n 1.4.4).

We s h a l l define R ~ ( Z . ,

,

. .

.

,

I$ recursively.

( 0 I f

p

i s a l e t t e r then ( Q u ( I + , , . .

,%))$

equals fl i t s e l f

if $

d u o

not occur i n the s t r i n g a

,

and i t equals

Zi

i f

P

i s t h e i - t h e l a a a t of t h e s t r i n g .

( i i ) I f

P

i s an i d e n t i f i e r , i f A,,

.

.

.

,%

a r e AL-expresaions (1

r

I), and A

-

p(+,..

.+,),

then

( 1 t may arouse a u r i o s i t y t h a t $ ? f Z f Z f o r m e d , but

in

our a p p l i c a t i o n s this

$ w i l l not be

a

m e m b e r of t h e e t r i n g o

).

(id

Let A l 4 be Al-expressions, l e t A be an indetezminate,

and

A j

-

[A&,

&.

It is possible that A occurs i n t h e a t r i n g a. We denote by

+

the a t r i n g obtained from

o

by d e l e t i n g

I

( s o r

-

o i f A does n d t oocud sad

1*.

+**.13

t h e a t r i n g obtained from El

,

.

.

.

,%

by d a l e t i n g t h e P

3

If' A happ.n.d

t o

be

t h e j-th entry

of

0

.

( ~ e n c e

1

-

k o r

1

-

k

-

1).

and

we d e f i n e

1,6,8 Admissible t r i p l e s . As i n 1.4.5,we shall d e f i n e a s e t of admissible t r i p l e s with respect t o a book A , , , . . , ) ' N by recursion. The c o n s t r u c t i o n i s more i n t r i c a t e i n the present case, i n p a r t i c u l a r s i n c e N w i l l be not constant d u r i n g the recursion: the question whether some t r i p l e s a r e admissible with r e s p e c t t o a given book w i l l sometimes be answered by a s k i n g whether some a l i g h t l y simpler t r i p l e w i l l be admissible with r e s p e c t t o a s l i g h t l y longer book.

We assume that

+,...,b

i s a p a r t l y c o r r e c t book. Let Px denote t h e

met

of

a l l AL-expressions formed by means of $l,...#N as i d e n t i f i e r e .

SN

i s

the set of a l l admissible s t r i n g s

of

A ? , .

.

.

,AN.

The admissible t r i p l e s f o r t h e b a ~ k A ~ , . . . & ~ all have t h e form (o&&'), where u E S P A €

Pp F-

sert

o r

F E pr ~ c t m ~ l y

i t rill t u r n out

that A has t h e empty s e t aa k i ~

met

of

t-

1 . f ~

f r e e v a r i a b l e s . The same t h i n g holds f o r

'

I

i f

it

i s not aort. We r e q u i r e

(ii) ( i d e n t i o a l t o ( i i ) of 1.4.5) ( i d e n t i c a l

t o

( i i i ) of 1.4.5)

(

i d

Let

A €

PN,

8 E

Pr

Let A be

an

indeterminate. Then the t r i p l e

(u,[A,A@,

e o r t )

i a admissible i f t h e following aonditione sue Both s a t i s f i e d : (a) ( u , & s o r t ) is an admissible t r i p l e t (b)

(o

',

@ l r )

(B

N+,))O, 8013) i s

an

admissible t r i p l e with reapeot t o t h e book A,,

.

. .

,

Ap

.

Hem AN+,

is

t h e l i n e with identifier

BN+l

,

and with ON+,

-

0#

$+,

-

-9

YN+,

- A , aad a* i s t h e

st*

o

+

{&+l}

( L e e t h e string obtained by p l a c i n g t h e

extra

dement

@ ~ + l a f t e r the s t r i n g = ) . F i n a l l y ,

{A}

atands f o r the string o m - s i s t i n g of the element A only.

(4

Let A

EPp

B EPN,

B

EP N

,

and l e t A be en indeterminate.

Then

t h e t r i p l e (U,[A,A]B,[A,A 12) i s admissible i f both (o,h,sort),

(P

)@,

Q { ~ J ( P , + , 12)

' A } N+1 axe admissible ( f o r n o t a t i o n s e e under (iv&

Let A EPN, 9 EPNe C EFpT, l e t o be a n admissible string, and

l e t Alp be d i s t i n o t v a r i a b l e s . Assume t h a t (a ,@ ,A) and ( a & ,[A,A 191

are admissible, We consider two cases:

( a ) C has the f o w , A ] E ~ . NOW we define that

(b) C does not start with

[

.

Then we, d e f i n e t h a t

(

4{@N

(@

18

i s an admissible t r i p l e .

1.6.9 Addition

law.

This law

i s l i t e r a l l y the same

as t h e one of 1.4.6, although t h e meaning of t h e words and symbols i s t h e one of s e c t i o n a 1r6.1-1.6.8 r a t h e r

thur

t h e one of s e c t i o n s 1 .4.1-1.4.5.

1.6.10 A book A l

,.

.

.

,AN

is milled a c o r r e c t LONGAL-book i f

it

i s

partly

oornot

and if, f o r j

-

2

,...,

N1 the j-th l i n e has been added t o t h e bookAl,...,5-, socording t o t h e a d d i t i o n l a w .

1.7. D e s c r i p t i o n of

BL.

An AL-book i s a LONGAL-book with abbreviated forms

of

t h e expressions, j u s t l i k e PAL a b b r e v i a t e s LONGPAL. The

rulee

a r e t h o s e of 1.5 p l u s some e x t r a ruleer

( i i i > If 0 and A

are

expressions f o r which T Q has been defined,

and

if C

-

[A #]E

,

then we d e f i n e

I f 8

and

Z

are

exprerssione

for

whioh

T

o

has

been defined,

if

I f 8 and C a r e expressions f o r which

T o

has been defined, and if