A note on weak diamond properties

A note on weak diamond properties

Citation for published version (APA):

Bruijn, de, N. G. (1978). A note on weak diamond properties. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 7808). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1978

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics Menorandurn 78-08. I s s u e d August 1978. A n o t e on weak diamond p r o p e r t i e s . by N.G. d e B r u i j n . Eindhoven U n i v e r s i t y of Technology Department of Mathematics P.O.Box 513 5600 MB Eindhoven The N e t h e r l a n d s .

A n o t e on weak diamond p r o p e r t i e s .

1 , I n t r o d u c t i o n . L e t S be a s e t w i t h a b i n a r y r e l a t i o n >. We assume i t

t o s a t i s f y x > x f o r a l l X E S. We a r e i n t e r e s t e d i n e s t a b l i s h i n g a

p r o p e r t y CR (named a f t e r i t s r e l e v a n c e f o r t h e Church-Rosser theorem of lambda c a l c u l u s , c f . [ I ] ) . We s a y t h a t x - y i f x > y o r y > x . We

*

s a y t h a t x > y i f t h e r e i s a f i n i t e sequence x , ,

...,

x w i t h x=x > x >

n 1 2

>

...

> xn=yY and a l s o i f x=y. We s a y t h a t S s a t i s f i e s CR i f f o r any sequence x l ,

...,

x w i t h n

*

*

t h e r e e x i s t an element X E S w i t h b o t h x > z and x > z. 1 n I t i s u s u a l t o s a y t h a t ( S , > ) h a s t h e diamond p r o p e r t y (DP) i f f o r a l l x , y , z w i t h x > y , x > z t h e r e e x i s t s a w w i t h y > w , z > w.

where x > y i s i n d i c a t e d by a l i n e from x downwards t o y , e t c . The l i t t l e c i r c l e s around y and z i l l u s t r a t e t h e l o g i c a l s i t u a t i o n : t h e diagram y& can b e c l o s e d by

Yt/Z

.

W It i s n o t h a r d t o show t h a t DP i m p l i e s CR. A s i m p l e way t o p r e s e n t a proof i s by c o u n t i n g " i n v e r s i o n s " i n sequences l i k e x > x < x 1 2 3 < X4> > x5 < x x7: i f i < j and x . < x 6 1 i + l Y X j > Xj + l ' t h e n we s a y t h a t t h e p a i r ( i , j ) forms an i n v e r s i o n . A p p l i c a t i o n s of DP, l i k e r e p l a c i n g x <

*

3 X4 > > x5 by x3 > x 4 < x d e c r e a s e t h e number of i n v e r s i o n s . Once a l l i n - 5 ' v e r s i o n s a r e gone, w e have e s t a b l i s h e d CR. The f o l l o w i n g p r o p e r t y WDP i s weaker t h a n DP. I t s a y s : " i f x > y 1

and x > z t h e n w e x i s t s such t h a t y >* .w and z >* w". I t i s v e r y f r u s t r a t - i n g i n a t t e m p s t o p r o v e t h e Church-Rosser theorem f o r v a r i o u s systems, t h a t

WDP

does n o t imply CR. A counterexample can b e o b t a i n e d by means of t h e

1

-

f o l l o w i n g pi'cture ( c f

.

[ 2 1 p. 49) : X 4 Y4 z 1 _{e t c .} 4

T h i s example a l s o shows t h a t CR n e i t h e r f o l l o w s from WDP2 where WDP 2 i s s l i g h t l y s t r o n g e r t h a n WDP and s a y s : " i f x > y and x > z t h e n w e x i s t s

*

*

I

such t h a t y > w and z > w and a t l e a s t one of y > w and z > w". S t r o n g e r a g a i n i s WDP3, e x p r e s s i n g : I t i f x > y and x > z t h e n w e x i s t s such t h a t y > * w and z > w." T h i s WDP3 does imply CR. A c t u a l l y WDP3 i m p l i e s WDP4, which s a y s :

*

*

*

*

" i f x > y and x > z t h e n w e x i s t s such t h a t b o t h y > w and z > w." T h i s

*

*

WDP i s t h e DP f o r ( S , > ) , and t h e r e f o r e i m p l i e s CR f o r (S,> ) , and t h a t 4

i s t h e same t h i n g a s CR f o r ( S , > ) . The d e r i v a t i o n of WDP4 from WDP i s 3

i l l u s t r a t e d by t h e f o l l o w i n g p i c t u r e ( c f . C29 p. 59) which s p e a k s f o r i t s e l f :

I n t h i s n o t e we go c o n s i d e r a b l y f u r t h e r . I n s t e a d of h a v i n g j u s t one r e l a t i o n > we c o n s i d e r a s e t of r e l a t i o n s > where m i s t a k e n from an i n d e x

m

s e t M. The i d e a behind t h i s i s t h a t i n t h e Church-Rosser theorem t h e r e l a t i o n s r e p r e s e n t lambda c a l c u l u s r e d u c t i o n s ; t h e r e may be r e d u c t i o n s of v a r i o u s t y p e s , and diamond p r o p e r t i e s may depend on t h e s e t y p e s . I t i s o u r p u r p o s e t o e s t a b l i s h weak diamond p r o p e r t i e s which g u a r a n t e e CR (where CR h a s t o b e i n t e r p r e t e d a s i n

s e c t i o n 4.

2 . The index s e t .

( M y < )

i s a w e l l - o r d e r e d s e t . T h a t i s , t h e o r d e r < i s t o t a l ( i . e . i t i s t r a n s i t i v e , and f o r a l l m ,m we have e x a c t l y one of m =m

1 2 1 2' m l < m2'

m < m and t h e r e a r e no i n f i n i t e d e s c e n d i n g c h a i n s m >

m

2 1

'

1 2 > m 3 > " '

Note t h a t i n M we do n o t have m

< m y

i n c o n s t r a s t t o what w i l l b e assumed i n S. There might b e u s e f o r c a s e s w i t h more g e n e r a l (M; <), l i k e p a r t i a l o r d e r w i t h d e s c e n d i n g c h a i n c o n d i t i o n . We s h a l l n o t s t u d y such e x t e n s i o n s i n t h i s n o t e .

3. The s e t S w i t h i t s o r d e r r e l a t i o n . S i s a s e t , and f o r each T E M t h e r e i s a r e l a t i o n > on S. The o n l y t h i n g we r e q u i r e i s t h a t x > x f o r a l l x E S and a l l

m m

m E M .

For a l l mcM we now i n t r o d u c e two f u r t h e r r e l a t i o n s > and >

.

We s a y t h a t

m+ m-

x > y i f t h e r e i s a f i n i t e c h a i n x = x O , x l ,

...

x =y ( p o s s i b l y n=O x=y) and elem-

m+ n

e n t s k , m y

...,

k

<

m such t h a t x

0 'kl x I > k 2

.

.

.

> x And we s a y t h a t

n k n '

x > y i f t h e r e i s a c h a i n x=x

0 > k l x I >k2

...

> x = y w i t h k e m y

m- k n

n 1

k < m . Again t h i s i n c l u d e s t h e c a s e t h a t n=O, x=y, even i n t h e c a s e t h a t n

m i s t h e minimal element of M and no k w i t h k ' < m e x i s t . Note t h a t i f x > m- y t h e n (x=y) v 3 ~ E M ~ I C < m 'k+ 4. The p r o p e r t i e s CR(m). We w r i t e x

-

y

if

k c M e x i s t s w i t h k

<

m and m x > _{y o r y >k x . We w r i t e x - y i f mcM e x i s t s w i t h x}

-

y. I n o t h e r k m

words, x - y means t h a t k c M e x i s t s such t h a t x >k y o r y > k x. L e t mcM. We s a y t h a t CR(m) h o l d s i f f o r e v e r y f i n i t e sequence x x

-

. . .

-

x t h e r e e x i s t s z such t h a t b o t h x >m+ z and x > z. 1 m 2 m m n 1 n m+ We s a y t h a t CR h o l d s i f f o r e v e r y f i n i t e sequence x l U x 2

...-

x n t h e r e e x i s t s z such t h a t b o t h x l >m+ z a n d x > Z f o r s o m e m . n m+ Obviously CR i s e q u i v a l e n t t o V CR(m)

.

meM

5. The b a s i c diamond p r o p e r t i e s . I f EM, t h e diamond p r o p e r t y D 1 (m) i s d e f i n e d by t h e f o l l o w i n g diagram.

D l (m) :

m+

9;-

m-

T h i s h a s t o b e r e a d a s f o l l o w s (and f u r t h e r diagrams have t o be i n t e r - p r e t e d a n a l o g o u s l y : I f x , y , z a r e such t h a t x >m y , x > z , t h e n u,v,w e x i s t

m such t h a t

( s o on t h e l e f t we have a c h a i n from y t o w w i t h a l l l i n k s 5 m; on t h e r i g h t we have a c h a i n from z t o w w i t h a l l l i n k s I m b u t w i t h a t most one = m).

Note t h a t Dl(m) i s a g e n e r a l i z a t i o n of WDP and n o t of WDP ( s e e s e c t i o n 3

-

2 1 ) . We g e t WDP a s a s p e c i a l c a s e of D (m) i f m i s t h e minimal element of M 3 1 and i f > i s j u s t w r i t t e n a s >

.

m

The second diamond p r o p e r t y t o b e c o n s i d e r e d depends on two e l e m e n t s m,k of M , w i t h k < m. I t s diagram i s

6. Some a u x i l i a r y diamond p r o p e r t i e s . We i n t e n d t o show t h a t Dl(m) and D,(m,k) ( f o r a l l m,k w i t h k < m ) l e a d t o CR. I n o r d e r t o a c h i e v e t h i s

L

we f o r m u l a t e a number of diamond p r o p e r t i e s t h a t w i l l p l a y a r 8 1 e i n t h e

The diagrams D and D w i l l p l a y t h e i r r 8 1 e o n l y i f k <

m,

and D o n l y i f

3 7 4

7. D e r i v a t i o n of CR from t h e b a s i c diamond p r o p e r t i e s . Throughout t h i s s e c t i o n we assume t h a t f o r a l l m E M we have D (m)

,

and f o r a l l k E M , m r M

1 w i t h k < m we have D (m,k).

2

F o r b r e v i t y we i n t r o d u c e C R * ( ~ ) by

Lemma 1 . L e t m , k ~ M , k < m. Then we have

P r o o f . I n t h e diagram f o r D j we s e e a b r a n c h k+. It c o n s i s t s of a number of s t e p s x

_o

'11 x 2 '12

...

> 1

x

n w i t h a l l l . ' s 1 2 k . The c a s e n=O i s t r i v i a l . n

We p r o v e t h e g e n e r a l c a s e by i n d u c t i o n w i t h r e s p e c t t o n. The b r a n c h k+ c a n b e s p l i t i n a b r a n c h 1 ( w i t h 1

r

k ) and a b r a n c h k+ (of n-l s t e p s ) . The f o l l o w i n g diagram now produces t h e p r o o f :

I n s i d e t h i s d i a g r a m we have g i v e n r e f e r e n c e t o a p p l i e d p r o p e r t i e s .

( 1 ) V 1 e ~ l l

r

k ' h c ~ l h < k D4(m*k91,h)

( n o t e t h a t i f t h e r e i s no h < k we j u s t a p p l y D ( m , l ) ) , 2

Lemma

2.

Let m y k y l y h ~ M y

h'

< k

<

m y 1

I

k. Assume D

3

(m,t) for all t e M

with

t <

k,and Dj(syt) for all s y t

E M

with

t' <

s

<

m. Furthermore assume

CR*(~). Then we have D4(m,kylyh

) .

Proof. If 1

I

h the argument is

with

(3) as above,

(4)

D3(m,h)

For h

<

1 we proceed by induction with respect to h

by

means of the

following

Lemma 3 . L e t me M. Assume C R * ( ~ ) and assume t h a t f o r a l l h E M w i t h h < m we have D (m,k). Then D5(m).

3

P r o o f . I t s u f f i c e s t o p r o v e t h e diagram

-

f o r a l l h < m. (Note t h a t i f m i s t h e minimal element of M t h e n t h e b r a n c h e s denoted by m- a r e empty and t h e r e i s n o t h i n g t o b e p r o v e d ) .

The proof i s g i v e n by t h e f o l l o w i n g p i c t u r e .

Lemma 4. I f meM t h e n D (m) i m p l i e s D (m).

5 6

P r o o f . The upper l e f t b r a n c h m+ can b e s p l i t i n t o a number of p i e c e s of t h e form (m-,m,m-) ( l i k e t h e b r a n c h e s o c c u r r i n g i n t h e diagram D (m); i f 5 no > o c c u r i n t h a t b r a n c h we i n t r o d u c e one a r t i f i c i a l l y , u s i n g x > x). m m To t h e s e p i e c e s we a p p l y D

(m)

i n s u c c e s s i o n . 5

Lemma 5. L e t m,k e M,

k

< m. Assume C R * ( ~ ) , D5(m)

,

D6(m)

,

and D (m,h) f o r 3

a l l h w i t h h < m. Then we have D (m,k)

.

7

P r o o f . We a p p l y i n d u c t i o n w i t h r e s p e c t t o k , assuming D (m,h) f o r a l l h 7

Lemma 6. L e t m e M and assume D (m,k) f o r a l l k < m. Then we have CR(m). 7 P r o o f . S i n c e D (m,k) f o r a l l k <

m,

we have t h e diagram D (m) ( i f t h e r e 7 8 a r e no k

<

m we a p p l y D (m)). The b r a n c h m+ c a n b e s p l i t i n t o a number 1 of p i e c e s (m-,m,m-), s o we c a n i n t e r p r e t D8(m) a s p r o p e r t y WDP f o r t h e 3

r e l a t i o n > which i s d e f i n e d a s f o l l o w s : x > y i f u and v e x i s t such t h a t

X >

m- u > m v > m- y. By t h e p r o c e d u r e d e s c r i b e d i n s e c t i o n 1 we now g e t t o CR f o r ( S , > ) , and t h a t means e x a c t l y t h e same t h i n g a s CR(m).

Theorem. F o r a l l meM we have CR(m).

*

P r o o f . Assume t h e theorem f a l s e . Then we have an m such t h a t CR (m) i s t r u e , b u t CR(m) f a l s e . We cannot have D (m,h) f o r a l l h w i t h h < m, f o r

3

t h e n we would have D (m) by lemma 3 , D6(m) by lemma 4 , and t h e n lemmas

5

5 and 6 would l e a d t o CR(m). So t h e r e i s some k w i t h k < m and D3(m,k) f a l s e . L e t n be t h e s m a l l e s t element of M f o r which j

E M

e x i s t s w i t h j < n and D ( n , j ) f a l s e . Next l e t

i

b e t h e s m a l l e s t i n d e x < n w i t h D ( n , i ) f a l s e .

3 3

By lemma 1 we have ( n o t e t h a t n I m) 1 and h such t h a t 1

<

i,

h < i ,

D ( n , i , l , h ) f a l s e . Now lemma 2 l e a d s t o a c o n t r a d i c t i o n . 4

References.

[I] Barendregt, H.P.: The Type Free Lambda Calculus, ch. D7 in Hand-

book of Mathematical Logic, ed. J.Barwise, North- Holland Publ.Comp., Amsterdam-New York

-

Oxford

1977.

Nederpelt, R.P.: Strong normalization in a typed lambda calculus with lambda structured types, Ph.D.Thesis, Eindhoven University of Technology, 1973.