Upper bounds for the length of normal forms and for the length of reduction sequence in lambda-typed lambda calculus

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Upper bounds for the length of normal forms and for the length

of reduction sequence in lambda-typed lambda calculus

Citation for published version (APA):

Bruijn, de, N. G. (1985). Upper bounds for the length of normal forms and for the length of reduction sequence in lambda-typed lambda calculus. Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1985

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August 12, 1985

M23

Revised form, September

20,

1985

Upper bounds for the length of normal forms and for the length of reduction sequences

in lambda-typed lambda calculus

by N.G. de Bruijn

Department of Mathematics and Computing Science Eindhoven University of Technology

PO

Box 513, 5600MB Eindhoven, The Netherlands

1. For notations and definitions, as far as not explained in this note itself, we refer to

[I].

Nevertheless it should be noted that for the material of the present note there is no substantial difference between the system of 111 and the system of

[4].

Almost all ideas

involved in this note are already in [4]; the only new

idea is to use Nederpelt's normalization proof for estimating the length of the normal forms. The estimate we give here for the length of the normal form is presented as a very fast growing function of the length of the given non-normal expression.

I r

is of the order of the diagonal of the Ackermann function.

2.

We consider the minireductions of

[l]

section 4, slightly generalized since we shall admit these operations with the

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u s e of AT-couples ( [ I ] s e c t i o n 4.3) i n s t e a d of j u s t AT-pairs ( [ I ] s e c t i o n 4.2).

A lambda t r e e w i l l be c a l l e d almost-normal i f e a c h one of

i t s AT-couples a d m i t s AT-removal ( [ I ] s e c t i o n 4.5). The i d e a w i l l be t h a t we s t a r t from a lambda t r e e , a p p l y a sequence of l o c a l b e t a - r e d u c t i o n s and hope t o end up w i t h an almost-normal lambda t r e e . From t h a t p o i n t onwards i t w i l l be a simple a c t i o n t o g e t t o a normal lambda t r e e , j u s t o m i t t i n g AT-couples u n t i l t h e r e a r e no more l e f t .

The t e r m "almost-normal" w i l l a l s o be used f o r a s u b s e t of t h e set of p o i n t s of a lambda t r e e . It w i l l mean t h a t t h i s s u b s e t c o n t a i n s no e n d - p o i n t s t h a t r e f e r t o t h e T-part of an AT-couple.

3 . The number of e n d - p o i n t s i n a lambda t r e e w i l l be c a l l e d i t s

l e n g t h . I f a lambda t r e e ( V , l a b ) i s s e m i c o r r e c t , i t has a norm, denoted by norm(V,lab) ( [ I ] s e c t i o n 5.9). T h i s i s a g a i n a lambda t r e e ; i t s l e n g t h i s l e n g t h ( n o r m ( V , l a b ) ) , and w i l l be a b b r e v i a t e d t o l n ( V , l a b ) .

The n o t i o n of norm was a l s o d e f i n e d f o r s u b t r e e s , and t h a t g i v e s r i s e t o t h e s i m i l a r e x t e n s i o n f o r I n .

I f a t r e e ( V , l a b ) i s r e p r e s e n t e d a s a formula P, we j u s t

w r i t e l n ( P ) i n s t e a d of l n ( V , l a b ) .

4. L e t u s d e v e l o p some t e r m i n o l o g y around t h e s t r u c t u r e of t h e main l i n e ( s e e [ l ] s e c t i o n 5.2) of a lambda tree. The r i g h t m o s t l a b e l on t h e main l i n e i s e i t h e r a dummy o r

t

,

a l l t h e o t h e r s a r e e i t h e r T o r A. I n t h e formula t h e s e T's and A's c o r r e s p o n d t o a b s t r a c t o r s

[...:...I

and a p p l i c a t o r s

<...>.

We c a n draw a g r a p h of t h e sequence i f we r e p r e s e n t T's and A's by l i n e segments w i t h p o s i t i v e and n e g a t i v e s l o p e , r e s p e c t i v e l y . An example i s shown i n f i g u r e

1.

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F i g u r e 1. Graph of TTATTAAATAATTTAATATTAATA

...

I n f i g u r e 1 we have drawn t h e T's a s heavy l i n e s i f t h e y a r e " v i s i b l e " from t h e l e f t , and t h e A's a s heavy l i n e s i f t h e y are v i s i b l e from t h e r i g h t . These v i s i b l e T's can be s a i d t o form t h e w e s t e r n f a c a d e , and t h e v i s i b l e A's t h e g a s t e r n f a c a d e . A l l t h e o t h e r A's and T's can be combined p a i r w i s e t o AT-couples.

This c o u p l i n g c a n be v i s u a l i s e d a s f o l l o w s : we c o n s i d e r t h e g r a p h s

as a l o n g i t u d i n a l s e c t i o n of a f o l d e d p a p e r s t r i p . Next w e a p p l y g l u e t o t h e u p p e r s i d e of t h e s t r i p and p r e s s t h e s t r i p

t o g e t h e r ( p r e s s i n g from t h e l e f t and from t h e r i g h t ) . Taking

t h e p r e s s u r e away, we s e e t h a t t h e AT-couples a r e p a s t e d t o g e t h e r , and t h a t w e s t e r n and e a s t e r n f a c a d e a r e u n i n t e r r u p t e d l i n e s

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F i g u r e 2. Showing w e s t e r n f a c a d e , e a s t e r n f a c a d e , and AT-couples.

To e v e r y T and t o e v e r y A i n t h e main l i n e sequence t h e r e c a n be a t t a c h e d a p o s i t i v e number: t h e l e n g t h of t h e norm of t h e l e f t - h a n d s u b t r e e . O r , i n t e r m s of e x p r e s s i o n s i n s t e a d of t r e e s , t o t h e a b s t r a c t o r [x:P] we a t t a c h t h e number l n ( P ) , and t o t h e a p p l i c a t o r <Q> we a t t a c h l n ( Q ) . Note t h a t t h e s e norms a r e d e f i n e d o n l y by v i r t u e of t h e lower p a r t of t h e t r e e ; t h a t

p a r t g i v e s a meaning t o t h e t y p e s of t h e dummies i n v o l v e d i n P and Q.

S i m i l a r l y we can a t t a c h a v a l u e of I n t o t h e r i g h t m o s t end-point of t h e lambda t r e e , e i t h e r l n ( x ) , where x i s a dummy, o r l n ( ' C ) ( = I ) .

L e t [ x l : P l ] , . . . , [ x k : P k ] be t h e a b s t r a c t o r s of t h e w e s t e r n f a c a d e , and <Ql>,...,<Qm> t h e a p p l i c a t o r s of t h e e a s t e r n f a c a d e . L e t u s c a l l

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--

5

--

the open in^ value of our lambda tree,

the deficit of the sequence of abstractors and aplicators. The length of the norm of the rightmost end-point will be called the closing value.

We note that all terminology developed thus far for lambda trees, can be used for its subtrees too.

5. If a lambda tree is norm-correct

([I]

section 5.9) then in every AT-couple <Q>[x:P] we have ln(Q) = ln(P). Moreover we have

ln([x:P]R) = ln(P)

+

ln(R), ln(<Q>R) = ln(R)

-

ln(Q)>.

By

the way, this shows that the purely accidental notational

similarity between "length of norm" and "natural logarithm" is not so unfortunate after all. The formation of [x:P]R can be felt as a kind of product, and <Q>R as a kind of quotient (with the denominator in front).

6.

The following theorem for norm-correct lambda trees can be proved by routine methods:

Theorem 1. Let

E

be a subtree in a norm-correct lambda tree,

-

let p be its opening value, d its deficit, q its closing value. Then we have

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7. We i n t r o d u c e a s l i g h t m o d i f i c a t i o n of t h e n o t i o n of a l m o s t - n o r m a l i t y , t o be c a l l e d f r a c t u r e d almost-normality. L e t P be a lambda t r e e , and l e t i t be w r i t t e n i n t h e form WUd where W and U a r e sequences of a b s t r a c t o r s and ,

a p p l i c a t o r s , and CI: i s e i t h e r o r a dummy. L e t t h e e a s t e r n f a c a d e of W c o n s i s t of t h e a p p l i c a t o r s <Ql>,...,<Qm>.

W e s a y t h a t P i s f r a c t u r e d a l m o s t normal w i t h r e s p e c t t o W i f a l l i t s AT-couples admit AT-removals, p o s s i b l y e x c e p t f o r AT-couples of which t h e a p p l i c a t o r i s one of

<Ql>,.*.,<Qm>.

So i n ( h i u ~ t h e r e may o c c u r bound i n s t a n c e s of dummies which a r e bound by a b s t r a c t o r s i n U which form AT-couples w i t h one

of <Ql>,...,<Qm>. I f s u c h bound i n s t a n c e s do n o t o c c u r i n U i t s e l f ( l e a v i n g t h e f i n a l x a s t h e o n l y p o s s i b i l i t y ) , we can e x p r e s s t h i s by s a y i n g t h a t

WU

i s almost-normal.

I f WIJU i s f r a c t u r e d almost-normal w i t h r e s p e c t t o W , and i f WIJ i s almost-normal, t h e n w e have two p o s s i b i l i t i e s :

( i ) The f i n a l & ' d o e s n o t admit l o c a l b e t a - r e d u c t i o n . I n o t h e r words, L'is e i t h e r C o r i t i s a dummy bound by an

a b s t r a c t o r t h a t d o e s n o t form a n AT-couple w i t h a n a p p l i c a t o r i n t h e e a s t e r n f a c a d e of W.

I n t h i s c a s e WUw. i s o b v i o u s l y almost-normal.

( i i ) The f i n a l c t ' i s a dummy x , bound by a n a b s t r a c t o r forming a n AT-couple w i t h some < Q j > from t h e e a s t e r n f a c a d e of W. L e t t h i s Qj have t h e form

V T .

( C i s t o r a dummy). W a p p l y e l o c a l b e t a - r e d u c t i o n , which t r a n s f o r m s WUx i n t o W W 6

.

I n t h i s s i t u a t i o n i t i s e a s y t o s e e t h a t

W U V C

i s f r a c t u r e d almost-normal w i t h r e s p e c t t o W. The dummies i n

V 6

bound by

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a b s t r a c t o r s o u s i d e

V T

cannot g i v e r i s e t o b e t a - r e d u c t i o n , s i n c e V 6 i s a t r a n s p l a n t a t i o n of a p i e c e of W, and W i s

almost-normal. But t h e dummies i n

V t

bound by a b s t r a c t o r s i n

V Q

i t s e l f , c a n g i v e r i s e t o l o c a l b e t a - r e d u c t i o n s ; t h o s e a b s t r a c t o r s form AT-couples w i t h a p p l i c a t o r s on t h e e a s t e r n f a c a d e of WU.

We s h a l l show t h a t

under t h e a s s u m p t i o n t h a t o u r lambda t r e e WUx i s norm-correct. According t o s e c t i o n 6 we have d e f i c i t ( U )

<

l n ( x ) . But t h e e a s t e r n f a c a d e of WU can c o n t a i n more t h a n t h e one of U :

i t a l s o c o n t a i n s t h a t p a r t of t h e e a s t e r n f a c a d e of W t h a t d o e s n o t form c o u p l e s w i t h p i e c e s of t h e w e s t e r n f a c a d e of Ux. One s u c h p i e c e of t h e e a s t e r n f a c a d e of W i s <Qj>, mentioned above. T h e r e f o r e

S i n c e WU i s norm-correct we have l n ( Q j ) = l n ( x ) . And by

Theorem 1 we have d e f i c i t ( U )

<

l n ( x ) . So d e f i c i t ( U )

<

l n ( Q j ) , and (1) f o l l o w s .

8. It was shown i n [ 2 ] t h a t a lambda t r e e P of l e n g t h n s a t i s f i e s

'h- I

l n ( P ) 2

.

It f o l l o w s t h a t i f a lambda t r e e has l e n g t h n t h e n

n-1

i t s d e f i c i t i s a t most n.2

.

T h i s e s t i m a t e i s rough; l e t

u s w r i t e H(n) f o r t h e b e s t p o s s i b l e e s t i m a t e .

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that any norm-correct lambda tree of length <n can be reduced, by repeated local beta-reduction (no AT-removals), to an

almost-normal form of length at most q. Then it is not hard to show that if

Rd

(where

R

is a sequence of abstractors and applicators and

id

is either -G or a dummy) is a norm-correct lambda tree of length n then we can reduce the sequence

R

to an almost-normal form of length at most (n-1)q.

10. We define the functions G and

V

by recursion.

The function G is a function of three integer variables

k , h, n, all having non-negative values only. The values of G are non-negative integers. We define them by recursion, according

to the lexicographic order of the pairs (k,n):

(i) if k = 0 or n = 0: G(k,h,n) = h .f ?z

+

I ,

(ii) if k

>

0, n

>

0:

G(k,h,n) = G(k

-

1, h

+

(n-l)G(k,h,n

-

I), h).

This definition is very similar to the one of the Ackermann function (see [ 3 ] ) .

It is easy to show that G is a monotonically increasing function of each one of its variables.

By means of this function

G

we define the function V: V(n) will be defined for all positive integers n, and its values will

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where

H

is the function mentioned in section 8.

11. We now come back to the situation sketched in section

7.

We shall prove the following

Theorem 2. Let WRrc! be a norm-correct lambda tree,

fractured almost-correct with respect to W. Let h be the length of W, n the length of

R,

k the deficit of W. Then

W R d can be reduced (by local beta-reductions, without AT-removal) to an almost-normal form of length 1 G(k,h,n).

Proof. We apply induction with respect to k , next for every value of

k

induction with repect to n, and once

k

and n have been fixed we prove the statement for all h simultaneously. If

k

= 0 or n=O then WVW is almost normal itself; its length is h

+

n

+

1.

Let k>O, and assume that the statement is true for all smaller values of

k

(with arbitrary h and n). We apply induction with respect to n. The case n=O has been settled, so we take n

>

0 , and we assume that the statement is true for all smaller

values of n.

We now apply the idea exposed in section

9,

with

q=G(k,h,n-1). The idea has to

be

slightly modified because of the W in front and the fractured almost-normality, but we shall leave this to the reader. Because of our induction assumption we conclude that our WR can be reduced to some almost-normal WU (the reduction cannot affect W)

,

where the length of

U

is InG(k,h,n-1).

We can now apply section

7.

In case (i) W U 3 is almost-normal, in case (ii) W U L ; reduces to W W C

,

the length of V is at most h (V being a copy of a part of W), W U V 6 is fractured almost-normal with respect to WU, and the

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d e f i c i t of WU i s l e s s t h e n k. So by i n d u c t i o n W U V C r e d u c e s t o

something of l e n g t h a t most G(k-l,h+(n-l)G(k,h,n-l),h). By t h e

d e f i n i t i o n of G this i s a t most G(k,h,n).

1 2 . W e c a n now prove o u r main r e s u l t .

Theorem 3 . A norm-correct lambda t r e e of l e n g t h m can be reduced ( l o c a l b e t a - r e d u c t i o n s , no AT-removals) t o a n a l m o s t - c o r r e c t lambda t r e e w i t h l e n g t h 5: V(m). 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 m. The c a s e m = l i s t r i v i a l . Having proved t h e r e s u l t f o r m

g

n , we s h a l l t r e a t t h e c a s e m = n

+

1. L e t t h e t r e e be p r e s e n t e d a s

Q3

(where c) i s e i t h e r t o r a dummy). So Q h a s l e n g t h n. According t o s e c t i o n 9 we can reduce

Q t o a n almost-normal form W of l e n g t h S ' n V ( n ) .

I f G, e q u a l s

r ,

o r a dummy t h a t does n o t admit b e t a - r e d u c t i o n ,

t h e n W c : i s almost-normal, s o i t s u f f i c e s t o remark t h a t nV(n)

+

1

C

V(n

+

1 ) .

If Gi i s a dummy that a d m i t s b e t a - r e d u c t i o n , t h e n t h i s r e d u c t i o n l e a d s t o W U c r (where a g a i n f,T i s e i t h e r

r

o r a dummy), and t h i s i s f r a c t u r e d almost-normal ( s e e s e c t i o n 7 ) ,

s o we c a n a p p l y s e c t i o n 11. The l e n g t h of W i s a t most nV(n).

F u r t h e r m o r e , U i s a copy of a s u b e x p r e s s i o n of W , s o i t s l e n g t h i s

<

nV(n) t o o ( b u t i t i s n o t hard t o show t h a t i t i s _( V(n)

i t s e l f ) . The d e f i c i t of W can be e s t i m a t e d by H(n) a c c o r d i n g t o s e c t i o n 8 (remark t h a t t h e d e f i c i t d i d n o t change by p a s s i n g from Q t o W; w i t h o u t t h a t remark we would have t o u s e t h e e s t i m a t e ~ ( n ~ ( n ) ) i n s t e a d of H ( n ) ) . Applying Theorem 2 we g e t an

almost-normal form of l e n g t h a t most G(H(n),nV(n),nV(n)), and this e q u a l s V(n+l).

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13.

We shall indicate how strong normalization for norm-correct lambda trees follows from Theorem

3.

First we mention that for our local beta-reduction the Church-Rosser property holds. This is essentially contained in Nederpelt ( 4 1 , where it is proved for

fl.

-reduction (

P

-reduction is the effect of local beta-reduction applied to all dummies bound by one and the same abstractor).

The normal forms corresponding to local beta-reduction are what we have called almost-normal. From Theorem 3 we have the

reducibility to such an almost-normal form, and the Church-Rosser property expresses the uniqueness.

If local beta-reduction transforms the lambda tree (V,lab) into a lambda tree (V0,lab') then (V,lab) is almost entirely embedded in (V',labO): we have VcV', and lab' coincides with lab on

V with the exception of the end-points to which the

beta-reduction was applied. It follows that (V,lab), and all its reducts (by repeated local beta-reduction), are embedded in the almost-normal form, if we take exception for the labels of end-points.

By a local beta-reduction the length of a lambda tree will increase, apart from a trivial case where it remains constant. The trivial case is that the AT-couple in question has its applicator in the form <x>, where x is either t o r a dummy.

This increase of length enables us to prove strong

normalization.

Theorem

4

(strong normalization for local beta-reduction). Let

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be such that every P . is obtained from P; by some local

L + I

3

beta-reduction. Then we have m S(V(n))

+

V(n).

Proof. Let Q be the almost-normal form of P. We have

We show that a sub-sequence P.,P

,...,

P.?L can have constant length

J

J*'

3

only if h

3

(~(n))'. In such a sub-sequence the tree is constant, only the labels at end-points can change. At such an end-point a dummy is sometimes replaced by another one whose abstractor lies lower in the tree, or by t

.

The number of candidates for the change is at most equal to the length of the tree, so at most V(n). A candidate can never turn up a second time for the

same position. Irrespective of the order in which these

2

replacements ar carried out, we conclude to h

5

(V(n))

.

In the sequence P,

,

...

,

Pm we have at most V(n)

-

1

steps where the length increases, and, at most V(n) subintervals

2

of length S (V(n)) where the length is constant. This proves the theorem.

14.

If we admit both local beta-reduction and AT-removal, we get to what is usually called the normal form. It is easy to understand that the longest reduction sequences are obtained by taking first all the local beta-reduction steps and then the AT-removals. A tree with length

5

V(n) can not have more than V(n) AT-couples, so a reduction sequence (starting from