Dalen, D. van

Schwichtenberg, Helmut

Troelstra, A.S.

English

Open Access LMU

STUDIES IN LOGIC AND THE FOUNDATIONS OF MATHEMATICS V O L U M E 110 Editors J. B A R W I S E , Stanford D. K A P L A N , Los Angeles H. J . K E I S L E R , Madison P. SUPPES, Stanford . S. T R O E L S T R A , Amsterdam N O R T H - H O L L A N D P U B L I S H I N G C O M P A N Y A M S T E R D A M · N E W Y O R K · O X F O R D THE L. Ε. J. BROUWER CENTENARY SYMPOSIUM Proceedings of the Conference heldinNoordwijkerhout, 8-13 June, 1981 Edited by A. S. TROELSTRA Universiteit van Amsterdam Amsterdam, The Netherlands and D. VAN DALEN Rijksuniversiteit Utrecht Utrecht, The Netherlands cm m 1982 N O R T H - H O L L A N D P U B L I S H I N G C O M P A N Y A M S T E R D A M · N E W Y O R K · O X F O R D CONTENTS PREFACE P.H.G. ACZEL The type t h e o r e t i c i n t e r p r e t a t i o n of c o n s t r u c t i v e set theory choice p r i n c i p l e s D.S. BRIDGES Recent progress i n c o n s t r u c t i v e approximation theory D. H.J. DE JONGH Formulas of one p r o p o s i t i o n a l v a r i a b l e i n i n t u i t i o n i s t i c a r i t h m e t i c * C.N.DELZELL Continuous sums of squares of forms S. FEFERMAN Monotone i n d u c t i v e d e f i n i t i o n s M.P. FOURMAN Notions of choice sequence M.P. FOURMAN, R.J. GRAYSON Formal spaces* L. GORDEEV Constructive models f o r set theory w i t h e x t e n s i o n a l i t y S. HAYASHI A note on the bar i n d u c t i o n r u l e * J.M.E. HYLAND The e f f e c t i v e topos G. KREISEL, A. MACINTYRE Constructive l o g i c versus a l g e b r a i z a t i o n I B.A. KÜHNER Some extensions of Markov's c o n s t r u c t i v e continuum and t h e i r a p p l i c a t i o n s to the theory of c o n s t r u c t i v e f u n c t i o n s * J. LAMBEK, I.MOERDIJK Two sheaf representations of elementary toposes* J.S. LODDER, D. VAN DALEN Lawlessness and independence* E. MARTINO Creative subject and bar theorem* G. METAKIDES, A. NERODE The i n t r o d u c t i o n of non-recursive methods i n mathematics R. MINES Alg e b r a i c number theory, a survey* Contents I . MOERDIJK* Glueing t o p o i and higher order d i s j u n c t i o n and existence* 359 G. E. REYES, Ngo VAN QUf Smooth functors and s y n t h e t i c calculus 377 F. RICHMAN F i n i t e dimensional algebras over d i s c r e t e f i e l d s 397 W. RUITENBURG P r i m a l i t y and i n v e r t i b i l i t y of polynomials* 413 A. S C E D R O V Independence of the fan theorem i n the presence of c o n t i n u i t y p r i n c i p l e s * 435 A. SCEDROV, P.J. SCOTT A note on the Friedman slash and Freyd covers* 443 H. SCHWICHTENBERG Complexity of no r m a l i z a t i o n i n the pure typed lambda-calculus 453 CA. SMORYNSKI Nonstandard models and c o n s t r u c t i v i t y 459 A.S. TROELSTRA On the o r i g i n and development of Brouwer's concept of choice sequence* 465 G. F. VAN DER HOEVEN An a p p l i c a t i o n of p r o j e c t i o n s of lawless sequences 487 W.P. VAN STIGT L.E.J. Brouwer, the s i g n i f i c i n t e r l u d e 505 W. VELDMAN On the c o n s t r u c t i v e c o n t r a p o s i t i o n s of two axioms of countable choice 513 An a s t e r i s k i n d i c a t e s a c o n t r i b u t e d paper. The f o l l o w i n g i n v i t e d speakers are not represented by a paper i n t h i s volume (the t i t l e of t h e i r i n v i t e d address be-tween parentheses): V . L i f s c h i t z ( C l a s s i c a l numbers from the c o n s t r u c t i v e stand-p o i n t ; P.Martin-Löf (About mathematical expressions and t h e i r synonymy); D.S.Scott (Some sheaf models f o r i n t u i t i o n i s t i c set theory). The programme of the c o n t r i b u t e d papers sessions l i s t e d the f o l l o w i n g t a l k s not represented by a paper i n t h i s volume: M. Beeson, Remarks on the s t r u c t u r e of the continuum; 0. Demuth, On a class of real s and i t s r o l e i n c o n s t r u c t i v e a n a l y s i s ; G.R. Gargov, On the modal l o g i c s of p r o v a b i l i t y i n some extensions of Heyting's a r i t h m e t i c ; J. Geiser, Tense l o g i c w i t h a p p l i c a t i o n s to c o n s t r u c t i v e a n a l y s i s ; M. Gelfond, A class of theorems w i t h v a l i d c o n s t r u c t i v e counterpart; N. Greenleaf, I d e n t i t y and e q u a l i t y i n c o n s t r u c t i v e set theory; G. Heinzmann, Les points de vue c o n s t r u c t i v i s t e s de l a philosophie des mathema-ti q u e s de Ferdinand Gonseth(1890-1975); THE LM.J. BROUWER CENTENARY SYMPOSIUM A.S. Troelstra and D. van Dalen (editors) © North-Holland Publishing Company, 1982 453 COMPLEXITY OF NORMALIZATION IN THE PURE TYPED LAMBDA - CALCULUS Hdbrut SclwxLcktznbeAg By the pure typed λ-calculus we mean as usual the system of terms b u i l t up from typed v a r i a b l e s x T , y T , . . . and maybe typed constants a T,b T,... by means of a p p l i ­c a t i o n {t0*Ts°) and λ-abstraction {\x°tT)°~*T. Here the types τ , σ , . . . are induc­t i v e l y generated from a ground type 0 by means of ( σ - > τ ) . I t is well-known ( c f . e.g. T r o e l s t r a [T] ) that any such term has a uniquely determined normal form w i t h respect t o so-called B-reductions . . . ( X x t ) s . . . -»·... t [s] ... , and that t h i s nor­mal form w i l l e v e n t u a l l y be reached no matter which sequence of reduction steps one chooses 1^. In t h i s paper we w i l l be concerned w i t h estimates f o r the number of re d u c t i o n steps necessary to reach the normal form. We w i l l give an & lower bound i n §1 by 2) w r i t i n g down terms t of length 3n and showing that i t takes at least 2 n_^ - η reduction steps ( w i t h 2_ := 1 , 2 , := 2 n ) t o br i n g t i n t o i t s normal form. In §2 0 n+i η ^ we describe a p a r t i c u l a r normalization procedure and give an £ upper bound ( i n terms of m a x ( l h ( t ) , L ( t ) ) , where 1h(t) denotes the length of t and L ( t ) denotes the inner type level of t , i . e . the maximum type level ^ of a subterm of t ) f o r the number o f reduction steps t h i s procedure w i l l c a rry out. The r e s u l t of §1 also f o l l o w s from Statman [S] , where i t i s shown more gene­r a l l y that the problem whether two terms t ^ and t ^ have the same normal form i s not elementary r e c u r s i v e . However, f o r the more s p e c i f i c question we are i n t e r e s ­ted in here i t is possible to give our much simpler proof. Also, the mere r e s u l t of §2, namely t h a t f o r some s p e c i f i c n ormalization procedure there is an & upper We make use here of the f o l l o w i n g conventions, ( l ) Type supersc r i p t s w i l l be omitted whenever they are clear from the context or i n e s s e n t i a l . (2) Terms t h a t d i f f e r only in the bound v a r i a b l e s used are i d e n t i f i e d . (3) S u b s t i t u t i o n i s de­noted by t [s] . (4) Brackets w i l l be omitted whenever possible; we w i l l w r i t e t s r for ( t s ) r . x 2) By the length of a term t we mean the number of occurrences of v a r i a b l e s or con­stants in t except those immediately behind a λ-symbol. ^The type l e v e l | τ | of a type τ i s defined i n d u c t i v e l y by 10 Ε : = 0, ! σ τ | = max ( Ι σ | + 1 , | τ | ) . 454 Η. SCHWICHTENBERG bound on the number of reduction steps, is c e r t a i n l y not new to any expert in the f i e l d . However, i t seems t h a t the simple e x p l i c i t d e s c r i p t i o n of the bounding f u n c t i o n obtained below is of some i n t e r e s t . I t should also be noted t h a t , by combining the r e s u l t s proved here w i t h those of Gandy [G], one can o b t a i n a universal & upper bound f o r the number of reduc­t i o n steps w i t h respect to any normalization procedure. This can be seen as f o l ­lows. For any term t of type τ, by Gandy's method one can define a closed type-0-term | t | w i t h the property tha t i t s numerical value i s a bound on the number of reduction steps, where i t does not matter in which way the reduction steps are choosen. Now to obtain a bound f o r the numerical value of | t | , we f i r s t note that by the argument of §2 we have an & bound on the number of reduction steps the s p e c i f i c n ormalization procedure given there w i l l carry out to produce the normal form of | t | ; t h i s bound is in terms of max(1h|11), L (111)). Since, by Gandy's con­s t r u c t i o n of | t | , l h ( | t | ) depends only l i n e a r l y on l h ( t ) , and L ( | t | ) = L ( t ) , we also have an upper bound on the number of reduction steps in terms of m a x ( l h ( t ) , l _ ( t ) ) . Next note that any reduction step at most squares the length of the o r i g ­in i n a l term. So we have an £ upper bound on the length (and hence on the numerical value) of the normal form of I t l , again in terms of max(1h(t), L ( t ) ) . This gives the desired r e s u l t . (The f a c t t h a t one can o b t a i n an £ upper bound from Gandy's work in [G] has been mentioned to me by G.E. Mine and R. Statman). §1. The pure types k are defined i n d u c t i v e l y by 0 : = 0, k +1 := k k . We de­f i n e j_t£rajtj_on jf^nc_tJjojTa_l_s I of pure type k + 2 by w i t h η occurrences of f a f t e r AfAx; here f,x are v a r i a b l e s of type k + 1,k, res­p e c t i v e l y . Let f°g be an a b b r e v i a t i o n f o r λχ f ( g x ) , and l e t t = s mean that t and s have the same normal form. With t h i s n o t a t i o n we can w r i t e w i t h η occurrences of f a f t e r Af. The main point of our argument i s the f o l l o w i n g simple lemma, which can be traced back t o Rosser ( c f . Church [C, p. 3 0 ] ) . η I :== AfAx f ( f ( . . . f ( f x ) . . . ) ) , η I = A f f o f o . . . o f η LEMMA. (I f)°(l f ) = I m η m+n f = I m η 1 1 = 1 m η m η Normalization in typed lambda-calculus 455 As an immediate consequence we have t n : s ι 2 ι 2 . . . ι 2 - . 2 2 (w i t h 2 n := 1, 2 1 := 2 ) . Now consider any sequence of reduction steps t r a n s -0 ' ' n+1 3 t i n t o i t s 3 η steps in t h i s sequence. forming i t s normal form, and l e t S denote the t o t a l number of reduction a η THEOREM. S >2 0 - n. η n-2 P^.00JS. The length of t is 3n. Note that any reduction step . . . (Axt) s . . . -> . . . ί χ [s] . .. can at most square the 1 ength of the or i gi na 1 term. Hence we have 2 < length of l _ (the normal form of t ) η ζ η η S 2 η < (length of t ) \ Π 2 n = ( 3 n ) 2 2 n + $ n 2 n < 1L (since 3n<2^ ) , and the theorem is proved. §2. Our aim here is to set up a s p e c i f i c normalization procedure f o r which an 8> upper bound on the number of reduction steps can be obtained e a s i l y . So l e t an a r b i t r a r y term be given. Our normalization procedure is an obvious one: we search f o r redexes of maximal type l e v e l , and among those we take the rightmost one and convert i t . Here by a redex we mean as usual an occurrence of a subterm ( λ χ σ ί τ ) σ Ίs°, and to convert i t means to replace i t by ΐ χ[δ]. i t s type level i s the type level of σ-*τ. In order to get an estimate f o r the number of reduction steps needed, we associate a number w i t h any given term and show that t h i s number decreases w i t h any reduction step. To o b t a i n such a number, we f i r s t assign to any term t a sequence a . ( t ) of numbers, as f o l l o w s : a . ( t ) is the number of redexes or o f v a r i ­ables in t w i t h type 1evel i + 1. Obv i ous1y only f i η i t e l y many a . ( t ) w i l l be d i f -ferent from 0. Now l e t us consider a reduction step and see how the assigned se­quence w i l l change: 456 Η. SCHWICHTENBERG . . . ( A x t ) s t [s ] . . c + a ( t ) + 1 c + a ( t ) m m m m 7 c m , +a ( t ) +a (s) < c + a m , ( t ) +a As)a At) m-l m-1 m-l m-1 m-1 m-1 n -1 c 0 + a 0 ( t ) + a 0 ( s ) < c 0 + a f l ( t ) + a 0 ( s ) a n . , ( t ) . Here m+1 is the level of the type σ + τ of Xxt, and η is the level of the type σ of s. (Note t h a t , i f n = 0, we have a.(s) =0 f o r a l l i ). c. i s the c o n t r i b u t i o n to our sequence of t h a t part of the terms above that is denoted by ... . More p r e c i s e l y , i f those terms are w r i t t e n as r ^ [ ( A x t ) s ] and r ^ f t ^ O ] ] , r e s p e c t i v e l y , then c. = a . ( r ) - 1 . ι ι Now denote the sequence associated w i t h the o r i g i n a l term by 9Γ and the sequence associated w i t h the reduced term ( f o r which we only gave an estimate) by 2T' . We want to have numbers |9ΐ| , |9ϊΊ assigned to ST,ST' such that |3ί| > IST'I . This can be done as f o l l o w s . Let 2> = (d.) be a sequence w i t h only f i n i t e l y many e n t r i e s d i f f e r e n t from 0. Let m be maximal w i t h d >0. Then d e f i n e \%\ = q(m,d ,d) w i t h d = max(d n,...,d , ) , =· ' m' [1 m-1 where 2 g(m,a+1,b) = g(m,a,b ) + 1 g(m+1,0,b) = g(m,b,b) g(0,0,b) = b. Note that g belongs to the class δ of Grzegorczyk [Gr] , and t h a t f o r any f i x e d m the f u n c t i o n g(m,.,.) belongs t o & ^ , i . e . is elementary re c u r s i v e . I t is easy to check t h a t g(m,a,b) i s monotone in a and b f o r any f i x e d m. Using t h i s , l e t us show that |W| > Ι3ΓΊ . Case 1: c +a ( t ) > 0 . 3 m m Then we have |8T| = g(m,c + a ( t ) + 1 ,m) wi th Μ := max (c. + a. ( t ) + a . (s)) m m 0<i<m 1 1 1 = g(m,c +a ( t ) ,M ) + 1 m m > g(m,c + a ( t ) , max (c. +a. ( t ) + a. {s)an 1 ( t ) ) ) + I . > W I + 1 Normalization in typed lambda-calculus 457 Case 2 : c +a ( t ) = 0 and M = 0 . Obvious. Case 3 : c + a ( t ) = 0 , i maximal such m m . m m that c . + a . ( t ) + a . ( s ) = 0 . Then we have Ι9ΓΙ = g(m,1 ,M) = g(m,0,M 2) + 1 = g(m-1,M2,M2) + 1 > g(i,M 2,M 2) > IST' I . Here we have made use of the obvious f a c t t h a t g(m,b,b) is monotone in m. This concludes the proof of |3ί| > |?ί'| . We can summarize our argument as f o l l o w s . THEOREM. For any given term t , the number of reduction steps f o r the proce­dure described above is <q(m,a ( t ) , a ( t ) ) . Here m+ 1 is the maximal type l e v e l of m ' r a redex in t , a ( t ) := Q ^ 3 ^ a j ( t ) and a . ( t ) is the number of redexes or of va r i a b l e s In t whose type level i s COROLLARY, ( l ) There is an δ f u n c t i o n f such that f o r a l l closed t y p e - 0 -terms t the above normalization procedure terminates in < f ( m a x ( 1 h ( t ) , L ( t ) ) ) steps. ( 2 ) For a l l m there is an elementary recursive f u n c t i o n g m such that f o r a l l closed type - 0-terms t w i t h L ( t ) < m the above normalization procedure t e r ­minates in < g ( 1 h ( t ) ) steps, m REFERENCES [ C] Church, Α.: The c a l c u l i of lambda-conversion. Annals of Math. Studies No. 6 . Princeton 1 9 ^ 1 . [ G] Gandy, R.O.: Proofs of strong n o r m a l i z a t i o n . In: To H.B. Curry: essays in combinatory l o g i c , lambda calculus and formalism. London 1 9 8 0 . [Gr] Grzegorczyk, Α.: Some classes of recursive f u n c t i o n s . Rozprawy Mat. 4 ( 1 9 5 3 ) . [ S] Statman, R.:The typed λ-calculus is not elementary recursive. Theor. Computer Science 9 ( 1 9 7 9 ) . [ T] T r o e l s t r a , A.S. ( e d i t o r ) : Metamathematical i n v e s t i g a t i o n of i n t u i t i o n i s t i c a r i t h m e t i c and an a l y s i s . Springer Lecture Notes in Math. No. Ihk. B e r l i n 1 9 7 3 . 

(editor): Metamathematical investigation of intuitionistic arithmetic and analysis.

Proofs of strong normalization. In: To H.B. Curry: essays in combinatory logic, lambda calculus and formalism.

R.:The typed λ-calculus is not elementary recursive.

Some classes of recursive functions.

The calculi of lambda-conversion.

Complexity of normalization in the pure typed Lambda-calculus

Universität München: Elektronischen Publikationen

STUDIES IN LOGIC 
AND 
THE FOUNDATIONS OF MATHEMATICS 
V O L U M E 110 
Editors 
J. B A R W I S E , Stanford 
D. K A P L A N , Los Angeles 
H. J . K E I S L E R , Madison 
P. SUPPES, Stanford 
. S. T R O E L S T R A , Amsterdam 
N O R T H - H O L L A N D P U B L I S H I N G C O M P A N Y 
A M S T E R D A M · N E W Y O R K · O X F O R D 
THE L. Ε. J. BROUWER 
CENTENARY SYMPOSIUM 
Proceedings of the Conference 
heldinNoordwijkerhout, 8-13 June, 1981 
Edited by 
A. S. TROELSTRA 
Universiteit van Amsterdam 
Amsterdam, The Netherlands 
and 
D. VAN DALEN 
Rijksuniversiteit Utrecht 
Utrecht, The Netherlands 
cm 
m 
1982 
N O R T H - H O L L A N D P U B L I S H I N G C O M P A N Y 
A M S T E R D A M · N E W Y O R K · O X F O R D 
CONTENTS 
PREFACE 
P.H.G. ACZEL 
The type t h e o r e t i c i n t e r p r e t a t i o n of c o n s t r u c t i v e set theory 
choice p r i n c i p l e s 
D.S. BRIDGES 
Recent progress i n c o n s t r u c t i v e approximation theory 
D. H.J. DE JONGH 
Formulas of one p r o p o s i t i o n a l v a r i a b l e i n i n t u i t i o n i s t i c 
a r i t h m e t i c * 
C.N.DELZELL 
Continuous sums of squares of forms 
S. FEFERMAN 
Monotone i n d u c t i v e d e f i n i t i o n s 
M.P. FOURMAN 
Notions of choice sequence 
M.P. FOURMAN, R.J. GRAYSON 
Formal spaces* 
L. GORDEEV 
Constructive models f o r set theory w i t h e x t e n s i o n a l i t y 
S. HAYASHI 
A note on the bar i n d u c t i o n r u l e * 
J.M.E. HYLAND 
The e f f e c t i v e topos 
G. KREISEL, A. MACINTYRE 
Constructive l o g i c versus a l g e b r a i z a t i o n I 
B.A. KÜHNER 
Some extensions of Markov's c o n s t r u c t i v e continuum and t h e i r 
a p p l i c a t i o n s to the theory of c o n s t r u c t i v e f u n c t i o n s * 
J. LAMBEK, I.MOERDIJK 
Two sheaf representations of elementary toposes* 
J.S. LODDER, D. VAN DALEN 
Lawlessness and independence* 
E. MARTINO 
Creative subject and bar theorem* 
G. METAKIDES, A. NERODE 
The i n t r o d u c t i o n of non-recursive methods i n mathematics 
R. MINES 
Alg e b r a i c number theory, a survey* 
Contents 
I . MOERDIJK* 
Glueing t o p o i and higher order d i s j u n c t i o n and existence* 359 
G. E. REYES, Ngo VAN QUf 
Smooth functors and s y n t h e t i c calculus 377 
F. RICHMAN 
F i n i t e dimensional algebras over d i s c r e t e f i e l d s 397 
W. RUITENBURG 
P r i m a l i t y and i n v e r t i b i l i t y of polynomials* 413 
A. S C E D R O V 
Independence of the fan theorem i n the presence of c o n t i n u i t y 
p r i n c i p l e s * 435 
A. SCEDROV, P.J. SCOTT 
A note on the Friedman slash and Freyd covers* 443 
H. SCHWICHTENBERG 
Complexity of no r m a l i z a t i o n i n the pure typed lambda-calculus 453 
CA. SMORYNSKI 
Nonstandard models and c o n s t r u c t i v i t y 459 
A.S. TROELSTRA 
On the o r i g i n and development of Brouwer's concept of choice 
sequence* 465 
G. F. VAN DER HOEVEN 
An a p p l i c a t i o n of p r o j e c t i o n s of lawless sequences 487 
W.P. VAN STIGT 
L.E.J. Brouwer, the s i g n i f i c i n t e r l u d e 505 
W. VELDMAN 
On the c o n s t r u c t i v e c o n t r a p o s i t i o n s of two axioms of 
countable choice 513 
An a s t e r i s k i n d i c a t e s a c o n t r i b u t e d paper. The f o l l o w i n g i n v i t e d speakers are 
not represented by a paper i n t h i s volume (the t i t l e of t h e i r i n v i t e d address be-
tween parentheses): V . L i f s c h i t z ( C l a s s i c a l numbers from the c o n s t r u c t i v e stand-
p o i n t ; P.Martin-Löf (About mathematical expressions and t h e i r synonymy); 
D.S.Scott (Some sheaf models f o r i n t u i t i o n i s t i c set theory). 
The programme of the c o n t r i b u t e d papers sessions l i s t e d the f o l l o w i n g t a l k s 
not represented by a paper i n t h i s volume: 
M. Beeson, Remarks on the s t r u c t u r e of the continuum; 
0. Demuth, On a class of real s and i t s r o l e i n c o n s t r u c t i v e a n a l y s i s ; 
G.R. Gargov, On the modal l o g i c s of p r o v a b i l i t y i n some extensions of Heyting's 
a r i t h m e t i c ; 
J. Geiser, Tense l o g i c w i t h a p p l i c a t i o n s to c o n s t r u c t i v e a n a l y s i s ; 
M. Gelfond, A class of theorems w i t h v a l i d c o n s t r u c t i v e counterpart; 
N. Greenleaf, I d e n t i t y and e q u a l i t y i n c o n s t r u c t i v e set theory; 
G. Heinzmann, Les points de vue c o n s t r u c t i v i s t e s de l a philosophie des mathema-
ti q u e s de Ferdinand Gonseth(1890-1975); 
THE LM.J. BROUWER CENTENARY SYMPOSIUM 
A.S. Troelstra and D. van Dalen (editors) 
© North-Holland Publishing Company, 1982 453 
COMPLEXITY OF NORMALIZATION IN THE PURE TYPED LAMBDA - CALCULUS 
Hdbrut SclwxLcktznbeAg 
By the pure typed λ-calculus we mean as usual the system of terms b u i l t up from 
typed v a r i a b l e s x T , y T , . . . and maybe typed constants a T,b T,... by means of a p p l i ­
c a t i o n {t0*Ts°) and λ-abstraction {\x°tT)°~*T. Here the types τ , σ , . . . are induc­
t i v e l y generated from a ground type 0 by means of ( σ - > τ ) . I t is well-known ( c f . 
e.g. T r o e l s t r a [T] ) that any such term has a uniquely determined normal form w i t h 
respect t o so-called B-reductions . . . ( X x t ) s . . . -»·... t [s] ... , and that t h i s nor­
mal form w i l l e v e n t u a l l y be reached no matter which sequence of reduction steps 
one chooses 1^. 
In t h i s paper we w i l l be concerned w i t h estimates f o r the number of re d u c t i o n 
steps necessary to reach the normal form. We w i l l give an & lower bound i n §1 by 
2) 
w r i t i n g down terms t of length 3n and showing that i t takes at least 2 n_^ - η 
reduction steps ( w i t h 2_ := 1 , 2 , := 2 n ) t o br i n g t i n t o i t s normal form. In §2 0 n+i η ^ 
we describe a p a r t i c u l a r normalization procedure and give an £ upper bound ( i n 
terms of m a x ( l h ( t ) , L ( t ) ) , where 1h(t) denotes the length of t and L ( t ) denotes 
the inner type level of t , i . e . the maximum type level ^ of a subterm of t ) f o r 
the number o f reduction steps t h i s procedure w i l l c a rry out. 
The r e s u l t of §1 also f o l l o w s from Statman [S] , where i t i s shown more gene­
r a l l y that the problem whether two terms t ^ and t ^ have the same normal form i s 
not elementary r e c u r s i v e . However, f o r the more s p e c i f i c question we are i n t e r e s ­
ted in here i t is possible to give our much simpler proof. Also, the mere r e s u l t 
of §2, namely t h a t f o r some s p e c i f i c n ormalization procedure there is an & upper 
We make use here of the f o l l o w i n g conventions, ( l ) Type supersc r i p t s w i l l be 
omitted whenever they are clear from the context or i n e s s e n t i a l . (2) Terms t h a t 
d i f f e r only in the bound v a r i a b l e s used are i d e n t i f i e d . (3) S u b s t i t u t i o n i s de­
noted by t [s] . (4) Brackets w i l l be omitted whenever possible; we w i l l w r i t e t s r 
for ( t s ) r . x 
2) 
By the length of a term t we mean the number of occurrences of v a r i a b l e s or con­
stants in t except those immediately behind a λ-symbol. 
^The type l e v e l | τ | of a type τ i s defined i n d u c t i v e l y by 10 Ε : = 0, ! σ τ | = max 
( Ι σ | + 1 , | τ | ) . 
454 Η. SCHWICHTENBERG 
bound on the number of reduction steps, is c e r t a i n l y not new to any expert in the 
f i e l d . However, i t seems t h a t the simple e x p l i c i t d e s c r i p t i o n of the bounding 
f u n c t i o n obtained below is of some i n t e r e s t . 
I t should also be noted t h a t , by combining the r e s u l t s proved here w i t h those 
of Gandy [G], one can o b t a i n a universal & upper bound f o r the number of reduc­
t i o n steps w i t h respect to any normalization procedure. This can be seen as f o l ­
lows. For any term t of type τ, by Gandy's method one can define a closed type-0-
term | t | w i t h the property tha t i t s numerical value i s a bound on the number of 
reduction steps, where i t does not matter in which way the reduction steps are 
choosen. Now to obtain a bound f o r the numerical value of | t | , we f i r s t note that 
by the argument of §2 we have an & bound on the number of reduction steps the 
s p e c i f i c n ormalization procedure given there w i l l carry out to produce the normal 
form of | t | ; t h i s bound is in terms of max(1h|11), L (111)). Since, by Gandy's con­
s t r u c t i o n of | t | , l h ( | t | ) depends only l i n e a r l y on l h ( t ) , and L ( | t | ) = L ( t ) , we 
also have an upper bound on the number of reduction steps in terms of m a x ( l h ( t ) , 
l _ ( t ) ) . Next note that any reduction step at most squares the length of the o r i g ­
in 
i n a l term. So we have an £ upper bound on the length (and hence on the numerical 
value) of the normal form of I t l , again in terms of max(1h(t), L ( t ) ) . This gives 
the desired r e s u l t . (The f a c t t h a t one can o b t a i n an £ upper bound from Gandy's 
work in [G] has been mentioned to me by G.E. Mine and R. Statman). 
§1. The pure types k are defined i n d u c t i v e l y by 0 : = 0, k +1 := k k . We de­
f i n e j_t£rajtj_on jf^nc_tJjojTa_l_s I of pure type k + 2 by 
w i t h η occurrences of f a f t e r AfAx; here f,x are v a r i a b l e s of type k + 1,k, res­
p e c t i v e l y . Let f°g be an a b b r e v i a t i o n f o r λχ f ( g x ) , and l e t t = s mean that t and 
s have the same normal form. With t h i s n o t a t i o n we can w r i t e 
w i t h η occurrences of f a f t e r Af. 
The main point of our argument i s the f o l l o w i n g simple lemma, which can be 
traced back t o Rosser ( c f . Church [C, p. 3 0 ] ) . 
η 
I :== AfAx f ( f ( . . . f ( f x ) . . . ) ) , η 
I = A f f o f o . . . o f 
η 
LEMMA. (I f)°(l f ) = I m η m+n 
f 
= I 
m η 
1 1 = 1 m η m η 
Normalization in typed lambda-calculus 455 
As an immediate consequence we have 
t n : s ι 2 ι 2 . . . ι 2 - . 2 
2 
(w i t h 2 n := 1, 2 1 := 2 ) . Now consider any sequence of reduction steps t r a n s -0 ' ' n+1 
3 t i n t o i t s 3 η 
steps in t h i s sequence. 
forming i t s normal form, and l e t S denote the t o t a l number of reduction a η 
THEOREM. S >2 0 - n. η n-2 
P^.00JS. The length of t is 3n. Note that any reduction step 
. . . (Axt) s . . . -> . . . ί χ [s] . .. can at most square the 1 ength of the or i gi na 1 term. 
Hence we have 
2 < length of l _ (the normal form of t ) 
η ζ η 
η S 
2 η 
< (length of t ) 
\ Π 
2 n 
= ( 3 n ) 2 
2 n + $ n 2 n 
< 1L (since 3n<2^ ) , 
and the theorem is proved. 
§2. Our aim here is to set up a s p e c i f i c normalization procedure f o r which 
an 8> upper bound on the number of reduction steps can be obtained e a s i l y . So l e t 
an a r b i t r a r y term be given. Our normalization procedure is an obvious one: we 
search f o r redexes of maximal type l e v e l , and among those we take the rightmost 
one and convert i t . Here by a redex we mean as usual an occurrence of a subterm 
( λ χ σ ί τ ) σ Ίs°, and to convert i t means to replace i t by ΐ χ[δ]. i t s type level i s 
the type level of σ-*τ. 
In order to get an estimate f o r the number of reduction steps needed, we 
associate a number w i t h any given term and show that t h i s number decreases w i t h 
any reduction step. To o b t a i n such a number, we f i r s t assign to any term t a 
sequence a . ( t ) of numbers, as f o l l o w s : a . ( t ) is the number of redexes or o f v a r i ­
ables in t w i t h type 1evel i + 1. Obv i ous1y only f i η i t e l y many a . ( t ) w i l l be d i f -
ferent from 0. Now l e t us consider a reduction step and see how the assigned se­
quence w i l l change: 
456 Η. SCHWICHTENBERG 
. . . ( A x t ) s t [s ] . . 
c + a ( t ) + 1 c + a ( t ) m m m m 7 
c m , +a ( t ) +a (s) < c + a m , ( t ) +a As)a At) m-l m-1 m-l m-1 m-1 m-1 n -1 
c 0 + a 0 ( t ) + a 0 ( s ) < c 0 + a f l ( t ) + a 0 ( s ) a n . , ( t ) . 
Here m+1 is the level of the type σ + τ of Xxt, and η is the level of the type σ 
of s. (Note t h a t , i f n = 0, we have a.(s) =0 f o r a l l i ). c. i s the c o n t r i b u t i o n 
to our sequence of t h a t part of the terms above that is denoted by ... . More 
p r e c i s e l y , i f those terms are w r i t t e n as r ^ [ ( A x t ) s ] and r ^ f t ^ O ] ] , r e s p e c t i v e l y , 
then c. = a . ( r ) - 1 . 
ι ι 
Now denote the sequence associated w i t h the o r i g i n a l term by 9Γ and the 
sequence associated w i t h the reduced term ( f o r which we only gave an estimate) 
by 2T' . We want to have numbers |9ΐ| , |9ϊΊ assigned to ST,ST' such that |3ί| > IST'I . 
This can be done as f o l l o w s . Let 2> = (d.) be a sequence w i t h only f i n i t e l y many 
e n t r i e s d i f f e r e n t from 0. Let m be maximal w i t h d >0. Then d e f i n e 
\%\ = q(m,d ,d) w i t h d = max(d n,...,d , ) , 
=· ' m' [1 m-1 
where 
2 
g(m,a+1,b) = g(m,a,b ) + 1 
g(m+1,0,b) = g(m,b,b) 
g(0,0,b) = b. 
Note that g belongs to the class δ of Grzegorczyk [Gr] , and t h a t f o r any f i x e d 
m the f u n c t i o n g(m,.,.) belongs t o & ^ , i . e . is elementary re c u r s i v e . 
I t is easy to check t h a t g(m,a,b) i s monotone in a and b f o r any f i x e d m. 
Using t h i s , l e t us show that |W| > Ι3ΓΊ . Case 1: c +a ( t ) > 0 . 3 m m 
Then we have 
|8T| = g(m,c + a ( t ) + 1 ,m) wi th Μ := max (c. + a. ( t ) + a . (s)) 
m m 0<i<m 1 1 1 
= g(m,c +a ( t ) ,M ) + 1 m m 
> g(m,c + a ( t ) , max (c. +a. ( t ) + a. {s)an 1 ( t ) ) ) + I . 
> W I + 1 
Normalization in typed lambda-calculus 457 
Case 2 : c +a ( t ) = 0 and M = 0 . Obvious. Case 3 : c + a ( t ) = 0 , i maximal such m m . m m 
that c . + a . ( t ) + a . ( s ) = 0 . Then we have 
Ι9ΓΙ = g(m,1 ,M) 
= g(m,0,M 2) + 1 
= g(m-1,M2,M2) + 1 
> g(i,M 2,M 2) 
> IST' I . 
Here we have made use of the obvious f a c t t h a t g(m,b,b) is monotone in m. This 
concludes the proof of |3ί| > |?ί'| . We can summarize our argument as f o l l o w s . 
THEOREM. For any given term t , the number of reduction steps f o r the proce­
dure described above is <q(m,a ( t ) , a ( t ) ) . Here m+ 1 is the maximal type l e v e l of 
m ' r 
a redex in t , a ( t ) := Q ^ 3 ^ a j ( t ) and a . ( t ) is the number of redexes or of 
va r i a b l e s In t whose type level i s 
COROLLARY, ( l ) There is an δ f u n c t i o n f such that f o r a l l closed t y p e - 0 -
terms t the above normalization procedure terminates in < f ( m a x ( 1 h ( t ) , L ( t ) ) ) 
steps. ( 2 ) For a l l m there is an elementary recursive f u n c t i o n g m such that f o r 
a l l closed type - 0-terms t w i t h L ( t ) < m the above normalization procedure t e r ­
minates in < g ( 1 h ( t ) ) steps, 
m 
REFERENCES 
[ C] Church, Α.: The c a l c u l i of lambda-conversion. Annals of Math. Studies No. 6 . 
Princeton 1 9 ^ 1 . 
[ G] Gandy, R.O.: Proofs of strong n o r m a l i z a t i o n . In: To H.B. Curry: essays in 
combinatory l o g i c , lambda calculus and formalism. London 1 9 8 0 . 
[Gr] Grzegorczyk, Α.: Some classes of recursive f u n c t i o n s . Rozprawy Mat. 4 ( 1 9 5 3 ) . 
[ S] Statman, R.:The typed λ-calculus is not elementary recursive. Theor. 
Computer Science 9 ( 1 9 7 9 ) . 
[ T] T r o e l s t r a , A.S. ( e d i t o r ) : Metamathematical i n v e s t i g a t i o n of i n t u i t i o n i s t i c 
a r i t h m e t i c and an a l y s i s . Springer Lecture Notes in Math. No. Ihk. B e r l i n 
1 9 7 3 . 


Complexity of normalization in the pure typed Lambda-calculus

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