Algebraic and Regular Trees

P UBLICATIONS DU D ÉPARTEMENT DE MATHÉMATIQUES DE L ^YON

B. C ^OURCELLE

Algebraic and Regular Trees

Publications du Département de Mathématiques de Lyon, 1985, fascicule 2B

« Compte rendu des journées infinitistes », , p. 91-95

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Numérisation de documents anciens mathématiques

A L G E B R A I C AND REGULAR TREES by B. COURCELLE

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T h e o r e t i c a l C o m p u t e r S c i e n c e 2 5 ( 1 9 8 3 ) 9 5 - 1 6 9 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

F U N D A M E N T A L P R O P E R T I E S O F I N F I N I T E T R E E S

B r u n o C O U R C E L L E

VER de Mathématiques et Informatique, Université Bordeaux-Ï, 33405 Talence. France

Introduction

Infinite trees naturally arise in mathematical investigations on the semantics of p r o g r a m m i n g languages. T h e y arise in essentially two ways: when o n e unloops or unfolds a p r o g r a m undefinitely. O n e obtains then either a tree of execution paths (infinite in genera!) in the case of a p r o g r a m written in an imperative language like F O R T R A N or an expression tree in the case of a p r o g r a m written in an applicative language like LISP. In the latter case, the expression tree is usually infinite although its value can be finitely c o m p u t e d in each case; this is possible by the use of if-then-else as a base function (like the addition of integers) and not as a piece of control s t r u c t u r e . O n c e again, the infiniteness of the tree c o r r e s p o n d s to the infiniteness of the set of possible c o m p u t a t i o n s .

In both cases, the semantics of the program is completely defined by the associated tree. H e n c e two p r o g r a m s are equivalent if the associated trees are the same (the converse being not true). Roughly speaking, this allows to distinguish between the equivalence of p r o g r a m s which is only due to the control structure (loops, recursive calls, etc. . . .) from the equivalence which also d e p e n d s on the p r o p e r t i e s of the d o m a i n s of c o m p u t a t i o n and the given base' functions on these d o m a i n s .

It should be n o t e d that these infinite tiees are finitely defined. H e n c e we are lead to try to decide w h e t h e r two infinite trees defined in some finitary way are equal.

91

T w o types of infinite trees will be considered: the regular trees which are defined by unlooping F O R T R A N - l i k e p r o g r a m or flowcharts and the algebraic trees which are defined by unfolding recursive program schemes m o r e or less derived from LISP p r o g r a m s .

W e shall introduce o p e r a t i o n s on trees: the first-order substitution which corre- sponds (roughly) t o the sequential composition of flowcharts (by the o p e r a t o r ; of A L G O L ) or to functional application (in the case of an applicative language). We shall also introduce t h e second-order substitution which c o r r e s p o n d s to the replace- ment of a function symbol in an expression tree by some expression tree intended to d e n o t e the c o r r e s p o n d i n g function.

H e r e is a brief survey of the content of the p a p e r which is i n t e n d e d to be a synthesis of several aspects of infinite trees usually defined and studied separately for different p u r p o s e s :

(1) Topological (i.e. metric) and order-theoretical properties of infinite trees are investigated in parallel in o r d e r to enlighten similarities and differences.

(2) First- and second-order substitutions are investigated in the two above frameworks.

(3) Regular trees, rational expressions defining them are studied. T h e concept of an iterative theory, d u e to C . C . Elgot, is one of the possible algebraic frameworks where to study infinite t r e e s : the set of regular trees forms the free iterative theory.

Regular trees also arise as most general first-order unifiers in a generalized sense.

(4) Algebraic trees play a similar role with respect to s e c o n d - o r d e r substitutions as regular trees with respect to first-order ones. Their combinatorial p r o p e r t i e s are sufficiently complicated to yield an open problem equivalent to the D P D A equivalence p r o b l e m .

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T H E SOLUTIONS OF TWO S T A R - H E I G H T PROBLEMS FOR R E G U L A R T R E E S

J P. B R A Q U E L A I R E a n d B. C O U R C E L L E

Department of Mathematics and Computer Science. Bordeaux-l University. 3340* Interne, trance

Introduction

R e g u l a r t r e e s , i.e., t r e e s w h i c h a r e e i t h e r finite o r infinite w i t h o n l y finitely m a n y d i s t i n c t s u b t r e e s , p l a y a n i m p o r t a n t r o l e in t h e t h e o r y of p r o g r a m s c h e m e s I h e y h a v e b e e n i n v e s t i g a t e d b y C o u s i n e a u [ 9 ] , J a c o b [ 1 6 ] , E l g o t et al. [ 1 3 ] a n d ( o u r c e l l e [ 8 ] .

S i n c e t h e y f o r m t h e free i t e r a t i v e t h e o r y ( g e n e r a t e d by s o m e r a n k e d a l p h a b e t F ) , t h e y a r e d e n o t e d by c e r t a i n iterative theory expressions ( s e e [ 2 , 1 3 , 14]) T h e s e i t e r a t i v e t h e o r y e x p r e s s i o n s i n c l u d e t h e rational expressions i n d e p e n d e n t l y d e f i n e d by C o u s i n e a u [ 9 ] . T h e r e l a t i o n b e t w e e n t h e s e t w o c l a s s e s of e x p r e s s i o n s h a s b e e n s h o w n b y C o u r c e l l e [ 8 ] .

All t h e s e e x p r e s s i o n s u s e a n i t e r a t i o n o p e r a t o r ( d e n o t e d t o r *) v e r y c l o s e t o K l e e n e \ * for l a n g u a g e s . T h e y r a i s e a star-height problem, i.e., t h e p r o b l e m of c o n s t r u c t i n g a r a t i o n a l e x p r e s s i o n of m i n i m a l s t a r - h e i g h t w h i c h d e f i n e s a g i v e n r e g u l a r t r e e

T h i s p r o b l e m is trivial for i t e r a t i v e t h e o r y e x p r e s s i o n s w h i c h u s e v e c t o r i t e r a t i o n s i n c e e v e r y r e g u l a r t r e e c a n b e d e f i n e d by s u c h a n e x p r e s s i o n w i t h o n e i t e r a t i o n if t h e t r e e is infinite a n d n o i t e r a t i o n if it is finite [ 1 2 ] . It is n o t if i t e r a t i v e t h e o r y e x p r e s s i o n s a r e r e s t r i c t e d s o as t o u s e o n l y scalar iteration. W e s o l v e it a n d w e s h o w t h a t t h e m i n i m a l s t a r - h e i g h t is e x a c t l y t h e r a n k of t h e m i n i m a l g r a p h of t h e t r e e ( t h e r a n k of a d i r e c t e d g r a p h h a s b e e n i n t r o d u c e d by E g g a n [ 1 0 ] f o r t h e s t u d y of r a t i o n a l e x p r e s s i o n s d e f i n i n g l a n g u a g e s a n d f u r t h e r i n v e s t i g a t e d b y M c N a u g h t o n [ 1 7 . 18] a n d C o h e n a n d B r z o z o w s k i [ 3 - 6 ] ) .

T h e s e e x p r e s s i o n s u s e a n o p e r a t i o n c a l l e d composition, a t y p i c a l c a s e of w h i c h is e.,

(e

. e

), w h i c h d e n o t e s t h e t r e e o b t a i n e d by t h e s u b s t i t u t i o n of V a l ( ? , ) . . Va!( e

) at c e r t a i n l e a v e s of V a l ( t ' ) ( w e d e n o t e b y V a l ( ^ ) t h e t r e e d e f i n e d by t h e e x p r e s s i o n e).

T h e m a j o r c o n t r i b u t i o n of C o u s i n e a u w a s t o s h o w t h a t t h e o p e r a t i o n of c o m p o s i - t i o n is d i s p e n s a b l e a n d t h a t t h e r e s u l t i n g e x p r e s s i o n s still g e n e r a t e all r e g u l a r t r e e s ( s e e ¡ 8 ] for a s i m p l e p r o o f ) . T h e s e r e s t r i c t e d e x p r e s s i o n s r a i s e a n o t h e r s t a r - h e i g h t p r o b l e m for w h i c h w e a l s o give t h e s o l u t i o n . T h e m i n i m a l s t a r - h e i g h t in t h i s s e n s e is a l s o o b t a i n e d f r o m t h e c o n s i d e r a t i o n of t h e m i n i m a l g r a p h of t h e t r e e .

94

F o r t e c h n i c a l r e a s o n s , w e shall w o r k n e i t h e r w i t h i t e r a t i v e t h e o r y e x p r e s s i o n s [ 2 , 8, 1 3 ] n o r w i t h r a t i o n a l e x p r e s s i o n s [ 8 , 9 ] b u t w i t h s l i g h t l y d i f f e r e n t e x p r e s s i o n s (still c a l l e d rational) w h i c h u s e t h e f o l l o w i n g c o n s t r u c t i o n s :

*,ie\. i t e r a t e V a l ( ^ ) w i t h r e s p e c t t o t h e v a r i a b l e v.

(^J.<< ^lK)(e^ . . . , e^k): s u b s t i t u t e V a l ( e , ) , . . . , V a l ( e^{f c}

) for Vi,...,v

in

O u r r e s u l t s will be o b t a i n e d for t h e s e r a t i o n a l e x p r e s s i o n s b u t t h e y t r a n s f e r e a s i l y t o t h e a b o v e m e n t i o n e d e x p r e s s i o n s .

T h e p r o o f s of o u r t w o r e s u l t s f o l l o w t h e s a m e p a t t e r n t h a t c a n b e s k e t c h e d a s f o l l o w s .

A r e g u l a r t r e e is m a n i p u l a t e d by m e a n s of a finite p o i n t e d g r a p h of w h i c h it is t h e infinite u n l o o p i n g . T h e s e g r a p h s c a n b e ' s t r u c t u r e d ' , in d i f f e r e n t w a y s , b u t e a c h ' s t r u c t u r i n g ' is c h a r a c t e r i z e d by a n i n t e g e r , its ' d e p t h ' .

F o r e a c h s t r u c t u r i n g of ' d e p t h ' n, o n e c a n c o n s t r u c t a r a t i o n a l e x p e s s i o n of s t a r - h e i g h t n. H e n c e , a c e r t a i n r a t i o n a l e x p r e s s i o n c a n b e a s s o c i a t e d w i t h a ' s t r u c t u r ­ i n g ' of m i n i m a l ' d e p t h ' of t h e m i n i m a l p o i n t e d g r a p h of t h e g i v e n t r e e .

It t u r n s o u t t h a t t h i s r a t i o n a l e x p r e s s i o n is t h e r i g h t o n e , i.e., is of m i n i m a l s t a r - h e i g h t a m o n g all t h o s e d e f i n i n g t h e g i v e n t r e e .

In o r d e r t o p r o v e t h i s , w e first d e f i n e s o m e s y n t a c t i c a l m a n i p u l a t i o n s p e r f o r m i n g s o m e s i m p l i f i c a t i o n s of r a t i o n a l e x p r e s s i o n s . T h e y t r a n s f o r m a r a t i o n a l e x p r e s s i o n i n t o a n e q u i v a l e n t o n e in normal form.

F r o m t h e s y n t a c t i c a l s t r u c t u r i n g of a m i n i m a l r a t i o n a l e x p r e s s i o n in n o r m a l f o r m , o n e c a n c o n s t r u c t a ' s t r u c t u r i n g ' of t h e m i n i m a l g r a p h of t h e t r e e w h o s e ' d e p t h ' is n o t less t h a n t h e s t a r - h e i g h t of t h e e x p r e s s i o n . A n d f r o m t h e first c o n s t r u c t i o n o n e g e t s a n e q u a l i t y a s r e q u i r e d .

References

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B. COURCELLE

U.E.R. de Mathematiques et Informatique Université de Bordeaux I - F-33405 TALENCE