Structured Numbers. Properties of a hierarchy of operations on binary trees.

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1 Structured Numbers. Properties of a hierarchy of operations on binary trees. Vincent D. Blondel February 4, 997 Abstract We introduce a hierarchy of operations on (nite and innite) binary trees. The operations are obtained by successive repetition of one initial operation. The rst three operations are generalizations of the operations of addition, multiplication and exponentiation for positive integers. Introduction The product of two positive integers a and b can be dened as the sum of b factors each equal to a. The bth exponent of a, denoted by a " b, can similarly be dened as the product of b factors each equal to a. The process of getting new operations by repeating old ones ends with exponentiation because this last operation is not associative. The denition a "" b = a " a " a " ::: " a b factors () is ambiguous since for example (4 " 4) " 4 6= 4 " (4 " 4). For the the right hand side of () to be well dened both the number of factors and the order in which the operations " are performed have to be specied. A way of doing this is to ask the second operand to carry not only a quantitative information, the number of times a is repeated, but also a structured information, the order in which the operations " are performed. Binary trees are naturally designated object to convey such a structured information. The number of external nodes (leaves) of a binary tree can be used to specify the number of factors, and the structure of the tree can then be used to specify the order in which the operations are performed. In this paper, we dene countably many internal operations on binary trees. The rst operation, which we denote by :, is obtained by forming the binary tree whose left and right subtrees are equal to the operands. This operation is not associative. The second operation 2 : is dened as follows: From the binary trees a and b we construct the binary tree a 2 : b by repeating the operation : on the tree a with the structure dictated by b. In the same way, we dene an operation 3 : by repeating 2 :, an operation 4 : by repeating 3 :, etc. We eventually obtain countably many internal operations ( k : for k ) with the denition a k : b = a k? : a k? k? : a :... k? : a b factors Institute of Mathematics, University of Liege, B-4000 Liege, Belgium. vblondel@ulg.ac.be. Parts of this work were completed while the author was at OCIAM Oxford, at KTH Stockholm and at INRIA Paris.

2 The number of external nodes of the binary tree resulting from the operation :, 2 : and 3 : are equal to the sum, product and exponentiation of the number of external nodes of the operands. These three operations are thought of as binary trees counterparts of the usual operations of addition, multiplication and exponentiation. The operations k : for k 4 have no natural number counterparts since for these cases the structure of the trees have to be taken into account to compute the number of external nodes of a k : -product. The object of this paper is to study some of the properties of the operations k : described above and formalized in the second section of the paper. In Section 3 the operations are shown to satisfy a range of algebraic properties that generalize elementary properties for integers. In Section 4 we show that binary trees can be decomposed in a unique way as products of prime binary trees. In the fth section we analyse the operations k : for k 4. In Section 6 we describe various integer valued functions associated to trees and show how these functions behave with respect to k : -products of binary trees. In a nal section we argue that the notions introduced for nite binary trees can be generalized for innite trees. This is achieved by formalizing binary trees by means of factorial languages. Dierent authors have proposed to continue the hierarchy +; ; " on natural numbers by introducing operations of \super-exponentiation". D. Knuth's recursive denition [8] is a "" b = a " (a " (a " (a... " a)...)) b a """ b = a "" (a "" (a "" (a... "" a)...)) b a " ::: " b = a " ::: "(a " ::: " a::: " ::: " a)...)) k k? k? k? b This denition coincide, modulo elementary notational modications, with the denition originally given by Ackermann of a recursive function that is not primitive recursive (see [3]). The denition has the disadvantage of making an arbitrary choice on how the non-associative operations are performed and, as a result, these operations exhibit poor algebraic properties (see, however, []). Operations on graphs, trees and binary trees constitute a classical object of study in theoretical computer science (see [20], [2], [9], [7, Vol. Section 2.3 ]) but we have found no reference that uses the particular structure of binary trees as a mean for dening repeated operations. The contribution that is probably closest to ours is the \arithmetic of shapes" developped by I. M. Etherington half a century ago in the context of genetics. The transmission of a probability distribution of genes by mating is an operation that is commutative but not associative. In order to describe this operation, Etherington has introduced in [6] operations on trees that are similar to : ; 2 : and 3 : and that have given rise to the widely studied genetic algebras (see [6], [5] and [5]). The operations k : for k 4 are not dened in the context of genetic algebras because the trees considered there are not ordered and there is no natural denition of k : for k 4 for unordered trees. Motivated by the remarks made in this introduction, binary trees are introduced in [2], [3] and [4] as one possible representation of the concept of \structured numbers". A motivation for this terminology is justied by the following observation: Free groups with a simple generator are isomorphic to (Z; +), free monoids with a single generator are isomorphic to (N; +) and free groupoids with a single generator are isomorphic to (BT; : ) where BT denotes the set of binary trees and : is the rst operation in our hierarchy. This analogy motivates the notation SN used in [2] for the set of binary trees which can be seen as one possible representation of the notion of structured number. 2

3 Figure : Binary trees and their corresponding paranthesed expressions. 2 The operations Let BT be the set of binary trees [7, Vol. Section 2.3 ] dened by the symbolic equation [9]: BT = +. = n BT BT A tree a 2 BT dierent from the one node tree is ordered in such a way that we can make a distinction between the left subtree a L 2 BT and the right subtree a R 2 BT. For notational convenience we represent a binary tree a 2 BT by its horizontal paranthesed expression (a) 2 f; (; )g, where : BT! f; (; )g is recursively dened by () =. ( = n ) = ((a L ) (a R )) a L a R In the sequel we identify a 2 BT with (a). A binary tree a 2 BT is represented by if it is the one node tree, and by (a L a R ) if it has the left and right subtree a L and a R, respectively. Some binary trees together with their paranthesed expressions are drawn in Figure. The number of external nodes (leaves) of a binary tree a 2 BT is called the weight of a. Denition The weight function weight : BT! N is dened inductively by if a = weight(a) = weight(a L ) + weight(a R ) if a = (a L a R ) : 3

4 Binary trees that are distinct but that have identical weight are said to dier by their shape. We dene countably many operations on binary trees. Denition 2 The operation : : BT BT! BT is dened by a : b = (ab). For k 2, the operations k : : BT BT! BT are dened by a k a if b = : b = (a k k? : b L ) : (a k : b R ) if b = (b L b R ) The integer k is called the index of the operation : k. k+ The operation : has a simple expression in terms of k :. Suppose a; b are binary trees and n is the weight of b. The binary tree c = a k+ : b is equal to the tree resulting from the k : -product of n trees a in an order prescribed by the shape of b. For example and k+ a : () = (a k : a) k+ a : ((())()) = ((a k : (a k : a)) k : (a k : a)) Elementary examples of binary trees resulting from the operations :, : 2 and : 3 are given in Figure 2. The tree a: b is obtained by constructing the tree whose left and right subtrees are a and b respectively. The tree a : 2 b is obtained by grafting a copy of the tree a at the leaves of the tree b. No such simple geometrical description is available for a : 3 b. From the denition of the function weight and of the operation : we deduce that weight(a : b) = weight(a) + weight(b). Since the operations of higher index are dened by successive repetition of : the next result is easely obtained. Proposition Let a; b be binary trees.. weight(a : b) = weight(a) + weight(b) 2. weight(a : 2 b) = weight(a):weight(b) 3. weight(a : 3 b) = weight(a) weight(b) In the sequel the three rst operations : ; 2 : and 3 : on binary trees will be called addition, multiplication and exponentiation. There exist no natural number counterpart to k : for k 4 because in this case the weight of a k : b depends on the weight of a and b but also on their respective shape. 3 Algebraic properties The properties of the operations : ; 2 : ; 3 : numbers. for binary trees are similar to those of +; :; " for natural Theorem. Let a be a binary tree and k 3.. a : 2 = a = : 2 a 2. a : k = a and : k a = 2. Let a; b; c be binary trees and k 2.. a k : (b : c) = (a k k? : b) : (a k : c) 2. a : k (b : 2 c) = (a : k b) : k c 4

5 Figure 2: Binary trees resulting from the operations of addition, multiplication and exponentiation. 5

6 3. Let a; b; c; d be binary trees.. Left cancellation: If a : 2 b = a : 2 c then b = c. 2. Right cancellation: If a : 2 b = c : 2 b then a = c. Proof.. The equalities a : k = a for k 2 follow from the denition of : k. We prove : 2 a = a by induction on a. The result is clearly true for a = so let a = (a L a R ) and assume that : 2 a L = a L and : 2 a R = a R. Then a = a L : a R = ( : 2 a L ) : ( : 2 a R ) = : 2 (a L : a R ) = : 2 a and the theorem is proved. The equalities : k a = a are proved by induction on a. 2. The rst property is a rephrasement of the denition of : k. We prove the second property by induction on c. When c = we have a : k (b : 2 c) = a : k (b : 2 ) = a : k b = (a : k b) : k = (a : k b) : k c so let c = (c L c R ) and assume that a : k (b : 2 c L ) = (a : k b) : k c L and a : k (b : 2 c R ) = (a : k b) : k c R. Then by successive applications of the rst property we obtain and thus (a k : (b 2 k? : c L )) : (a k : (b 2 : c R )) = ((a k : b) k k? : c L ) : ((a k : b) k : c R ) a : k ((b : 2 c L ) : (b : 2 c R )) = (a : k b) : k (c L : c R ) a : k (b : 2 (c L : c R )) = (a : k b) : k (c L : c R ) a : k (b : 2 c) = (a : k b) : k c: 3. We prove the right cancellation rule only, the proof of the left cancellation rule is similar. We proceed by induction on b. The result is clearly true for b = so let b = (b L b R ) and assume that the result holds for b L and b R. If a : 2 b = c : 2 b, then (a : 2 b L ) : (a : 2 b R ) = (c : 2 b L ) : (c : 2 b R ) and a : 2 b L = c : 2 b L. By the induction hypothesis we are lead to the conclusion. 2 When evaluated with k = 2 the properties 2. and 2.2 give. a:(b + c) = a:b + a:c 2. a:(b:c) = (a:b):c whereas an evaluation with k = 3 gives. a (b+c) = a b :a c 2. a (b:c) = (a b ) c These four usual identities in Z have thus counterparts for binary trees. Central in the sequel is the fact that multiplication is associative. Counterexamples. Commutativity. The operations k : are non commutative. For k = and k = 2 this can be seen from the examples () : 6= : () and () 2 : (()) 6= (()) 2 : () whereas for the operations of higher index it suces to notice that a k : 6= k : a for any binary tree a dierent from. Associativity. The operation : is not associative as is easily seen from the example : (: ) 6= (: ):. The operations : k for k 3 are not associative either. For example, (a : k ) : k a 6= a : k ( : k a) when a 6=. Thus, by Theorem, the only associative operation is : 2. Cancellation rule. The left cancellation rule does not hold for the operations k : when k 3. Indeed, for any binary tree a and k 3 we have k : a =. Thus k : a = k : b for all binary trees a and b which clearly shows that the left cancellation rule does not hold. In Section 5 we give necessary and sucient conditions for the equality a k : b = a k : d. The right cancellation rule for k 3 is much harder to analyse. It is known to hold for k = 3 and k = 4 but the general case is yet unsettled. We conjecture here that it holds for all k : when k. 6

7 4 Prime trees and prime decomposition Denition 3 The binary tree a is prime if it is dierent from the one node tree and if a = b : 2 c implies that b = or c =. Trees that are not prime are composite. The weight of a product of binary trees is equal to the product of the weights. It is therefore clear that any binary tree whose weight is a prime number is automatically prime. The converse of this statement is not true. The binary tree ((())) has weight 4 and is a prime binary tree. It is easy to see that four of the ve binary trees of weight four are prime and that, in general, a natural number n is prime if and only if all binary trees of weight n are prime. In Table we give, for the rst values of n, the number C n of binary trees of weight n, the number I n of composite trees of weight n, and the number P n of prime trees of weight n. It is well-known that the number of binary trees of weigth n is equal to the nth Catalan number C n = (2n?2)!=(n!(n?)!) (see [4], [2], [] or [2]). No simple expression for I n or P n seems available. The sequence P n does not appear in the recent encyclopedic list of integer sequences [22]. Ph. Flajolet has shown [9] that T n? I n is equal to 0 if n is prime, is equal to (C p ) 2 if n = p 2 is the square of a prime, and is otherwise asymptotic to 2C p C n=p where p is the smallest prime factor of n. Binary trees dierent from the one node tree can be decomposed into products of prime binary trees. The decomposition is unique up to and including the sequence in which the factors appear. We rst need a lemma for proving this. Lemma Let a ; a 2 ; b ; b 2 be binary trees such that a : 2 a 2 = b : 2 b 2. If a and b (or a 2 and b 2 ) are prime, then a = b and a 2 = b 2. Proof. If a = b, then the left cancellation rule for multiplication shows thata 2 = b 2. We proceed by induction on a 2 to prove that a = b. If a 2 = then a = b : 2 b 2. Since a is prime and b is dierent from we must have b 2 = and thus a = b. Assume now that a : 2 a 2 = b : 2 b 2 and a 2 = (a 2L a 2R ). If b 2 =, then a : 2 a 2 = b and the theorem is proved, so assume b 2 = (b 2L b 2R ). We have (a : 2 a 2L ) : (a : 2 a 2R ) = (b : 2 b 2L ) : (b : 2 b 2R ). By the cancellation rule for addition and the induction hypothesis we then conclude a = b as requested. 2 It is now easy to show: Theorem 2 (Existence and uniqueness of prime decomposition) Let a be a binary tree dierent from. Then a = a : 2 a 2 : 2 : 2 a n for some n and prime binary trees a i. If a = b : 2 b 2 : 2 : 2 b m is another such decomposition. Then n = m and a i = b i for i = ; :::; n. Proof. Let a be a binary tree dierent from. If a is prime, then n = and a = a is the decomposition sought. If a is composite, there exists a and a 2 dierent from and such that a = a 2 : a 2. The factors can then be further decomposed until prime factors are reached. Since the weight of the factors are positive and strictly decreasing, the procedure must end after a nite number of steps. Thus a = a 2 : a 2 2 : 2 : a n for some n and a i prime binary trees. Assume now that a = b 2 : b 2 2 : 2 : b m is another such decomposition. By the lemma we must have a = b and a 2 2 : a 2 3 : a n = b 2 2 : b 2 3 : b m. But then by successive repetition of the same argument we are lead to the conclusion. 2 The decomposition given in the theorem is a prime decomposition. The ith factor in the decomposition is uniquely determined and is the ith factor of a. We can characterise the binary trees whose sum is prime. Theorem 3 Let a; b be binary trees dierent from the one node tree. Then a : b is prime if and only if the rst factors of a and b are distinct. 7

8 weight n C n I n P n Table : Number of binary trees, composite trees and prime binary trees of given weight. Proof. (Necessity) Let the rst factors of a and b be distinct and assume by contradiction that a : b is not prime. Then a = c : 2 c 2 for some c and c 2 dierent from. Since c 2 is dierent from we may write c 2 = c 2L : c 2R. Hence a : b = (c : 2 c 2L ) : (c : 2 c 2R ). By the cancellation rule this leads to a = c : 2 c 2L and b = c : 2 c 2R. But since c is dierent from these last identies show that the rst factors of a and b are identical and a contradiction is thus attained. (Suciency) Let a; b be distinct from and assume by contradiction that a = c : 2 a 0 and b = c : 2 b 0 for some c dierent from. Then a : b = (c : 2 a 0 ) : (c : 2 b 0 ) = c : 2 (a 0 : b 0 ) = c : 2 c 0 with c and c 0 dierent from. A contradiction is achieved and the theorem is proved. 2 5 The operations : k for k 3 Multiplication of binary trees is associative. The result of a : 3 b therefore depends on a and on the weight of b but not otherwise on the shape of b. Proposition 2 Let a; b ; b 2 be binary trees and assume that weight(b ) = weight(b 2 ). Then a : 3 b = a : 3 b 2. We use this result for introducing a new notation. Let n and a 2 BT. By a: 3 n we mean the binary tree a : 3 b where b is any binary tree of weight n. A similar construction is possible for operations of higher index. Denition 4 Let k 3. The operations 3 k : BT BT! N are dened inductively by. a3 3 b = weight(b) 2. a3 k b = With this purpose-built denition we have: if b = (a3 k b L ):((a : k b L )3 k? (a : k b R )) if b = (b L b R ) Theorem 4 Let a; b be binary trees and k 3. Then a : k b = a : 3 (a3 k b). Proof. We proceed by induction on k. For k = 3 the result is contained in Proposition 2. We assume k? that the result holds for : and show, by induction on b, that it also holds for k :. If b =, then 2 8

9 a : k = a = a : 3 = a : 3 = a : 3 (a3 k ) so let b = (b L b R ) and assume that a : k b L = a : 3 (a3 k b L ) and a : k b R = a : 3 (a3 k b R ). We have a : k b = a : k (b L b R ) = (a k k? : b L ) : (a k : b R ) = (a : k b L ) : 3 ((a : k b L )3 k? (a : k b R )) = (a : 3 (a3 k b L )) : 3 ((a : k b L )3 k? (a : k b R )) = a : 3 ((a3 k b L )((a : k b L )3 k? (a : k b R ))) = a : 3 (a3 k b) and the theorem is proved. 2 Cancellation rules for : and : 2 are analysed in Section 3. We have shown that the left cancellation rule does not hold for k 3 since, for example, : k a = : k b for any binary trees a and b. With the help of Theorem 4 our argument can now be made precise. Theorem 5 Let a; b; d be binary trees dierent from and k 3. Then a : k b = a : k d if and only if a3 k b = a3 k d. Proof. (Necessity) By Theorem 4 we know that a k : b = a 3 : (a3 k b) and a k : d = a 3 : (a3 k d). Hence a 3 : (a3 k b) = a 3 : (a3 k d). But then the prime decomposition theorem leads to a3 k b = a3 k d as requested. (Suciency) This part is trivial. If a3 k b = a3 k d, then a k : b = a 3 : (a3 k b) = a 3 : (a3 k d) = a k : d. 2 Corollary Let a; b; d be binary trees dierent from. Then a : 3 b = a : 3 d if and only if weight(b) = weight(d). Some of the algebraic properties of the operations : k are summarized in Table 2 Commutativity Associativity neutral cancellation rules : no no no a : b = c : d, a = c and b = d 2 : no yes a 2 : = a a 2 : b = c 2 : b, a = c 2 : a = a a 2 : b = a 2 : d, b = d 3 : no no a 3 : = a a 3 : b = c 3 : b, a = c 3 : a = a 3 : b = a 3 : d, weight(b) = weight(d) 4 : no no a 4 : = a a 4 : b = c 4 : b, a = c 4 : a = a 4 : b = a 4 : d, a3 4 b = a3 4 d k : no no a k : = a Conjecture: a k : b = c k : b, a = c (k 5) : k a = a : k b = a : k d, a3 k b = a3 k d Table 2: Algebraic properties of the operations : k. 6 Valuations A valuation is a function dened on binary trees and taking values in Z. Operations on integers can be used in a natural way to dene valuations. Denition 5 Associated to : Z Z! Z and e 2 Z is a valuation : BT! Z dened inductively by 9

10 e if a = (a) = (a L )(a R ) if a = (a L a R ) Valuations that can be obtained in this way are called inductive. We immediatly recognise that the weight function is an inductive valuation obtained by setting ab = a + b and e =. Other inductive valuations are given next. Maximal height. If we set ab = + max(a; b) and e = 0 the valuation obtained gives the maximal height of all external nodes. This quantity is refered too as the height of the tree and the corresponding valuation is denoted by height. Minimal height. The valuation obtained with ab = + min(a; b) and e = 0 gives the minimal height of all external nodes and is denoted minheight. Strahler number. The valuation obtained with ab = [a = b] + max(a; b) and e = 0 appear in various contexts (the expression [a = b] outputs when a = b and outputs 0 otherwise). In [8] it is called the \register function" and is used to calculate the minimal number of registers needed to evaluate an arithmetic expression (see also [6]). The same function is known in hydrology as the Horton-Strahler function and is used to describe characteristics of river ows (see [23], [24] and references cited therein). The Strahler number of a tree is equal to the height of the maximal complete tree that can be embedded in the tree [8]. We denote this valuation by Strahler. 2-bud. The valuation obtained with ab = [a = b] + min(a; b) and e = 0 (note the similarity with the denition of the Strahler number) has, to our knowledge, never been analysed. With denote this function by 2? bud. The 2-bud of a binary tree a can be shown equal the largest n 0 for which the nth rst factors of a are equal to (). The 2? bud of a binary tree a is thus equal to the largest n for which a = (() 3 : n) 2 : b for some tree b. Because of the grafting interpretation of the operation 2 : this corresponds also to the height n of the largest complete binary tree () 3 : n that appear everywhere on the boundary of a. Boolean valuation. The valuation obtained with ab = [a = b] and e = 0 outputs if the weight of the tree is even and outputs 0 if it is odd. In Table 3 we list some possible choices of operation and their resulting valuations. Several of these inductive valuations have remarkable properties with respect to : k. Theorem 6 Assume : Z Z! Z, e = 0, and let be the inductive valuation associated to and e. If a + (bc) = (a + b)(a + c) for all a; b; c 2 Z, then. (a : 2 b) = (a) + (b) 2. (a : 3 b) = (a):(b) 3. (a : 4 b) = (a) (b) Proof. We prove the rst identity by induction on b. For b = we have (a : 2 ) = (a) = (a) + () so let b = (b L b R ) and assume that (a : 2 b L ) = (a) + (b L ) and (a : 2 b R ) = (a) + (b R ). We have then (a : 2 b) = (a : 2 (b L : b R )) = ((a : 2 b L ) : (a : 2 b R )) = (a : 2 b L )(a : 2 b R ) = ((a) + (b L ))((a) + (b R )) 0

11 ab e resulting valuation a + b weight + max(a; b) 0 height + min(a; b) 0 minheight [a = b] + max(a; b) 0 strahler [a = b] + min(a; b) 0 2? bud [a = b] 0 if weight is even 0 if weight is odd Table 3: Some inductive valuations. = (a) + ((b L )(b R )) = (a) + (b) The other identities are similarly proved. 2 Corollary 2 Let be one of the valuations height; minheight; Strahler or 2? bud. Then. (a 2 : b) = (a) + (b) 2. (a 3 : b) = (a):(b) 3. (a 4 : b) = (a) (b) Proof. The corresponding pairs (; e) satisfy the conditions of Theorem Innite trees The operations k : introduced for nite binary trees can be extended to innite binary trees. For convenience we shall look at innite trees as languages over 2-letter alphabets. Let be a nite alphabet and L ; L 2 be two languages over (for denitions see [9]). The product of L and L 2 is the language L L 2 = fx x 2 : x 2 L ; x 2 2 L 2 g. Given x 2, the language x L and the residual x? L of L by x are dened by x L = fxg L and x? L = fy 2 : x y 2 Lg, respectively. If n 0, the truncation of L at size n is the language dle n = fx 2 L : jxj ng where jxj is the length of x. For a language L and n 0 we dene L 0 = f!g (! denotes the empty word), L n+ = L n L and L = S i=0 Li. Finally, a language L over is factorial if x; v 2 and x v 2 L implies that x 2 L. Factorial languages always contain! unless they are empty. Proposition 3 Let L; L ; L 2 ; ::: be factorial languages over, x 2 and n 0. The languages x? L; dle n ; L = S i=0 L i; L = T i=0 L i; L L 2 and L are factorial. A binary tree is entirely specied by its set of internal nodes. Any internal node can be reached by starting from the root and specifying the nite sequence of left and right movements needed to reach it. Let us denote these movements by a and b respectively. A node can thus be seen as a word over the alphabet = fa; bg. A nite (innite) tree will therefore have a representation as a nite (innite) language over. To the one node binary tree corresponds the empty language L = ;, the tree () is represented by L = f!g and the two binary trees of weight 3 have f!; ag and f!; bg as corresponding languages. It is easy to see that languages generated by binary trees are factorial. The converse is also true: Any factorial language over a 2-letter alphabet can be seen as a representation 2

12 of a particular binary tree, the corresponding binary tree is innite when the language is. We denote by F L the set of factorial languages and by F F L the set of nite factorial languages over the two letter alphabet fa; bg. There is an obvious bijection between F F L and BT. Denition 6 The operation : : F F L F F L! F F L is dened by L : L 2 = f!g [ a L [ b L 2. For k 2, the operations : k : F F L F F L! F F L are dened inductively by L k L if L : L 2 = 2 = ; (L k : a? k? L 2 ) : (L k : b? L 2 ) if L 2 6= ; We now remove the niteness condition on the languages L and L 2 and dene, for nite or innite languages, the operations k : (k 0) by L : k L 2 = [ i=0(dl e i : k dl 2 e i ) Proposition 4 The operations : k are operations from F L F L to F L. Proof. The statement is clearly true for nite languages because any k : product of two nite languages can be expressed in terms of nitely many operations of the form given in Proposition 3. We complete the proof by observing that the result of a k : product of innite languages is dened by a countable union of nite factorial languages. 2 Most properties of the operations k : for nite binary trees were proved by induction. This principle does not anymore hold for innite binary trees and many of the properties shown for nite binary trees do therefore not hold for innite binary trees. It is nevertheless possible to show that the properties 2. and 2.2 of Theorem remain satised for innite binary trees. On the other hand the properties 3. and 3.2 in the same theorem may be violated by innite binary trees. Assume for example that L =, L 2 = f!g and L 3 = f!; a; bg. Then L 2 : L 2 = L 2 : L 3 but L 2 6= L 3. The result of operations of index greater or equal to 3 can be expressed as 2 : products of identical factors. Innitely many factors are multiplied when the second operand is innite. The operation 2 : correspond to the usual multiplication of languages of computer science and the operation 3 : therefore degenerates into the Kleene star operation when the second operand is innite. Theorem 7 Let L ; L 2 be two factorial languages. Then. L : 2 L 2 = L 2 L 2. L n L 3 : L 2 = L some n if L 2 is nite if L 2 is innite Proof. We prove the result for L 2 nite by induction on L. The result is clearly true for L = ;; so let L 2 = L 0 2 : L 00 2 and assume that L 3 : L 0 2 = L n0 and L 3 : L 00 2 = L n00 for some n 0 ; n We have then L 3 : L 2 = L 3 : (L 0 2 : L 00 2) = (L 3 : L 0 2) 2 : (L 3 : L 00 2) = L n0 L n00 = L n0 +n 00 as requested. Assume now that L 2 is innite. We show that the languages L and L : 3 L 2 coincide. Indeed, if x 2 and x 2 L : 3 L 2, then x 2 (dl e i : 3 dl 2 e i ) for some i. But then by the nite case there exist some n for which x 2 (dl e i ) n L n L. Assume now that x 2 L. Then x 2 L n for some n. But then x 2 L : 3 dl 2 e i for some i and thus x 2 L : 3 L 2 as requested. 2 2

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