Choffrut, C., Karhumaki, J.: Combinatorics on Words, 329--438, In [18].  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... . Each hole means an unknown symbol of S. A (partial) word u = u 1 : u n , where u i are symbols, is called (locally) p periodic if u i = u i p for all i 2 f1; n pg such that u i 6= and u i p 6= The following result is a generalization of the classical Fine and Wilf s theorem [9, 6]: Theorem 3 ( 4] Let u be a partial word of length n which is p periodic and q periodic. If u contains only one hole, and if n p q, then u is gcd(p;q) periodic. Now let us start the proof of Theorem 2 and first consider the easiest case: Lemma 1 If the symmetric morphism j is defined by j(0) ....

C. Choffrut and J. Karhumaki. Combinatorics on words. In G. Rozenberg and A. Salomaa, eds., Handbook of Formal Languages, v. 1, chapter 6. Springer-Verlag, 1997.

....is the length of w, jwj a is the number of occurrences of a in w, and ae(w) denotes the primitive root of w. The conjugacy relation is denoted ; for two words u; v, u v iff u = pq; v = qp, for some p; q. For basic notions and results of combinatorics on words and formal languages we refer to [4, 10] and [14] respectively. For a language L, denote #L (n) card(fw 2 L j jwj = ng) this is referred to as the complexity (function) of L. For an integer k 0, L is called k poly slender if #L (n) O(n k ) L is poly slender iff it is k poly slender, for some k 0. A language L Sigma is ....

....(Clearly, z is not unique. Also, h will stand for h 1;1; 1 and will be called an underlying morphism of D. D is a Dyck loop iff it is a k Dyck loop, for some k. Notice that if l k, then any l Dyck loop is also a k Dyck loop. Example 2. For the underlying Dyck word z = 1 ] 1 [ 2 [ 3 ] 3 [ 4 ] 4 ] 2 , we can construct the Dyck loop L 1 = fa n 1 bb(abb) n 2 ab 4n 3 a n 4 bba n 4 ba(aab) 3n 2 j n i 0g: 2) The underlying morphism h is clear if we just mention that the images of ] 1 and ] 3 are empty. Example 3. The following more intricate example will be used also later. ....

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Choffrut, C., and J. Karhumaki, Combinatorics of Words, in G. Rozenberg, A. Salomaa, eds., Handbook of Formal Languages (Springer-Verlag, Berlin, Heidelberg, 1997) 329 -- 438.

....relation. Consequently, the dimension property provided by the defect theorem is rather weak. Another fundamental result of words revealed in 1985 by Albert and Lawrence [1] and by Guba [9] is the compactness property of free semigroups (also known as Ehrenfeucht s Conjecture) see also [6], 12] 13] each independent set of equations of words is finite. This result, contrary to the above examples, shows that dependencies of words do imply some, although weak, dimension properties. As a conclusion our goal is to point out that the defect theorem formalizes a dimension property of ....

C. Choffrut and J. Karhumaki, Combinatorics of words, in Handbook of Formal Languages, Vol. 1, (A. Salomaa and G. Rozenberg, eds.), Springer-Verlag, 1997, pp. 329 -- 438.

....use. Let Sigma be a finite alphabet. We denote by card( Sigma) the number of elements of Sigma, by Sigma the set of finite words over Sigma, and by 1 the empty word. For w 2 Sigma , jwj denotes the length of w. For any notions and results of combinatorics on words we refer to [Lo] and [ChKa]. Let Sigma and Xi be two disjoint alphabets, of constants and variables, respectively. By convention, lower case letters denote constants whereas capitals denote variables. A word equation e is a pair of words ; 2 ( Sigma [ Xi) denoted e : A solution of e is a morphism h : ....

....relations, which we shall denote 6 p , 6 f , 6 suf , respectively, are expressed by the variables X and Y in the equation Y = XZ, Y = WXZ, and Y = ZX, respectively. Example 2. The conjugacy relation is expressed by the variables X and Y in the system ae X = PQ Y = QP It is known (see, e.g. [ChKa]) that it is then possible to find from this system a single equation expressing the conjugacy relation. 3 Subwords We consider in this section the subword relation 6 s defined, for u; v 2 Sigma , by u 6 s v iff u = a 1 a 2 : a n ; n 0; a i 2 Sigma; 1 6 i 6 n; v = v 1 a 1 v 2 a 2 : ....

Choffrut, C., Karhumaki, J., Combinatorics of words, in (G. Rozenberg, A. Salomaa, eds.) Handbook of Formal Languages, Vol. I, SpringerVerlag, Berlin, Heidelberg, 1997.

....that the empty language is confluent w.r.t.any quasi order on Sigma . Also, for any two quasi orders 1 and 2 on Sigma , if 1 2 , then any language confluent w.r.t. 1 is confluent also w.r.t. 2 . Some of the most important quasi orders on Sigma in combinatorics on words, 2 see [CK], are as follows, where we suppose that u; v are arbitrary finite words over Sigma: prefix: u p v iff there is w 2 Sigma such that v = uw; factor: u f v iff there is w; z 2 Sigma such that v = wuz; subword: u s v iff u = a 1 a 2 : a n ; n 0; a i 2 Sigma; 1 i n; v = v 1 a 1 v ....

....a star language, L = L , then L is confluent w.r.t. any quasi order f . 3 Well quasi orders The concept of a well quasi order has been frequently discovered; see, for instance, ER] Hi] Kr1] Ha] A complete account in this matter is given by [Kr2] The definitions we use here are from [CK]. Let be a quasi order on Sigma . A set L Sigma is an antichain of if all elements in L are pairwise incomparable w.r.t. that is, for any u; v 2 L, neither u v nor v u. The quasi order is called well founded if there is no infinite descending sequence w 1 Gamma1 w 2 ....

C. Choffrut, J. Karhumaki, Combinatorics of Words, in: G. Rozenberg, A Salomaa, eds., Handbook of Formal Languages (Springer-Verlag, Berlin, Heidelberg, 1997).

....for these relations which generalizes the well known theorem of Higman. Some language theoretic gaps for infinite antichains are also presented. 1 Introduction Consider a finite alphabet Sigma. Two of the most common relations on words are the factor and the subword partial orders, see [ChKa], Lo] defined, respectively, for two words u; v 2 Sigma , by u f v iff v = xuy; for some x; y 2 Sigma ; u s v iff v = v 0 u 1 v 1 u 2 v 2 : u k v k ; u = u 1 u 2 : u k ; k 1; v 0 ; u i ; v i 2 Sigma ; 1 i k: Roughly speaking, a word is a factor of another one it the ....

C. Choffrut, J. Karhumaki, Combinatorics of Words, in Handbook of Formal Languages, (G. Rozenberg, A Salomaa, eds.), Springer-Verlag, Berlin, Heidelberg, 1997.

....periodicity lemma, cf. FiWi] We shall denote by the empty word. For two words u; v, we say that u is a prefix of v, denoted u v, if v = ux, for some x 2 A . A word u is primitive if there is no word v such that u = v k , where k 2. For basic notions and results on words we refer to [ChKa] and [Lo] 2. Properties of words and periods In this section we give first the announced proof for Fine and Wilf s lemma and then prove some properties of words and periods needed in the proof of the main theorem in the next section. Lemma 1. If a word w has periods p and q and jwj = p q ....

C. Choffrut, J. Karhumaki, Combinatorics of Words, in G. Rozenberg, A. Salomaa, eds., Handbook of Formal Languages, Vol. 1 (Springer-Verlag, Berlin, Heidelberg, 1997) 329 -- 438.

....cases with we do not consider the words of the form (uv) k u with k 2. The function Phi(k) is defined by the recurrence Phi(1) 2; Phi(2) 5; Phi(n 2) Phi(n 1) Phi(n) case X result reference S; S f X (k) 2k Gamma 1 [Si] SM;SM 2k Gamma 1 f X (k) Phi(k) Gamma 1 [Si] [CK] F; F f X (k) k F ; F f X (k) 2k Gamma 1 [Lo] FM f X (k) k FM f X (k) 2k Gamma 1 Table 1 Characteristic functions 3 3 Properties of spectrums From Table 1 we can notice that in the cases S, SM and F the result is the same for the proper and full spectrums. The reason for this ....

....that if we know the spectrum with multiplicity, we can find out also the same spectrum without multiplicity. So in this case we have immediately the lower bound from the previous case: f SM (k) f SM (k) 2k Gamma 1. The upper bound follows by considering the following sequences of words, cf. [CK], u 0 = aba; v 0 = 1; w 0 = baa u 1 = ab; v 1 = ba; w 1 = ba u k 2 = w k u k 1 ; v k 2 = u k 1 u k ; w k 2 = u k 1 w k ; if k is even, u k 2 = u k 1 w k ; v k 2 = u k u k 1 ; w k 2 = w k u k 1 ; if k is odd. Since SSM (u k ; k) SSM (v k ; k) and obviously u k 6= v k and ju k j = jv k j = ....

Choffrut, C., Karhumaki, J., Combinatorics of words, in: G. Rozenberg and A. Salomaa (eds), Handbook of Formal Languages, Vol. I, Springer, 329--438, 1997.

....the language equation LK = KL holds true. We also present several open problems that involve the mentioned questions on semilinearity, language equivalence and commutation of languages. 2 Preliminaries We refer to [Sa73] or [Har78] to the basic definitions on automata and languages, and to [ChK97] or [L83] to those on words. Let be an alphabet, that is, a finite set of symbols, and denote by the set of all words over including the empty word, denoted by . Let w 2 be a word. Then jwj denotes the length of w, and jwj a denotes the number of occurrences of the letter a 2 ....

C. Choffrut and J. Karhumaki, Combinatorics of words, in Handbook of Formal Languages, Vol. 1, (A. Salomaa and G. Rozenberg, eds.), SpringerVerlag, 1997, pp. 329 -- 438.

...., we say that a language L Sigma is confluent w.r.t. if and only if for any x; y 2 L there is z 2 L such that x z and y z. Notice that the empty language is confluent w.r.t. any quasi order on Sigma . Some of the most important partial orders on Sigma in combinatorics on words, see [CK], are as follows, where we suppose that u; v are arbitrary 2 finite words over Sigma: prefix: u p v iff there is w 2 Sigma such that v = uw; factor: u f v iff there are w; z 2 Sigma such that v = wuz; subword: u s v iff u = a 1 a 2 : a n ; n 0; a i 2 Sigma; 1 i n; v = v 1 a 1 ....

C. Choffrut, J. Karhumaki, Combinatorics of Words, in G. Rozenberg, A. Salomaa, eds., Handbook of Formal Languages, Vol 1 (SpringerVerlag, Berlin, Heidelberg, 1997) 329--438.

....is rational, the most general result known on Conway s problem. Closely related to commutation are the notions of root and primitive root. In general, X is said to be a root of Y if Y = X , for some n 1; X is primitive if it has no roots other than itself. For words it is well known, see [11], or [47] that any nonempty word has a unique primitive root; for sets of words on the other hand, such a property does not hold in general, despite the fact that the notion of root has been extended in two different ways to sets of words, to the notions of root and premotif, see [7a] and [3] ....

.... to the left sequence of letters from E: V = a 3a 2a 1, with a n E, for all n 1. We denote the set of all left infinite words over E by E. For a finite nonempty word u E , we denote u = uuu. and v: u: For other notions and results on Combinatorics on Words we refer to [11], 47] and [48] 19 2.2.2 Languages A language over the alphabet E is a set of words over E, i.e. a subset of E . The empty language is denoted by 0. The operation of union, intersection, and difference are defined in the usual way for sets. We denote the union of two languages L and R by L ....

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C. Choffrut, J. Karhum/iki, Combinatorics of Words. In G. Rozenberg, A. Salomaa (eds.), Handbook of Formal Languages, vol. 1: 329-438, Springer-Verlag, 1997.

....is either of the form i2I for some I N, or of the form i2I t for some I N, respectively. Proof. If there exists a word t such that X t the result is a simple combinatorial application of Lemma 1. In the other case we know that X = fx; yg satisfies the condition xy 6= yx, cf. [2]. Consequently, the set = xy yx) X(xy yx) is marked, i.e. the words start with different letters. Further, for any X a Sigma and any Y XY = Y X if and only if a Xaa Y a = a Y aa Xa; implying that Z(a Xa) a Z(X)a: It follows that Z(X ) xy yx) ....

....languages. For example, the nonconstructive rationality of an equality set of two prefix morphisms is such an example, cf. 16] Another example is Higman s theorem stating that for any language its upwards closure under the partial ordering of being a scattered factor is rational, cf. 9] or [2]. ....

C. Choffrut and J. Karhumaki, Combinatorics of Words, In: G. Rozenberg and A. Salomaa (eds), Handbook of Formal Languages, vol. 1, 329--438, Springer, 1997.

....(u; v) if and only if (u) v) Assume that T has q states. Let K 0 (A) ker( be obtained from the computations of T of length at most 2q. It can be now shown using a combinatorial lemma on words that K 0 (A) is a required finite equivalent subrelation. For more details of this, see e.g. [4]. ut The second claim of Lemma 2.1 is an effective special case of the Ehrenfeucht s Compactness Property, for the proof of which we refer to Albert and Lawrence [1] or Guba [7] This result states that for each R X there exists a finite subrelation R 0 R that is equivalent to R. Note, ....

C. Choffrut and J. Karhumaki, Combinatorics of words, in Handbook of Formal Languages Vol. I, Springer, 1996, to appear.

....languages based on the number of auxiliary unknowns. Or even more strongly, for each k 1, there exists a language expressible by a word equation with k 1 unknowns, but not with k unknowns. 2 Preliminaries We assume that the reader is familiar with the basics of combinatorics of words, cf. [5] Let Sigma be an alphabet of constants and Theta be an alphabet of variables. We assume that these alphabets are disjoint. We use the convention that lower case letters represent constants and capital letters represent variables. We assume an unusual, but for our purposes usefull, convention ....

....complexity is Theta(n log n) see [9] Example 3.5. Consider the languege L = fabcbc bc : bc : n 1g: The language L is generated by D0L system (fa; b; cg; h; a) where h(a) abc, h(b) bc, h(c) c. The subword complexity of the language L is Theta(n ) see Example 9. 8 in [5]. Hence, L is not expressible by word equations. For a language L, denote by c L (n) the number of words in L of length n. Clearly, for languages which are not pattern free, i.e. contain a pattern language, we have c L (an b) 2 Omega Gamma for some constants a, b. In our second tool for ....

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Choffrut, C., and Karhumaki, J., Combinatorics of words, in G.Rozenberg and A.Salomaa (eds), Handbook of Formal Languages, Springer, 1997.

....Science 1 Introduction Several authors in the existing literature, cf. 15] used word equations in order to describe properties and relations of words, but, to our knowledge no attempt to synthesis or of a systematization of this topic has been done. This was emphasized also in a recent survey [5] where some results of the field were collected. Classical relations on words that are characterized as solutions sets of word equations are for instance, two words X and Y are powers of a same word if and only if they constitute a solution of the equation XY = Y X, and two words X and Y are ....

....certain properties by equations. As an illustration we recall the following. The union of solution sets of two equations can be expressed as a solution set of one equation, as was shown in [3] using 4 additional variables, and later improved to require only 2 additional ones by [8] cf. also [5]. Similarly, the inequality, that is the set of t tuples of words which does not satisfy a given equation e with t variables, can be expressed as a union of the solution sets of a finitely many equations each of those using 3 extra variables, cf. e.g. 5] and consequently the inequality is ....

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Choffrut, C., and Karhumaki, J., Combinatorics of words, in G.Rozenberg and A.Salomaa (eds), Handbook of Formal Languages, Springer, 1997.

....Structures of Computer Science 1 Introduction Defect theorem is one of the fundamental results on words, cf [Lo] Intuitively it states that if n words satisfy a non trivial relation, then these words can be expressed as products of at most n Gamma 1 words. Actually, as discussed in [CK], for example, there does not exist just one defect theorem but several ones depending on restrictions put on the required n Gamma 1 words. It is also well known that the nontrivial relation above can be replaced by a weaker condition, namely by the nontrivial one way infinite relation. The goal ....

....factorizations of words defect theorem can be stated as follows: Let X Sigma be a finite set of words. If there exists a word w 2 Sigma having two different X factorizations, then the rank of X is at most card(X) Gamma 1. Here the rank of X can be defined in different ways, cf again [CK]. For example, it can be defined as a combinatorial rank r c (X) denoting the smallest number k such that X Y with card(Y ) k. To describe our results let w be a bi infinite word, i.e. an element of Sigma Z , and X a finite subset of Sigma . We say that w has an X factorization ....

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Choffrut, C., Karhumaki, J., Combinatorics of words, in: G. Rozenberg and A. Salomaa (eds), Handbook of Formal Languages, Vol. I, Springer, 329--438, 1997.

....for languages commuting with a three element code. Several open problems are proposed in Section 6. 1 2 Preliminaries and background In this section we fix our terminology, and recall several known results related to this work. For further details in Combinatorics on Words, we refer to [2]. For two words u; v 2 Sigma , we say that u is a prefix of v if v = uw, for some w 2 Sigma , and write u v, and u = vw Gamma1 ; u is a proper prefix of v if both u and w are nonempty words. We say that u is a suffix of v if v = wu, for some w 2 Sigma , in which case we write u ....

....the nondirected graph G, whose vertices are the elements of Xi, and whose edges are the pairs ( i ; j ) 2 Xi Theta Xi, with i and j appearing as the first letters of the left and right handsides of some equation of S, respectively. The following basic result on combinatorics of words, cf. [2], is very useful and efficient in our later considerations. Lemma 1. Graph Lemma) Let S be a system and let X ae Sigma be a subset satisfying it. If the dependence graph of S has p connected components, then there exists a subset F of cardinality p such that X F . Note that in the ....

C. Choffrut, J. Karhumaki, Combinatorics on Words. In G. Rozenberg, A.Salomaa, eds., Handbook of Formal Languages, vol 1: 329-438, SpringerVerlag, 1997.

....one of the fundamental properties of words. It states that if a set of n words satisfies a nontrivial relation, then these words can be expressed as products of at most n Gamma 1 words, i.e. any nontrivial relation on words implies a defect effect, cf. Lo] Actually, as emphasized in [HK] and [CK], there does not exist just one, but several theorems which formalize the above defect effect, depending on requirements put on these n Gamma 1 words. As argued in [HKP] defect theorems can be viewed as a weak dimension property of words. It is weak since a finite set X of words can satisfy ....

....dual, problems. First, how large independent systems of relations on n words can exist that they do not force a defect effect larger than k, i.e. allow a set X of n words of rank at least n Gamma k to satisfy these relations, cf. KP] and [HKP] Here rank can be defined in a number of ways, see [CK], for example as the combinatorial rank which is the smallest number of words needed to express all words of X as product of these words. The second problem area asks to find conditions (on relations or sets of words) which imply a cumulative defect effect, i.e. if the set X of n words satisfy k ....

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Choffrut, C., Karhumaki, J., Combinatorics of words, in: G. Rozenberg and A. Salomaa (eds), Handbook of Formal Languages, Vol. I, Springer, 329--438, 1997. 10

....the combinatorial theory of words (cf. 8] It states that if a set of n words satisfies a nontrivial relation, then these words can be expressed as a catenation of at most n Gamma 1 words. Actually there does not exist just one, but several results which formalize the above defect effect (cf. [4, 6]) Also, instead of finite relations, one way infinite ones can be used (cf. 3, 4] and, very recently, a defect theorem for two way infinite relations was proved in [7] Research on combinatorial problems of node labeled k ary trees, viewed as extensions of words, was initiated by Nivat in ....

....a nontrivial relation, then these words can be expressed as a catenation of at most n Gamma 1 words. Actually there does not exist just one, but several results which formalize the above defect effect (cf. 4, 6] Also, instead of finite relations, one way infinite ones can be used (cf. [3, 4]) and, very recently, a defect theorem for two way infinite relations was proved in [7] Research on combinatorial problems of node labeled k ary trees, viewed as extensions of words, was initiated by Nivat in [11] One of such natural problems is to ask whether defect theorems hold for trees. ....

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C. Choffrut, J. Karhumaki. Combinatorics on words. In: G. Rozenberg and A. Salomaa (eds.), Handbook of Formal Languages vol.1, 329--438 (Springer), 1997. 17

....Science 1 Introduction Studying regularities in strings has a long tradition in combinatorics on words. Starting from the fundamental work of Thue [23, 24] who considered the existence of infinite repetition free words, many authors investigated such words in various contexts, see for a review [7]. In the case of infinite words which are defined by iterated morphisms the problems of repetition freeness become difficult even for a restricted class of morphisms [6] On the other hand detecting whether a morphism is square free is simple, while, surprisingly, detecting whether it is cube free ....

Choffrut C., Karhumaki J., Combinatorics of words. In G. Rozenbeg and A. Salomaa (eds), Handbook of Formal Languages, vol.1-3, Springer, 1997.

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Choffrut, C., Karhumaki, J.: Combinatorics on Words, 329--438, In [18].

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C. Choffrut and J. Karhumaki. Combinatorics on words. pages 329--438. In [36].

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C.Choffrut, J.Karhum aki, Combinatorics of words, in [15], 329{ 438.

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C. Choffrut and J. Karhumaki, Combinatorics on words. In: Handbook of Formal Languages, Vol. I, (G. Rozenberg, A. Salomaa, Eds.), Springer-Verlag 1997, pp. 329-- 438.

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C. Choffrut and J. Karhumaki. Combinatorics on words. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 1, chapter 6. Springer-Verlag, 1997.

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