J. Karhumaki, J. Manuch and W. Plandowski, On defect effect of biinfinite words, Lect. Notes Comp. Sci. 1450, (1998), 674 -- 682. Home/Search Document Details and Download
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Many Aspects of Defect Theorems - Harju, Karhumäki (Correct)
....word s 2 fw 1 ; v 1 g Z is 9 factorizable over fw 2 ; v 2 g. It is clear that the bi infinite words in fw 1 ; v 1 g Z need not be periodic. In the above example the combinatorial rank of X is strictly smaller than rank(X ) This observation, indeed, leads to a general result shown in [19]. Theorem 5.3. Let s 2 A Z be a bi infinite word that has two different factorizations over a finite set X A . Then s is periodic or r c (X) card(X) In particular, if X consists of two words, then s is periodic. Proof. The claim is clear, if X is not a code. Assume thus that X is a ....
J. Karhumaki, J. Manuch and W. Plandowski, On defect effect of biinfinite words, Lect. Notes Comp. Sci. 1450, (1998), 674 -- 682.
Multiple Factorizations of Words and Defect Effect - Karhumäki, Manuch (2000) (1 citation) Self-citation (Karhumaki Manuch) (Correct)
....of relations is restricted. A similar deep result is proved in [Br] extending ideas of [Ka] and [Ho] where it is shown that if X is a code and has unbounded synchronizing delay in both directions, then the rank of X is at most card(X) Gamma 2. Our starting point is a recent result proved in [KMP1], see also [KMP2] stating that if a non periodic bi infinite word possesses two different X factorizations, then the rank of X is at most card(X) Gamma 1. As emphasized in [KMP1] it is essential to use the notion of combinatorial rank described above. We ask the following: Problem 1. Let X be ....
....in both directions, then the rank of X is at most card(X) Gamma 2. Our starting point is a recent result proved in [KMP1] see also [KMP2] stating that if a non periodic bi infinite word possesses two different X factorizations, then the rank of X is at most card(X) Gamma 1. As emphasized in [KMP1] it is essential to use the notion of combinatorial rank described above. We ask the following: Problem 1. Let X be a set of n words and w a non periodic bi infinite word. Is it true that if w possesses k disjoint X factorizations, for k card(X) then the combinatorial rank of X is at most ....
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Karhumaki, J., Manuch, J., Plandowski, W., On defect effect of bi-infinite words, in: Proceedings MFCS'98, 23rd International Symposium, LNCS 1450, Springer, 674--682, 1998.
Defect Theorems for Trees - Karhumäki, Mantaci Self-citation (Karhumaki) (Correct)
....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. This question was answered affirmatively in [10, 9] by showing that the most basic ....
J. Karhumaki, J. Manuch, W. Plandowski. On defect effect of bi-infinite words. Proceedings of MFCS'98. Lecture Notes in Computer Science 1450, 674--682, 1998.
The Commutation of Finite Sets: A Challenging Problem - Choffrut, Karhumäki, al. (1999) (1 citation) Self-citation (Karhumaki) (Correct)
....refer to [4] for a more complete exposition of the subject. The idea is, given a set X of words, to state conditions on relations satisfied by words so that these words may be expressed with as few parameters as possible. The relations in question are stated in terms of one way infinite words. In [10], they are stated in terms of two way infinite words. Let Xi be a set of variables in one to one correspondence with a subset of nonempty words X A , say i x i for some fixed enumeration of X . An equation over the set Xi is a pair L( Xi) R( Xi) more traditionally denoted as L( Xi) ....
....will proceed if we want to prove that the words of a set X are all powers of a same word: it will suffice to find enough equations, possibly by introducing some new words, in such a way that the corresponding graph satisfy the condition of the proposition. This proposition was used effectively in [10] to derive a defect theorem for bi infinite words, which, in turn, is essential for some of our results. In order to formulate it we recall the following notion: the combinatorial rank of a finite set X A is minfjjF jj j X F g; i.e. the smallest number of words needed to express all ....
J. Karhumaki, J. Manuch, and W. Plandowski. On defect effect of bi-infinite words. In MFCS'98, volume 1450, pages 674--682. Lecture Notes in Computer Science, 1998.
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