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### Periodicity and unbordered segments of words

Andrzej Ehrenfeucht, D.M. Silberger
1979 Discrete Mathematics
%en T also is unbordered. unbordered left (right) segment of a. Let T be such Theorem. Let (II be finite and not of the form 0" for n > 1. Let L be a letter in the word ~1.  ...  A nonempty word 0 is said to be a border of a word ar if and only if (Y = hp = @p for some nonempty words A and p.  ...  Acknowledgement We are grateful. for the suggestions of L.W. Baggett and Jan Mycielski.  ...

### Relationship between the period of a finite word and the length of its unbordered segments

J.P. Duval
1982 Discrete Mathematics
A nonempty word is unbordered if and only if it has no proper period. We study in this paper the relationship between the smallest period of a wor6 and the maximum length of its unbordered segments.  ...  We established: If the length of a word is greater or eyuaE to farrdr times: tk maximum length of its unbordered segments, then this maximum length is the smallest period of the word.  ...  For each word f, one has the inequality h(f) 2 p(f). Proof. Let w be a segment of f and assume that Iw] > A(f). Then hCfi is a period of f and a period of w.  ...

### Abelian bordered factors and periodicity [article]

Emilie Charlier, Tero Harju, Svetlana Puzynina, Luca Zamboni
2015 arXiv   pre-print
Ehrenfeucht and Silberger proved that an infinite word is purely periodic if and only if it contains only finitely many unbordered factors.  ...  A finite word u is said to be bordered if u has a proper prefix which is also a suffix of u, and unbordered otherwise.  ...  n−1 ) and (x n , y n ) are connected with a line segment.  ...

### Page 1981 of Mathematical Reviews Vol. , Issue 80E [page]

1980 Mathematical Reviews
M. 80e:68175 Periodicity and unbordered segments of words. Discrete Math. 26 (1979), no. 2, 101-109.  ...  Then 7 also is unbordered. Theorem: Let a be finite and not of the form 8", for n> 1 and let L be a letter in the word a.  ...

### Page 1289 of Mathematical Reviews Vol. , Issue 84d [page]

1984 Mathematical Reviews
-P. 84d:05001 Relationship between the period of a finite word and the length of its unbordered segments. Discrete Math. 40 (1982), no. 1, 31-44.  ...  A word f is unbordered if A(f)=|f|. The maximal length of an unbordered segment of f is denoted p(f). This paper deals with relationships between A(f) and p(f).  ...

### A Characterization of Cellular Automata Generated by Idempotents on the Full Shift [article]

Ville Salo
2012 arXiv   pre-print
We prove a characterization of products of idempotent CA, and show examples of CA which are not easy to directly decompose into a product of idempotents, but which are trivially seen to satisfy the conditions  ...  Our proof uses ideas similar to those used in the well-known Embedding Theorem and Lower Entropy Factor Theorem in symbolic dynamics.  ...  Acknowledgements I would like to thank Ilkka Törmä for his idea of also discussing eventually idempotent cellular automata, and for his useful comments on an early version of this article.  ...

### Page 1301 of Mathematical Reviews Vol. , Issue 81D [page]

1981 Mathematical Reviews
Let p(f) be the maximum length of the unbordered segments of f and | f| the length of f.  ...  Author’s summary: “A nonempty word u from the free monoid A* is called a border of a word f if f&uA*N A*u. A word is said to be unbordered if it has no other border than f itself.  ...

### A Characterization of Cellular Automata Generated by Idempotents on the Full Shift [chapter]

Ville Salo
2012 Lecture Notes in Computer Science
We prove a characterization of products of idempotent CA, and show examples of CA which are not easy to directly decompose into a product of idempotents, but which are trivially seen to satisfy the conditions  ...  Our proof uses ideas similar to those used in the well-known Embedding Theorem and Lower Entropy Factor Theorem in symbolic dynamics.  ...  Acknowledgements I would like to thank Ilkka Törmä for his idea of also discussing eventually idempotent cellular automata, and for his useful comments on an early version of this article.  ...

### Finding the Leftmost Critical Factorization on Unordered Alphabet [article]

Dmitry Kosolobov
2016 arXiv   pre-print
We present a linear time and space algorithm computing the leftmost critical factorization of a given string on an unordered alphabet.  ...  Shur for helpful discussions and the invaluable help in the preparation of this paper.  ...  By Lemma 3, the string w[i..i+µ(i)−1] is unbordered and its minimal period is µ(i).  ...

### Finding the leftmost critical factorization on unordered alphabet

Dmitry Kosolobov
2016 Theoretical Computer Science
We present a linear time and space algorithm computing the leftmost critical factorization of a given string on an unordered alphabet.  ...  Shur for helpful discussions and the invaluable help in the preparation of this paper.  ...  By the same argument, one can show that i h does not lies in the segments [c s t−1 + |abaa|..c s t−1 + |abaab|), [c s t−1 + |abaabaa|..c s t−1 + |abaabaab|), . . .; in other words, i h cannot lie in an  ...

### Combinatorics of the Tautological Lamination [article]

Danny Calegari
2021 arXiv   pre-print
Denote by N_q(n,m) the number of times 2^mq^-n arises in the nth partition. We prove a recursion formula for N_q(n,0), and a gap theorem: N_q(n,n)=1 and N_q(n,m)=0 for ⌊ n/2 ⌋ < m < n.  ...  In each degree q the tautological lamination defines an iterated sequence of partitions of 1 (one for each integer n) into numbers of the form 2^m q^-n.  ...  These lengths are all of the form 2. Unbordered words Some words end like they begin, such as abra·cad·abra and b·aoba·b. Such words are said to be bordered. Others (most) are unbordered.  ...

### Conjugacy of reversible cellular automata and one-head machines [article]

Ville Salo
2022 arXiv   pre-print
We also prove that the Brin-Thompson group 2V and groups of reversible Turing machines have undecidable conjugacy problems, and show that the word problems of the automorphism group and the topological  ...  For many automorphism groups of subshifts, as well as the group of asynchronous transducers and the homeomorphism group of the Cantor set, our result implies the existence of two elements such that every  ...  The length of a word is always a period of it, and a word is unbordered if and only if its minimal period is its length.  ...

### Arithmetical Complexity of the Language of Generic Limit Sets of Cellular Automata [article]

Solène J. Esnay, Alonso Núñez, Ilkka Törmä
2022 arXiv   pre-print
Introduced by Milnor, it has been studied in the context of one-dimensional cellular automata by Djenaoui and Guillon, Delacourt, and Törmä.  ...  The generic limit set of a dynamical system is the smallest set that attracts most of the space in a topological sense: it is the smallest closed set with a comeager basin of attraction.  ...  Acknowledgements The first two authors would like to thank Mathieu Sablik for his wise, kind and steady help and guidance.  ...

### Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081)

Maxime Crochemore, Lila Kari, Mehryar Mohri, Dirk Nowotka, Marc Herbstritt
2011 Dagstuhl Reports
Scientists working in the fields of Combinatorics on Words, Computational Biology, Stringology, Natural Computing, and Machine Learning were invited to consider the seminar's topic from a wide range of  ...  This report documents the program and the outcomes of Dagstuhl Seminar 11081 "Combinatorial and Algorithmic Aspects of Sequence Processing".  ...  Each such power is numerically expressed by its exponent: exp(w) = |w|/π(w), where |w| is the length of the word w, and π(w) is the smallest period of w.  ...

### Universal groups of cellular automata [article]

Ville Salo
2018 arXiv   pre-print
This result follows from a case study of groups of RCA generated by symbol permutations and partial shifts with respect to a fixed Cartesian product decomposition of the alphabet.  ...  For prime alphabets, we show that this group is virtually cyclic, and that for composite alphabets it is non-amenable. For alphabet size four, it is a linear group.  ...  Acknowledgements I have studied the linear part (in the CA sense) of PAut(A) for |A| = 4 with Pierre Guillon and Guillaume Theyssier, and the linear case of Lemma 9 is due to Theyssier.  ...
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