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Periodicity and unbordered segments of words

1979
*
Discrete Mathematics
*

%en T also is

doi:10.1016/0012-365x(79)90116-x
fatcat:m5syrfidrbbf7pexkjb7oomcbq
*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. ...##
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Relationship between the period of a finite word and the length of its unbordered segments

1982
*
Discrete Mathematics
*

A nonempty

doi:10.1016/0012-365x(82)90186-8
fatcat:yzf4k4o3ifcyzb6bskzlqojhkq
*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. ...##
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Abelian bordered factors and periodicity
[article]

2015
*
arXiv
*
pre-print

Ehrenfeucht

arXiv:1501.07464v1
fatcat:nkv7tatpljfb5mq5uge5nwbtga
*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*. ...##
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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. ...##
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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). ...##
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A Characterization of Cellular Automata Generated by Idempotents on the Full Shift
[article]

2012
*
arXiv
*
pre-print

We prove a characterization

arXiv:1206.0585v1
fatcat:zwm4zkcdpbbgfpxhnr7uqd3p7a
*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. ...##
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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. ...##
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A Characterization of Cellular Automata Generated by Idempotents on the Full Shift
[chapter]

2012
*
Lecture Notes in Computer Science
*

We prove a characterization

doi:10.1007/978-3-642-30642-6_27
fatcat:dxubqj7upfdmxoktqayglrgd4e
*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. ...##
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Finding the Leftmost Critical Factorization on Unordered Alphabet
[article]

2016
*
arXiv
*
pre-print

We present a linear time

arXiv:1509.01018v2
fatcat:iisu6jld3zh45oxd5h33sxyzpe
*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). ...##
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Finding the leftmost critical factorization on unordered alphabet

2016
*
Theoretical Computer Science
*

We present a linear time

doi:10.1016/j.tcs.2016.04.037
fatcat:gobruz2cl5hrpawyw7gwmtdfbq
*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 ...##
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Combinatorics of the Tautological Lamination
[article]

2021
*
arXiv
*
pre-print

Denote by N_q(n,m) the number

arXiv:2106.00578v2
fatcat:eg6je6x3tbd7bf3qz2ocdwjkmu
*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*. ...##
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Conjugacy of reversible cellular automata and one-head machines
[article]

2022
*
arXiv
*
pre-print

We also prove that the Brin-Thompson group 2V

arXiv:2011.07827v2
fatcat:luvv7wqifzav7d6bkaf5odst2i
*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. ...##
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Arithmetical Complexity of the Language of Generic Limit Sets of Cellular Automata
[article]

2022
*
arXiv
*
pre-print

Introduced by Milnor, it has been studied in the context

arXiv:2204.06215v1
fatcat:q3myzzjajjdrddjvj27wunem6u
*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. ...##
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Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081)

2011
*
Dagstuhl Reports
*

Scientists working in the fields

doi:10.4230/dagrep.1.2.47
dblp:journals/dagstuhl-reports/CrochemoreKMN11
fatcat:s3v55bqr4fbdfo5kztirfoljqm
*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. ...##
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Universal groups of cellular automata
[article]

2018
*
arXiv
*
pre-print

This result follows from a case study

arXiv:1808.08697v2
fatcat:e3lftkzji5aczfn4a4v4xub2je
*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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