In combinatorics, a squarefree word is a word (a sequence of symbols) that does not contain any squares. A square is a word of the form , where is not empty. Thus, a squarefree word can also be defined as a word that avoids the pattern .
Finite squarefree words
editBinary alphabet
editOver a binary alphabet , the only squarefree words are the empty word and .
Ternary alphabet
editOver a ternary alphabet , there are infinitely many squarefree words. It is possible to count the number of ternary squarefree words of length .
n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
c(n) | 1 | 3 | 6 | 12 | 18 | 30 | 42 | 60 | 78 | 108 | 144 | 204 | 264 |
This number is bounded by , where [2]. The upper bound on can be found via Fekete's Lemma and approximation by automata. The lower bound can be found by finding a substitution that preserves squarefreeness[2].
Alphabet with more than three letters
editSince there are infinitely many squarefree words over three-letter alphabets, this implies there are also infinitely many squarefree words over an alphabet with more than three letters.
The following table shows the exact growth rate of the k-ary squarefree words:
alphabet size (k) | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|
growth rate | 2.6215080 | 3.7325386 | 4.7914069 | 5.8284661 | 6.8541173 | 7.8729902 |
alphabet size (k) | 10 | 11 | 12 | 13 | 14 | 15 |
growth rate | 8.8874856 | 9.8989813 | 10.9083279 | 11.9160804 | 12.9226167 | 13.9282035 |
2-dimensional words
editConsider a map from to , where is an alphabet and is called a 2-dimensional word. Let be the entry . A word is a line of if there exists such that , and for [3].
Carpi[4] proves that there exists a 2-dimensional word over a 16-letter alphabet such that every line of is squarefree. A computer search shows that there are no 2-dimensional words over a 7-letter alphabet, such that every line of is squarefree.
Generating finite squarefree words
editShur[5] proposes an algorithm called R2F (random-t(w)o-free) that can generate a squarefree word of length over any alphabet with three or more letters. This algorithm is based on a modification of entropy compression: it randomly selects letters from a k-letter alphabet to generate a (k+1)-ary squarefree word.
algorithm R2F is input: alphabet size , word length output: a (k+1)-ary squarefree word of length . (Note that is the alphabet with letters .) (For a word , is the permutation of such that precedes in if the right most position of in is to the right of the rightmost position of in . For example, has .) choose in uniformly at random set to followed by all other letters of in increasing order set the number of iterations to 0 while do choose in uniformly at random append to the end of update shifting the first elements to the right and setting increment by if ends with an square of rank do delete the last letters of return
Every (k+1)-ary squarefree word can be the output of Algorithm R2F, because on each iteration it can append any letter except for the last letter of .
The expected number of random k-ary letters used by Algorithm R2F to construct a (k+1)-ary squarefree word of length is Note that there exists an algorithm that can verify the squarefreeness of a word of length in time. Apostolico and Preparata[6] give an algorithm using suffix trees. Crochemore[7] uses partitioning in his algorithm. Main and Lorentz[8] provide an algorithm based on the divide-and-conquer method. A naive implementation may require time to verify the squarefreeness of a word of length .
Infinite squarefree words
editThere exist arbitrarily long squarefree words in any alphabet with three or more letters, as proved by Axel Thue[9].
Examples
editFirst difference of the Thue–Morse sequence
editOne example of an infinite squarefree word over an alphabet of size 3 is the word over the alphabet obtained by taking the first difference of the Thue–Morse sequence [9]. That is, from the Thue–Morse sequence
one forms a new sequence in which each term is the difference of two consecutive terms of the Thue–Morse sequence. The resulting squarefree word is
Another example found by John Leech[10] is defined recursively over the alphabet . Let be any squarefree word starting with the letter . Define the words recursively as follows: the word is obtained from by replacing each in with , each with , and each with . It is possible to prove that the sequence converges to the infinite squarefree word
Generating infinite squarefree words
editInfinite squarefree words can be generated by squarefree morphism. A morphism is called squarefree if the image of every squarefree word is squarefree. A morphism is called k–squarefree if the image of every squarefree word of length k is squarefree.
Crochemore[11] proves that a uniform morphism is squarefree if and only if it is 3-squarefree. In other words, is squarefree if and only if is squarefree for all squarefree of length 3. It is possible to find a squarefree morphism by brute-force search.
algorithm squarefree_morphism is output: a squarefree morphism with the lowest possible rank . set while True do set to the list of all squarefree words of length over a ternary alphabet for each in do for each in do for each in do if do break from the current loop (advance to next ) if and do if is squarefree for all squarefree of length do return increment by
Over a ternary alphabet, there are exactly 144 uniform squarefree morphisms of rank 11 and no uniform squarefree morphisms with a lower rank than 11.
To obtain an infinite squarefree words, start with any squarefree word such as , and successively apply a squarefree morphism to it. The resulting words preserve the property of squarefreeness. For example, let be a squarefree morphism, then as , is an infinite squarefree word.
Note that, if a morphism over a ternary alphabet is not uniform, then this morphism is squarefree if and only if it is 5-squarefree[11].
Letter combinations in squarefree words
editAvoid two-letter combinations
editOver a ternary alphabet, a squarefree word of length more than 13 contains all the squarefree two-letter combinations[12].
This can be proved by constructing a squarefree word without the two-letter combination . As a result, is the longest squarefree word without the combination and its length is equal to 13.
Note that over a more than three-letter alphabet there are squarefree words of any length without an arbitrary two-letter combination.
Avoid three-letter combinations
editOver a ternary alphabet, a squarefree word of length more than 36 contains all the squarefree three-letter combinations[12].
However, there are squarefree words of any length without the three-letter combination .
Note that over a more than three-letter alphabet there are squarefree words of any length without an arbitrary three-letter combination.
Density of a letter
editThe density of a letter in a finite word is defined as where is the number of occurrences of in and is the length of the word. The density of a letter in an infinite word is where is the prefix of the word of length [13].
The minimal density of a letter in an infinite ternary squarefree word is equal to [13].
The maximum density of a letter in an infinite ternary squarefree word is equal to [14].
Related concepts
editA cube-free word is one with no occurrence of www for a factor w. The Thue-Morse sequence is an example of a cube-free word over a binary alphabet.[15] This sequence is not squarefree but is "almost" so: the critical exponent is 2.[16] The Thue–Morse sequence has no overlap or overlapping square, instances of 0X0X0 or 1X1X1:[15] it is essentially the only infinite binary word with this property.[17] Dejean's theorem characterizes the minimum possible critical exponents for each alphabet size.[18]
The Thue number of a graph G is the smallest number k such that G has a k-coloring for which the sequence of colors along every simple path is squarefree.
The Kolakoski sequence is an example of a cube-free sequence.
An abelian p-th power is a subsequence of the form where each is a permutation of . There is no abelian-squarefree infinite word over an alphabet of size three: indeed, every word of length eight over such an alphabet contains an abelian square. There is an infinite abelian-squarefree word over an alphabet of size five.[19]
Notes
edit- ^ "A006156 - OEIS". oeis.org. Retrieved 2019-03-28.
- ^ a b c Shur, Arseny (2011). "Growth properties of power-free languages". Computer Science Review: 28–43.
- ^ Berthe, Valerie; Rigo, Michel (eds.), "Preface", Combinatorics, Words and Symbolic Dynamics, Cambridge University Press, pp. xi–xviii, ISBN 9781139924733, retrieved 2019-04-08
- ^ Carpi, Arturo (1988). "Multidimensional unrepetitive configurations". Theoretical Computer Science. 56 (2): 233–241. doi:10.1016/0304-3975(88)90080-1. ISSN 0304-3975.
- ^ Shur, Arseny (2015). "Generating square-free words efficiently". Theoretical Computer Science. 601: 67–72.
- ^ Apostolico, A.; Preparata, F.P. (Feb 1983). "Optimal off-line detection of repetitions in a string". Theoretical Computer Science. 22 (3): 297–315. doi:10.1016/0304-3975(83)90109-3. ISSN 0304-3975.
- ^ Crochemore, Max (Oct 1981). "An optimal algorithm for computing the repetitions in a word". Information Processing Letters. 12 (5): 244–250. doi:10.1016/0020-0190(81)90024-7. ISSN 0020-0190.
- ^ Main, Michael G; Lorentz, Richard J (Sep 1984). "An O(n log n) algorithm for finding all repetitions in a string". Journal of Algorithms. 5 (3): 422–432. doi:10.1016/0196-6774(84)90021-x. ISSN 0196-6774.
- ^ a b Berstel, Jean (1994). Axel Thue's papers on repetitions in words a translation. Départements de mathématiques et d'informatique, Université du Québec à Montréal. ISBN 2892761409. OCLC 494791187.
- ^ Leech, J. (1957). "A problem on strings of beads". Math. Gazette. 41: 277–278. Zbl 0079.01101.
- ^ a b Berstel, Jean (April 1984). "Some Recent Results on Squarefree Words" (PDF). Annual Symposium on Theoretical Aspects of Computer Science: 14–25.
- ^ a b Zolotov, Boris. "Another Solution to the Thue Problem of Non-Repeating Words" (PDF). ArXiv 2015.
- ^ a b Khalyavin, Andrey (2007). "The minimal density of a letter in an infinite ternary square-free word is 883/3215" (PDF). Journal of Integer Sequences. 10(2): 3.
- ^ Ochem, Pascal (2007). "Letter frequency in infinite repetition-free words". Theoretical Computer Science. 380 (3): 388–392. doi:10.1016/j.tcs.2007.03.027. ISSN 0304-3975.
- ^ a b Lothaire (2011) p.113
- ^ Krieger, Dalia (2006). "On critical exponents in fixed points of non-erasing morphisms". In Ibarra, Oscar H.; Dang, Zhe (eds.). Developments in Language Theory: Proceedings 10th International Conference, DLT 2006, Santa Barbara, CA, USA, June 26-29, 2006. Lecture Notes in Computer Science. Vol. 4036. Springer-Verlag. pp. 280–291. ISBN 3-540-35428-X. Zbl 1227.68074.
- ^ Berstel et al (2009) p.81
- ^ Rampersad, Narad; Shallit, Jeffrey (2016). "Repetitions in words". Combinatorics, words and symbolic dynamics. Encyclopedia Math. Appl. Vol. 159. Cambridge Univ. Press, Cambridge. pp. 101–150. MR 3525483.
- ^ Blanchet-Sadri, Francine; Simmons, Sean (2011). "Avoiding Abelian Powers in Partial Words". In Mauri, Giancarlo; Leporati, Alberto (eds.). Developments in Language Theory. Proceedings, 15th International Conference, DLT 2011, Milan, Italy, July 19-22, 2011. Lecture Notes in Computer Science. Vol. 6795. Berlin, Heidelberg: Springer-Verlag. pp. 70–81. doi:10.1007/978-3-642-22321-1_7. ISBN 978-3-642-22320-4. ISSN 0302-9743.
References
edit- Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009). Combinatorics on words. Christoffel words and repetitions in words. CRM Monograph Series. Vol. 27. Providence, RI: American Mathematical Society. ISBN 978-0-8218-4480-9. Zbl 1161.68043.
- Lothaire, M. (1997). Combinatorics on words. Cambridge: Cambridge University Press. ISBN 0-521-59924-5..
- Lothaire, M. (2011). Algebraic combinatorics on words. Encyclopedia of Mathematics and Its Applications. Vol. 90. With preface by Jean Berstel and Dominique Perrin (Reprint of the 2002 hardback ed.). Cambridge University Press. ISBN 978-0-521-18071-9. Zbl 1221.68183.
- Pytheas Fogg, N. (2002). Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, Anne (eds.). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. Vol. 1794. Berlin: Springer-Verlag. ISBN 3-540-44141-7. Zbl 1014.11015.