A Fine and Wilf s theorem for pseudoperiods and Justin s formula for generalized pseudostandard words

Transcription

1 A Fine and Wilf s theorem for pseudoperiods and Justin s formula for generalized pseudostandard words A. Blondin Massé 1,2, G. Paquin 2 and L. Vuillon 2 1 Laboratoire de Combinatoire et d Informatique Mathématique Université du Québec à Montréal 2 Laboratoire de mathématique, Université de Savoie Journées montoises 2010 September 7th, 2010 Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

2 Outline 1 Introduction 2 Definitions 3 Pseudoperiodicity 4 Palindromic closure 5 Computing generalized pseudostandard words 6 Standard Rote words 7 Concluding remarks Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

3 Outline 1 Introduction 2 Definitions 3 Pseudoperiodicity 4 Palindromic closure 5 Computing generalized pseudostandard words 6 Standard Rote words 7 Concluding remarks Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

4 Discrete planes Is there a way to generate discrete planes by iterated pseudopalindromic closure? Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

5 Discrete planes Is there a way to generate discrete planes by iterated pseudopalindromic closure? Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

6 Discrete planes Is there a way to generate discrete planes by iterated pseudopalindromic closure? Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

7 Outline 1 Introduction 2 Definitions 3 Pseudoperiodicity 4 Palindromic closure 5 Computing generalized pseudostandard words 6 Standard Rote words 7 Concluding remarks Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

8 Words Words are (finite or infinite) sequences of symbols called letters , ababababab and are all words. The reversal of a word w = w 1 w 2 w n, denoted by R(w), is the word R(w) = w n w n 1 w 1. Over binary alphabets, the complement of a word w = w 1 w 2 w n, denoted by w, is the word w = w 1 w 2 w n obtained by swapping each letter w i. A morphism ϕ is an operator on words compatible with concatenation, i.e. ϕ(uv) = ϕ(u)ϕ(v) for any words u, v. An antimorphism is an operator on words satisfying ϕ(uv) = ϕ(v)ϕ(u). The operator is a morphism while R is an antimorphism. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

9 Palindromes and pseudopalindromes A palindrome, such as and , is a word which reads the same forward and backward. R(00100) = A pseudopalindrome or ϑ-palindrome is a word fixed by an involutory antimorphism ϑ. For instance, is an E-palindrome (or antipalindrome), where E = R, since E(001011) = R(110100) = Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

10 Examples R F R L F L tenet ressasser nagubugan r d r r d r reconocer Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

11 Complexity A word u is called a factor of another word w is there exist words p and s such that w = pus. Example: 011 is a factor of but 111 is not. Given a word w, its set of factors of length n is denoted by F n (w). The language of w, denoted by F (w), is the set of all factors occurring in w. Let w be a (finite or infinite) word. The factor complexity (or simply complexity) of w is the function f w : N N n Card(F n (w)) Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

12 Periodic words Consider the periodic word Then (0010) ω = F w (0) = {ε} F w (1) = {0, 1} F w (2) = {00, 01, 10} F w (3) = {000, 001, 010, 100} F w (4) = {0001, 0010, 0100, 1000} F w (5) = {00010, 00100, 01000, 10000}... One might verify that f w (0) = 1, f w (1) = 2, f w (2) = 3 and f w (n) = 4 for all integers n 3. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

13 Sturmian words Sturmian words are infinite aperiodic binary words having complexity f (n) = n + 1. The Fibonacci word is a Sturmian word. It is the limit of the sequence f n defined by f 0 = b, f 1 = a and f n = f n 1 f n 2 for all integers n 2. f 0 = b f 1 = a f 2 = ab f 3 = aba f 4 = abaab f 5 = abaababa... Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

14 Geometric representation Geometrically, Sturmian words are discrete approximations of lines having irrational slope. Sturmian word Standard Sturmian word Standard Sturmian words correspond to lines having intercept 0. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

15 Rote words In 1992, Rote used a general approach to investigate words having complexity 2n. In particular, he restricted his attention to the subset of these words having a language invariant under the complement operator, called complementary-symmetric words. Consider the operator on {0, 1} defined by (w 1 w 2 w n ) = (w 2 w 1 )(w 3 w 2 ) (w n w n 1 ), where the substraction is taken modulo 2. For example, (00101) = 0111 and (00100) = Theorem (Rote, 1992) Let r be an infinite binary word. Then r is a complementary-symmetric word if and only if (r) is a Sturmian word. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

16 Geometric interpretation Rote words encode the height modulo 2 of a discrete line. a b a b a a a b 3 a b a b a b Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

17 The Thue-Morse word The Thue-Morse word is the limit of the following sequence: t 0 = 0 t 1 = 01 t 2 = 0110 t 3 = t 4 = It has remarkable properties such as being aperiodic and overlap-free. In particular, t n is a palindrome if n is even and is an antipalindrome if n is odd. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

18 Outline 1 Introduction 2 Definitions 3 Pseudoperiodicity 4 Palindromic closure 5 Computing generalized pseudostandard words 6 Standard Rote words 7 Concluding remarks Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

19 Fine and Wilf s theorem Theorem (Fine and Wilf, 1965) Let w be a finite word and p, q be two periods of w. If w p + q gcd(p, q), then gcd(p, q) is also a period of w. This bound is tight. Example The word w = has period 3 and 7, but does not have the period gcd(3, 7) = 1. This does not contradict the theorem since w = 8 < 9 = Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

20 Pseudoperiods Definition Let w be a word and σ be an involutory permutation. A positive integer p is called a σ-period of w if w[i] = σ(w[i + p]) for i = 1, 2,..., w p. Remark We write E-period as a synonym of -periods, where denotes the complement operator on binary alphabet. Example R-periods correspond exactly to the usual periods. Let w = Then 4 is an E-period of w. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

21 Words having two pseudoperiods Example The word w = , admits the E-periods 4, 12 and 20 and the R-periods 8 and 16. Let us try to construct a word of length 9 admitting 3 as an E-period and 5 as an R-period. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

22 Words having two pseudoperiods Example The word w = , admits the E-periods 4, 12 and 20 and the R-periods 8 and 16. Let us try to construct a word of length 9 admitting 3 as an E-period and 5 as an R-period. a Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

23 Words having two pseudoperiods Example The word w = , admits the E-periods 4, 12 and 20 and the R-periods 8 and 16. Let us try to construct a word of length 9 admitting 3 as an E-period and 5 as an R-period. a a a Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

24 Words having two pseudoperiods Example The word w = , admits the E-periods 4, 12 and 20 and the R-periods 8 and 16. Let us try to construct a word of length 9 admitting 3 as an E-period and 5 as an R-period. a a a a a a Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

25 Words having two pseudoperiods Example The word w = , admits the E-periods 4, 12 and 20 and the R-periods 8 and 16. Let us try to construct a word of length 9 admitting 3 as an E-period and 5 as an R-period. a a a a a a a a Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

26 Words having two pseudoperiods Example The word w = , admits the E-periods 4, 12 and 20 and the R-periods 8 and 16. Let us try to construct a word of length 9 admitting 3 as an E-period and 5 as an R-period. a a a a a = a a a a a Contradiction! Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

27 A Fine and Wilf s theorem for pseudoperiods Theorem (Blondin Massé, Paquin and V., 2010) Let w be a finite binary word. Let p be a ϑ 1 -period of w and q be a ϑ 2 -period of w, with (ϑ 1, ϑ 2 ) (R, R). If w p + q, then gcd(p, q) is an E-period of w. This bound is tight. Example Let w = Then w has the E-periods 3 and 4 but gcd(3, 4) = 1 is not an E-period of w. This does not contradict the theorem since w = 6 < 7 = Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

28 Outline 1 Introduction 2 Definitions 3 Pseudoperiodicity 4 Palindromic closure 5 Computing generalized pseudostandard words 6 Standard Rote words 7 Concluding remarks Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

29 Palindromic closure Let w be a word. The palindromic closure of w, denoted by w (+), is the shortest palindrome having w has a prefix. For instance, if w = Then w (+) = Palindromes are fixed by (+). w = w (+) = p s R(p) Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

30 Iterated palindromic closure Let w be a finite word and α be a letter. The iterated palindromic closure of w, denoted by ψ(w), is defined by ψ(ε) = ε and ψ(wα) = (ψ(w)α) (+). In particular, w is called the directive word of ψ(w). Let w = Then so that ψ(ε) = ε ψ(0) = (ψ(ε) 0) (+) = 0 (+) = 0 ψ(00) = (ψ(0) 0) (+) = (00) (+) = 00 ψ(001) = (ψ(00) 1) (+) = (001) (+) = ψ(w) = (ψ(001) 0) (+) = (001000) (+) = Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

31 Standard Sturmian words The Fibonacci word may be obtained from the directive word (ab) ω : Palindromes correspond to paths fixed by rotations of angle π. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

32 Standard Sturmian words The Fibonacci word may be obtained from the directive word (ab) ω : a Palindromes correspond to paths fixed by rotations of angle π. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

33 Standard Sturmian words The Fibonacci word may be obtained from the directive word (ab) ω : a b Palindromes correspond to paths fixed by rotations of angle π. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

34 Standard Sturmian words The Fibonacci word may be obtained from the directive word (ab) ω : a b a Palindromes correspond to paths fixed by rotations of angle π. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

35 Standard Sturmian words The Fibonacci word may be obtained from the directive word (ab) ω : a b a b Palindromes correspond to paths fixed by rotations of angle π. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

36 Pseudopalindromic closure Palindromic closure extends naturally to pseudopalindromes. More precisely, the ϑ-palindromic closure of w, denoted by w (+) ϑ, is the shortest ϑ-palindrome having w as a prefix. Let w = and E be the exchange antimorphism, i.e. E = R. Then w (+) E = w = w (+) E = p s E(p) ϑ-palindromes are fixed by (+) ϑ. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

37 Generalized pseudostandard words Let (Θ, w) be a finite bi-sequence, where Θ is a sequence of involutory antimorphisms and w is a sequence of letters having the same length. Let ϑ be an involutory antimorphism and α be a letter. Then the iterated pseudopalindromic closure of (Θ, w), denoted by ψ Θ (w) is defined inductively by ψ ε (ε) = ε and ψ Θ ϑ(w α) = ( ψ Θ (w )α ) (+) ϑ. The bi-sequence (Θ, w) is called directive bi-sequence. ψ (RRERE) (00100) = 0 Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

38 Generalized pseudostandard words Let (Θ, w) be a finite bi-sequence, where Θ is a sequence of involutory antimorphisms and w is a sequence of letters having the same length. Let ϑ be an involutory antimorphism and α be a letter. Then the iterated pseudopalindromic closure of (Θ, w), denoted by ψ Θ (w) is defined inductively by ψ ε (ε) = ε and ψ Θ ϑ(w α) = ( ψ Θ (w )α ) (+) ϑ. The bi-sequence (Θ, w) is called directive bi-sequence. ψ (RRERE) (00100) = 00 Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

39 Generalized pseudostandard words Let (Θ, w) be a finite bi-sequence, where Θ is a sequence of involutory antimorphisms and w is a sequence of letters having the same length. Let ϑ be an involutory antimorphism and α be a letter. Then the iterated pseudopalindromic closure of (Θ, w), denoted by ψ Θ (w) is defined inductively by ψ ε (ε) = ε and ψ Θ ϑ(w α) = ( ψ Θ (w )α ) (+) ϑ. The bi-sequence (Θ, w) is called directive bi-sequence. ψ (RRERE) (00100) = Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

40 Generalized pseudostandard words Let (Θ, w) be a finite bi-sequence, where Θ is a sequence of involutory antimorphisms and w is a sequence of letters having the same length. Let ϑ be an involutory antimorphism and α be a letter. Then the iterated pseudopalindromic closure of (Θ, w), denoted by ψ Θ (w) is defined inductively by ψ ε (ε) = ε and ψ Θ ϑ(w α) = ( ψ Θ (w )α ) (+) ϑ. The bi-sequence (Θ, w) is called directive bi-sequence. ψ (RRERE) (00100) = Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

41 Generalized pseudostandard words Let (Θ, w) be a finite bi-sequence, where Θ is a sequence of involutory antimorphisms and w is a sequence of letters having the same length. Let ϑ be an involutory antimorphism and α be a letter. Then the iterated pseudopalindromic closure of (Θ, w), denoted by ψ Θ (w) is defined inductively by ψ ε (ε) = ε and ψ Θ ϑ(w α) = ( ψ Θ (w )α ) (+) ϑ. The bi-sequence (Θ, w) is called directive bi-sequence. ψ (RRERE) (00100) = Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

42 Generalized pseudostandard words Let (Θ, w) be a finite bi-sequence, where Θ is a sequence of involutory antimorphisms and w is a sequence of letters having the same length. Let ϑ be an involutory antimorphism and α be a letter. Then the iterated pseudopalindromic closure of (Θ, w), denoted by ψ Θ (w) is defined inductively by ψ ε (ε) = ε and ψ Θ ϑ(w α) = ( ψ Θ (w )α ) (+) ϑ. The bi-sequence (Θ, w) is called directive bi-sequence. ψ (RRERE) (00100) = Definition (de Luca, De Luca, 2006) Infinite words obtained by iterated pseudopalindromic closure from bi-sequences are called generalized pseudostandard words. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

43 Examples The class of generalized pseudostandard words includes well-known families of words. All words of the form ψ ϑ ω(wα ω ) are periodic. Standard Sturmian and Episturmian words are exactly the aperiodic infinite words of the form ψ R ω(w), where w is not ultimately constant. Words of the form ψ ΘR ω(w) and ψ ΘE ω(w) are all quasi-sturmian words, i.e. having complexity f (n) = n + k for some constant k (Bucci et al., 2008). The Thue-Morse word is a generalized pseudostandard word (de Luca, De Luca, 2006). ψ (RE) ω(01 ω ) = What are the possible complexity functions for GPS words? Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

44 Outline 1 Introduction 2 Definitions 3 Pseudoperiodicity 4 Palindromic closure 5 Computing generalized pseudostandard words 6 Standard Rote words 7 Concluding remarks Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

45 The brute-force algorithm 1: function ψ(ϑ 1 ϑ 2 ϑ n, w 1 w 2 w n ) 2: Input: A finite bi-sequence (Θ, w) = (ϑ 1 ϑ 2 ϑ n, w = w 1 w 2 w n ) 3: Output: The iterated palindromic closure of (Θ, w) 4: u ε 5: for i {1, 2,..., n} do 6: u u w i 7: Compute the length j of the longest ϑ i -palindromic suffix of u. 8: u u ϑ i (Pref u j (u)) 9: end for 10: return u 11: end function Line 7 might be inefficient. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

46 Justin s formula Proposition (Justin, 2005) Let α A and w A. Then ψ(ε) = ε and { ψ(w)αψ(w) if w α = 0, ψ(wα) = ψ(w)ψ(w 1 ) 1 ψ(w) otherwise, with w = w 1 αw 2, w 2 α = 0. ψ(ε) = ε ψ(0) = ψ(ε) 0 ψ(ε) = 0 ψ(00) = ψ(0) ψ(ε) 1 ψ(0) = 00 ψ(001) = ψ(00) 1 ψ(00) = ψ(0010) = ψ(001) ψ(0) 1 ψ(001) = =... =... Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

47 Efficient Computation 1: function ψ(w 1 w 2 w n ) 2: Input: A finite word w = w 1 w 2 w n 3: Output: The iterated palindromic closure of w 4: LastLength[α] 1, α A 5: u ε 6: for i {1, 2,..., n} do 7: Length = u 8: if LastLength[w i ] = 1 then 9: u u w i u 10: else 11: u Pref u LastLength[wi ](u) u 12: end if 13: LastLength[w i ] = Length 14: end for 15: return u 16: end function Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

48 Generalizing Justin s formula Theorem (Blondin Massé, Paquin and V., 2010) Let (Θ, w) be a normalized finite directive bi-sequence of length n over {0, 1} and for i = 1, 2,..., n, let ψ i = ψ ϑ1 ϑ 2 ϑ i (w[1... i]), ψ 0 = ε and α i be the last letter of ψ i. (i) If w[1]w[2] w[n 1] w[n] = 0 or ϑ 1 ϑ 2 ϑ n 1 ϑn = 0, then ψ n 1 w[n]ψ n 1 if ϑ n = R, ψ n = ψ n 1 ψ n 1 if ϑ n = E and α n 1 w[n], ψ n 1 w[n]w[n] ψ n 1 if ϑ n = E and α n 1 = w[n]. (ii) If one can write (Θ, w) = (Θ ϑ nθ, w (ϑ n 1 ϑ n)(w[n])w ) such that i := w = Θ + 1 with Θ maximum, then ( ) ψ n = ψ n 1 (ϑ n 1 ϑ n) ψ 1 i ψ n 1. (iii) Otherwise, ψ n = { ψ n 1 ϑ n(ψ n 1 ) ψ n 1 w[n](ϑ n 1 ϑ n) (w[n]ψ n 1 ) if ϑ n = R or α n 1 w[n], if ϑ n = E and α n 1 = w[n]. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

49 Generalizing Justin s formula Theorem (Blondin Massé, Paquin and V., 2010) Let (Θ, w) be a normalized finite directive bi-sequence of length n over {0, 1} and for i = 1, 2,..., n, let ψ i = ψ ϑ1 ϑ 2 ϑ i (w[1... i]), ψ 0 = ε and α i be the last letter of ψ i. (i) If w[1]w[2] w[n 1] w[n] = 0 or ϑ 1 ϑ 2 ϑ n 1 ϑn = 0, then ψ n 1 w[n]ψ n 1 if ϑ n = R, ψ n = ψ n 1 ψ n 1 if ϑ n = E and α n 1 w[n], ψ n 1 w[n]w[n] ψ n 1 if ϑ n = E and α n 1 = w[n]. (ii) If one can write (Θ, w) = (Θ ϑ nθ, w (ϑ n 1 ϑ n)(w[n])w ) such that i := w = Θ + 1 with Θ maximum, then ( ) ψ n = ψ n 1 (ϑ n 1 ϑ n) ψ 1 i ψ n 1. (iii) Otherwise, ψ n = { ψ n 1 ϑ n(ψ n 1 ) ψ n 1 w[n](ϑ n 1 ϑ n) (w[n]ψ n 1 ) if ϑ n = R or α n 1 w[n], if ϑ n = E and α n 1 = w[n]. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

50 Missing pseudopalindromes (RREERR, ) (R, 0) = 0 (RR, 00) = 00 (RRE, 001) = 0011 (RREE, 0010) = missed (RREER, 00101) = (RREERR, ) = missed Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

51 Normalization Definition A directive bi-sequence (Θ, w) is normalized if it describes all the pseudopalindromic prefixes of ψ Θ (w). Theorem (Blondin Massé, Paquin and V., 2010) Let (Θ, w) be a directive bi-sequence. Then there exists a normalized directive bi-sequence (Θ, w ) such that ψ Θ (w) = ψ Θ (w ). Moreover, in order to get the normalized directive bi-sequence (Θ, w ) from (Θ, w), it is sufficient to replace the prefix (if it is of one of the following forms): (i) (RR, aa) by (RER, aaa); (ii) (R i 1 E, a i ) by (R i E, a i a); (iii) (R i EE, a i a a) by (R i ERE, a i a aa); for i 1 and then, to replace from left to right any factor (ϑ 1 ϑ 2 ϑ 2, abb) by (ϑ 1 ϑ 2 ϑ 1 ϑ 2, abbb), where ϑ 1, ϑ 2 {R, E}, ϑ 1 ϑ 2 and a, b {0, 1}. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

52 Outline 1 Introduction 2 Definitions 3 Pseudoperiodicity 4 Palindromic closure 5 Computing generalized pseudostandard words 6 Standard Rote words 7 Concluding remarks Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

53 Standard Rote words Consider again the Fibonacci word s = Using Rote s Theorem, we can construct the sequence r = which satisfies (r) = s by inverting the operator, which consists in computing partial sums. Definition A standard Rote word is a complementary-symmetric word r such that (r) is a standard Sturmian word. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

54 Relation between directive sequences The directive sequence of the Fibonacci word is (01) ω : s = On the other hand, the directive bi-sequence of the word r = satisfying (r) = s is ((RRE) ω, (001110) ω ). Is there a simple link between the two? The answer is yes. It suffices to look at the palindrome prefixes of s and the pseudopalindrome prefixes of r. ε Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

55 Three types of palindromes We divide in three categories the palindromes occurring in Sturmian words: (i) Even palindromes (e), (ii) Odd palindromes with central letter 0 (o). (iii) Odd palindromes with central letter 1 (a). Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

56 Palindrome types and iterated palindromic closure (ε, e) 0 1 (0, eo) (1, ea) (0 0, eoe) (0 1 0, eoa) (1 0 1, eae) (1 1, eao) (0 0 0, eoeo) ( , eoea) ( , eoae) ( , eoao) Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

57 Main result Theorem (Blondin Massé and V., 2010) Let w the the directive word of some standard Sturmian word s and r be the infinite word starting with 0 and satisfying (r) = s. If the transducer in the next slide receives w as input, then the generated output is exactly the directive bi-sequence of r. Input: The directive word of s. Output: The directive bi-sequence of r. Corollary Standard Rote words are normalized. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

58 Transducer oa0 0/(E, 0) 0/(R, 1) 0/(E, 0) 1/(R, 0) eo1 1/(R, 1) 1/(R, 1) ao0 0/(R, 1) 1/(E, 1) ae1 1/(R, 1) oe1 start ea1 1/(R, 0) 1/(R, 1) 0/(E, 0) 1/(E, 1) i 0/(R, 1) ea0 ε/(r, 0) 0/(R, 0) oe0 0/(R, 0) ae0 1/(R, 0) 0/(R, 0) 0/(E, 0) ao1 0/(R, 0) eo0 1/(E, 1) 1/(R, 0) 0/(R, 1) 1/(E, 1) oa1 Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

59 Outline 1 Introduction 2 Definitions 3 Pseudoperiodicity 4 Palindromic closure 5 Computing generalized pseudostandard words 6 Standard Rote words 7 Concluding remarks Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

60 Summary In this talk, we discussed generalized pseudostandard words. We presented a linear algorithm that allows one to compute GPS words over binary alphabets. We have exhibited new properties about the palindrome types occurring in Sturmian words. We have provided a transducer linking directive word of standard Sturmian words with directive bi-sequence of standard Rote words. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

61 Open problems Generalize Justin s formula for arbitrary alphabet and arbitrary antimorphisms. What is the maximum complexity of a generalized pseudostandard word? Computer exploration suggests that it is at most linear. Moreover, for binary alphabets, we have observed that complexity might be greater than 4n, but it seems to oscillates around it, which suggests the following: Conjecture (Blondin Massé and V., 2010) Let w be a generalized pseudostandard word. Then for any constant k > 4, there exists a positive integer n 0 such that f w (n) kn, for all n > n 0. Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45

62 Using Sage sage: A = Words([0,1]) sage: R = WordMorphism({0:0,1:1}, codomain=a) sage: E = WordMorphism({0:1,1:0}, codomain=a) sage: T = [R,E,E,E,E,E,R,R,R,E] + [R,E] sage: w = A([0,0,0,1,1,0,1,0,1,1] + [0,0]) sage: u = iterated_right_palindromic_closure(w, T) sage: v = u[:2000] sage: [round(float(v.number_of_factors(i) / (4 * i)),3) for i in range(1,200)] [0.5, 0.5, 0.5, 0.5, 0.5, 0.583, 0.643, 0.688, 0.722, 0.8, 0.864, 0.917, 0.962, 1.0, 1.033, 1.063, 1.088, 1.111, 1.132, 1.15, 1.167, 1.182, 1.196, 1.208, 1.22, 1.212, 1.204, 1.196, 1.19, 1.183, 1.177, 1.172, 1.167, 1.162, 1.157, 1.153, 1.149, 1.145, 1.141, 1.138, 1.134, 1.131, 1.128, 1.125, 1.122, 1.12, 1.117, 1.115, 1.112, 1.11, 1.108, 1.106, 1.104, 1.102, 1.1, 1.098, 1.096, 1.095, 1.093, 1.092, 1.09, 1.089, 1.087, 1.086, 1.085, 1.083, 1.082, 1.081, 1.08, 1.079, 1.077, 1.076, 1.075, 1.074, 1.073, 1.072, 1.071, 1.071, 1.07, 1.069, 1.068, 1.067, 1.066, 1.065, 1.065, 1.064, 1.063, 1.063, 1.062, 1.061, 1.06, 1.06, 1.059, 1.059, 1.058, 1.057, 1.057, 1.056, 1.056, 1.055, 1.054, 1.054, 1.053, 1.053, 1.052, 1.052, 1.051, 1.051, 1.05, 1.05, 1.05, 1.049, 1.049, 1.048, 1.048, 1.047, 1.047, 1.047, 1.046, 1.046, 1.045, 1.045, 1.045, 1.044, 1.044, 1.044, 1.043, 1.043, 1.043, 1.042, 1.042, 1.042, 1.041, 1.041, 1.041, 1.04, 1.04, 1.04, 1.04, 1.039, 1.039, 1.039, 1.038, 1.038, 1.038, 1.038, 1.037, 1.037, 1.037, 1.036, 1.036, 1.036, 1.036, 1.035, 1.034, 1.032, 1.03, 1.028, 1.027, 1.025, 1.023, 1.021, 1.02, 1.018, 1.017, 1.015, 1.013, 1.012, 1.01, 1.009, 1.007, 1.006, 1.004, 1.003, 1.001, 1.0, 0.999, 0.997, 0.996, 0.994, 0.993, 0.992, 0.99, 0.989, 0.988, 0.987, 0.985, 0.984, 0.983, 0.982, 0.98, 0.979, 0.978, 0.977, 0.976, 0.975, 0.973, 0.972] Blondin Massé et al. (LaCIM, LAMA) On the generalized pseudostandard words JM / 45