On factor and permutation complexity of infinite words Alexandr Valyuzhenich Sobolev Institute of Mathematics May 9, 2018 Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Basic definitions Let Σ be a finite alphabet. An infinite word over Σ is a sequence of the form w = w 1 w 2 w 3 . . . where w i ∈ Σ . Definition A finite word u is called a factor or subword of length n of an infinite word w if u = w i + 1 . . . w i + n for some i ≥ 0. Let F w ( n ) be the set of all distinct factors of w of length n . Definition The factor complexity (or subword complexity) of w is f w ( n ) = | F w ( n ) | . Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Basic definitions Definition An infinite word w is called periodic if w = uvvv . . . for some finite words u and v . Definition An infinite word w is called aperiodic if w is not a periodic word. We note that if w is a periodic word, then f w ( n ) ≤ C for some constant C . Theorem (Morse, Hedlund, 1940) Let w be an infinite aperiodic word. Then f w ( n ) ≥ n + 1. Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Sturmian words Definition An infinite word w is called a Sturmian word if f w ( n ) = n + 1 for arbitrary n . So, Sturmian words have the minimum factor complexity in the class of all infinite aperiodic words. Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Morphisms Definition A map ϕ : Σ ∗ − → Σ ∗ is called a morphism if ϕ ( xy ) = ϕ ( x ) ϕ ( y ) for any words x , y ∈ Σ ∗ . Let u = u 1 u 2 . . . u n be a word. Then ϕ ( u ) = ϕ ( u 1 ) ϕ ( u 2 ) . . . ϕ ( u n ) . So, every morphism is uniquely determined by the images of letters. Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Construction of fixed point of morphisms Consider a morphism ϕ : Σ ∗ − → Σ ∗ . Let a ∈ Σ and ϕ ( a ) = ax for some nonempty word x . Then ϕ 2 ( a ) = ϕ ( ϕ ( a )) = ϕ ( ax ) = ϕ ( a ) ϕ ( x ) = ax ϕ ( x ) and ϕ 3 ( a ) = ϕ ( ϕ 2 ( a )) = ax ϕ ( x ) ϕ 2 ( x ) . We see that ϕ n ( a ) = ax ϕ ( x ) . . . ϕ n − 1 ( x ) . We have that ϕ n − 1 ( a ) is a prefix ϕ n ( a ) for any n . We define the infinite word w as follows: let the prefix w 1 . . . w | ϕ n ( a ) | of w is ϕ n ( a ) . The word w is denoted n →∞ ϕ n ( a ) . We note that w = ϕ ( w ) . by lim Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Fixed point of morphisms Definition An infinite word w is a fixed point of a morphism ϕ if w = ϕ ( w ) . Proposition There exists only one fixed point w of a morphism ϕ that starts n →∞ ϕ n ( a ) . with symbol a . Moreover, w = lim Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Fixed point of morphisms Definition An infinite word w is a fixed point of a morphism ϕ if w = ϕ ( w ) . Proposition There exists only one fixed point w of a morphism ϕ that starts n →∞ ϕ n ( a ) . with symbol a . Moreover, w = lim Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Important morphisms Definition The morphism ϕ ( 0 ) = 01, ϕ ( 1 ) = 0 is the Fibonacci morphism. A fixed point of the Fibonacci morphism is called the Fibonacci word. The Fibonacci word is a Sturmian word. Definition The morphism ϕ ( 0 ) = 01, ϕ ( 1 ) = 10 is the Thue-Morse morphism. A fixed point of the Thue-Morse morphism is called the Thue-Morse word. Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Morphisms Definition A finite word u is called a square (cube) if u = vv ( u = vvv ) for some word v . Definition An infinite word w is called square-free (cube-free) if all factors of w are not squares (cubes). Problem Does there exist an infinite square-free (cube-free) word? Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Morphisms Theorem (Thue, 1906) There exists an infinite cube-free word over an alphabet of size two. For example, the Thue-Morse word is a cube-free word. Theorem (Thue, 1906) There exists an infinite square-free word over an alphabet of size three. For example, the fixed point of the morphism ϕ ( a ) = abc , ϕ ( b ) = ac and ϕ ( c ) = b is a square-free word. Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Factor complexity For the problem of finding the factor complexity of fixed point morphisms was developed a good general approach: In 1997 Cassaigne developed an algorithm for the calculating the factor complexity of a fixed point of biprefix morphisms. In 1998 Avgustinovich and Frid found the exact formula for the factor complexity of a fixed point of biprefix morphisms. Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Linear order on the shifts Let w = w 1 w 2 w 3 . . . be an infinite aperiodic word and < L is a lexicographic order on Σ . The word w i w i + 1 . . . is denoted by w [ i ] . We will write w [ i ] < w [ j ] if w [ i ] = xa . . . , w [ j ] = xb . . . and a < L b . Definition A permutation π = π 1 . . . π n of numbers { 1 , . . . , n } is a subpermutation of an infinite aperiodic word w if there exists i ≥ 0 such that π k < π m iff w [ i + k ] < w [ i + m ] . Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Example of subpermutation Let T be the Thue-Morse word: T = 0110100110010110 . . . We have that T [ 4 ] = 010 . . . , T [ 5 ] = 100 . . . , T [ 6 ] = 001 . . . and T [ 7 ] = 011 . . . . So T [ 6 ] < T [ 4 ] < T [ 7 ] < T [ 5 ] . Therefore π = 2413 is a subpermutation of the Thue-Morse word. Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Permutation complexity Let P w ( n ) be the set of all distinct subpermutations of w of length n . Definition The permutation complexity of w is p w ( n ) = | P w ( n ) | . Permutation complexity of aperiodic words is a relatively new notion in combinatorics on words. This complexity was introduced by Makarov: M. A. Makarov. On permutations generated by infinite binary words. Sib. Elektron. Mat. Izv., 3 (2006), 304–311. Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Valid permutations Definition A finite permutation π is called a valid if π is a subpermutation of some infinite binary word w . Example. π = 132 is a valid permutation. Indeed, we consider the word w = 01011101 5 . . . 01 2 n + 1 . . . . We have w [ 3 ] = 011 . . . , w [ 4 ] = 111 . . . and w [ 5 ] = 110 . . . . So w [ 3 ] < w [ 5 ] < w [ 4 ] . Therefore π = 132 is valid. Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Valid permutations We note that not all permutations are valid. For example, consider permutation π = 2134. Suppose that there exists a binary word w such that 2134 is a subpermutation of w . Then for some i we have w [ i + 2 ] < w [ i + 1 ] < w [ i + 3 ] < w [ i + 4 ] . Since w [ i + 1 ] > w [ i + 2 ] , w [ i + 2 ] < w [ i + 3 ] and w [ i + 3 ] < w [ i + 4 ] , we see that w i + 1 = 1, w i + 2 = 0 and w i + 3 = 0. Hence w [ i + 1 ] = 1 . . . > w [ i + 3 ] = 0 . . . and we obtain a contradiction. Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Valid permutations Let p ( n ) be the number of all distinct valid permutations of length n . Theorem (Makarov, 2006) n � Ψ( t ) · 2 n − t p ( n + 1 ) = t = 1 for n ≥ 1. Corollary (Makarov, 2006) p ( n + 1 ) = 2 n ( n − c + O ( n 2 − n / 2 )) . So the maximum permutation complexity of an infinite binary aperiodic word is 2 n ( n − c + O ( n 2 − n / 2 )) . Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Permutation complexity Theorem (Makarov, 2009) Let w be a Sturmian word. Then p w ( n ) = n . So, for the Sturmian words we have f w ( n − 1 ) = p w ( n ) . Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Permutation complexity Period doubling word is the fixed point of the morphism ϕ ( 0 ) = 0100, ϕ ( 1 ) = 0101. Theorem (Makarov, 2010) Let D be the period doubling word. Then � n + 6 · 2 t − 1 , if 5 · 2 t + 1 ≤ n ≤ 6 · 2 t and t ≥ 1; p D ( n ) = 2 n + 2 · 2 t − 2 , if 6 · 2 t + 1 ≤ n ≤ 10 · 2 t and t ≥ 0. for n ≥ 7. Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Permutation complexity Theorem (Widmer, 2011) Let T be the Thue-Morse word, n = 2 a + b and 0 < b ≤ 2 a . Then p T ( n ) = 2 ( 2 a + 1 + b − 2 ) for n ≥ 6. Alexandr Valyuzhenich On factor and permutation complexity of infinite words
Morphisms A morphism is called l –uniform if its blocks are of the same length l . A morphism ϕ : Σ ∗ − → Σ ∗ is called a marked if its blocks are of the form ϕ ( a i ) = b i x i c i , where x i is an arbitrary word, b i and c i are symbols of the alphabet Σ , and all b i (as well as all c i ) are distinct. A morphism ϕ : Σ ∗ − → Σ ∗ where Σ = { 0 , 1 } is called binary. Alexandr Valyuzhenich On factor and permutation complexity of infinite words
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