On factor and permutation complexity of infinite words Alexandr - PowerPoint PPT Presentation

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