Maximal Pattern Complexity Research Articles

Abelian maximal pattern complexity of words

AbstractIn this paper, we study the maximal pattern complexity of infinite words up to Abelian equivalence. We compute a lower bound for the Abelian maximal pattern complexity of infinite words which are both recurrent and aperiodic by projection. We show that in the case of binary words, the bound is actually achieved and gives a characterization of recurrent aperiodic words.

Maximal pattern complexity, dual system and pattern recognition

For a family Ω of sets in R2 and a finite subset S of R2, let pΩ(S) be the number of distinct sets of the form S∩ω for all ω∈Ω. The maximum pattern complexity pΩ∗(k) is the maximum of pΩ(S) among S with #S=k. The S attaining the maximum is considered as the most effective sampling to distinguish the sets in Ω. We obtain the exact values or at least the order of pΩ∗(k) in k for various classes Ω. We also discuss the dual problem in the case that #Ω=∞, that is, consider the partition of R2 generated by a finite family T⊂Ω. The number of elements in the partition is written as pR2(T) and pR2∗(k) is the maximum of pR2(T) among T with #T=k. Here, pΩ∗(k)=pR2∗(k) does not hold in general.For the general setting that Ω is an infinite subset of AΣ, where A is a finite alphabet, Σ is an arbitrary infinite set, and pΩ∗(k)=max#S=k#Ω∣S, it is known that the entropy h(Ω):=limk→∞logpΩ∗(k)/k exists and takes value in {log1,log2,…,log#A}. In this paper, we prove that the entropy h(Σ) of the dual system coincides with h(Ω).

Infinite permutations of lowest maximal pattern complexity

An infinite permutation α is a linear ordering of N . We study properties of infinite permutations analogous to those of infinite words, and show some resemblances and some differences between permutations and words. In this paper, we define maximal pattern complexity p α ∗ ( n ) for infinite permutations and show that this complexity function is ultimately constant if and only if the permutation is ultimately periodic; otherwise its maximal pattern complexity is at least n , and the value p α ∗ ( n ) ≡ n is reached exactly on a family of permutations constructed by Sturmian words.

On maximal pattern complexity of some automatic words

AbstractThe pattern complexity of a word for a given pattern S, where S is a finite subset of {0,1,2,…}, is the number of distinct restrictions of the word to S+n (with n=0,1,2,…). The maximal pattern complexity of the word, introduced in the paper of T. Kamae and L. Zamboni [Sequence entropy and the maximal pattern complexity of infinite words. Ergod. Th. & Dynam. Sys.22(4) (2002), 1191–1199], is the maximum value of the pattern complexity of S with #S=k as a function of k=1,2,…. A substitution of constant length on an alphabet is a mapping from the alphabet to finite words on it of constant length not less than two. An infinite word is called a fixed point of the substitution if it stays the same after the substitution is applied. In this paper, we prove that the maximal pattern complexity of a fixed point of a substitution of constant length on {0,1} (as a function of k=1,2,…) is either bounded, a linear function of k, or 2k.

Partitions by congruent sets and optimal positions

AbstractLet X be a metrizable space with a continuous group or semi-group action G. Let D be a non-empty subset of X. Our problem is how to choose a fixed number of sets in {g−1D; g∈G}, say σ−1D with σ∈τ, to maximize the cardinality of the partition ℙ({σ−1D; σ∈τ}) generated by them. Let An infinite subset Σ of G is called an optimal position of the triple (X,G,D) if holds for any k=1,2,… and τ⊂Σ with #τ=k. In this paper, we discuss examples of the triple (X,G,D) admitting or not admitting an optimal position. Let X=G=ℝn (n≥1) , where the action g∈G to x∈X is the translation x−g. If D is the n-dimensional unit ball, then holds and the triple (X,G,D) admits an optimal position. In fact, if n≥2 and Σ is an infinite subset of G such that for some δ with 0<δ<1 , Σ⊂{x∈ℝn ; ‖x‖=δ}, and that any subset of Σ with cardinality n+1 is not on a hyperplane, then Σ is an optimal position of the triple (X,G,D) . We have determined the primitive factor of the uniform sets coming from these optimal positions. We also show that in the above setting with n=2 and the unit square D′ in place of the unit disk D, the maximal pattern complexity is unchanged and p*X,G,D′(k)=k2 −k+2 , but there is no optimal position.

Maximal pattern complexity of higher dimensional words

This paper studies the pattern complexity of n-dimensional words. We show that an n-recurrent but not n-periodic word has pattern complexity at least 2 k, which generalizes the result of [T. Kamae, H. Rao, Y.-M. Xue, Maximal pattern complexity of two dimension words, Theoret. Comput. Sci. 359 (1–3) (2006) 15–27] on two-dimensional words. Analytic directions of a word are defined and its topological properties play a crucial role in the proof. Accordingly n-dimensional pattern Sturmian words are defined. Irrational rotation words are proved to be pattern Sturmian. A new class of higher dimensional words, the simple Toeplitz words, are introduced. We show that they are also pattern Sturmian words.

Super-stationary set, subword problem and the complexity

Let Ω ⊂ { 0 , 1 } N be a nonempty closed set with N = { 0 , 1 , 2 , … } . For N = { N 0 < N 1 < N 2 < ⋯ } ⊂ N and ω ∈ { 0 , 1 } N , define ω [ N ] ∈ { 0 , 1 } N by ω [ N ] ( n ) ≔ ω ( N n ) ( n ∈ N ) and Ω [ N ] ≔ { ω [ N ] ∈ { 0 , 1 } N ; ω ∈ Ω } . We call Ω a super-stationary set if Ω [ N ] = Ω holds for any infinite subset N of N . Denoting Ω ′ the derived set (i.e. the set of accumulating points) of Ω and deg Ω = inf { d ; Ω ( d + 1 ) = 0̸ } with Ω ( 1 ) = Ω ′ , Ω ( 2 ) = ( Ω ′ ) ′ , … , it is known [T. Kamae, Uniform set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)] that for any nonempty closed subset Ω of { 0 , 1 } N such that there exists an infinite subset N of N with deg Ω [ N ] < ∞ , there exists an infinite subset M such that Ω [ M ] is a super-stationary set. Moreover, if deg Ω [ N ] = ∞ for any infinite subset N of N , then the maximal pattern complexity [T. Kamae, Uniform set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)] p Ω ∗ ( k ) is 2 k ( k = 1 , 2 , … ) . Thus, the uniform complexity functions are realized by the super-stationary sets [T. Kamae, Uniform set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)]. We call ξ ∈ { 0 , 1 } ∗ a super-subword of ω ∈ { 0 , 1 } N if there exists S = { s 1 < s 2 < ⋯ < s k } with k = | ξ | such that ξ = ω [ S ] ≔ ω ( s 1 ) ω ( s 2 ) ⋯ ω ( s k ) . Let P ( ξ ) be the set of ω ∈ { 0 , 1 } N having no super-subword ξ . Denote Q ( Ξ ) = ∪ ξ ∈ Ξ P ( ξ ) and P ( Ξ ) = ∩ ξ ∈ Ξ P ( ξ ) , where Ξ ⊂ { 0 , 1 } ∗ . In this paper, we prove that the class of super-stationary sets other than { 0 , 1 } N coincides with the class of Q ( Ξ ) for nonempty finite sets Ξ ⊂ { 0 , 1 } + . Moreover, it also coincides with the class of P ( L ( Ξ ) ) for nonempty finite sets Ξ ⊂ { 0 , 1 } + , where L ( Ξ ) is the set of minimal covers of Ξ . Using these expressions, we can calculate the complexity of super-stationary sets and prove that the complexity function of a super-stationary set in k is either 2 k or a polynomial function of k for large k . We also discuss the word problems related to the super-subwords.

Uniform sets and complexity

For a nonempty closed subset Ω of { 0 , 1 } Σ , where Σ is a countably infinite set, let p Ω ( S ) ≔ # π S Ω be the complexity function depending on the nonempty finite sets S ⊂ Σ , where # denotes the number of elements in a set and π S : { 0 , 1 } Σ → { 0 , 1 } S is the projection. Define the maximal pattern complexity function p Ω ∗ ( k ) ≔ sup S ; # S = k p Ω ( S ) as a function of k = 1 , 2 , … . We call Ω a uniform set if p Ω ( S ) depends only on # S = k , and the complexity function p Ω ( k ) ≔ p Ω ( S ) as a function of k = 1 , 2 , … is called the uniform complexity function of Ω . Of course, we have p Ω ( k ) = p Ω ∗ ( k ) in this case. Such uniform sets appear, for example, as the partitions generated by congruent sets in a space with optimal positionings, or they appear as the restrictions of a symbolic system to optimal windows. Let Ω ′ be the derived set (i.e. the set of accumulating points) of Ω and deg Ω ≔ inf { d ; Ω ( d + 1 ) = 0̸ } with Ω ( 1 ) = Ω ′ , Ω ( 2 ) = ( Ω ′ ) ′ , … . We prove that for any nonempty closed subset Ω of { 0 , 1 } N , where N = { 0 , 1 , 2 , … } , such that deg ( Ω ∘ ρ ) < ∞ for some injection ρ : N → N , there exists an increasing injection ϕ : N → N such that Ω ∘ ϕ ∘ ψ = Ω ∘ ϕ for any increasing injection ψ : N → N . Such a set Ω ∘ ϕ is called a super-stationary set. Moreover, if deg ( Ω ∘ ρ ) = ∞ for any injection ρ : N → N , then p Ω ∗ ( k ) = 2 k ( k = 1 , 2 , … ) holds. A uniform set Ω ⊂ { 0 , 1 } Σ is said to have a primitive factor [ Ω ∘ ϕ ] if there exists an injection ϕ : N → Σ such that Ω ∘ ϕ is a super-stationary set, where [ Ω ∘ ϕ ] is the isomorphic class containing Ω ∘ ϕ . Then, any uniform set has at least one primitive factor, and hence, any uniform complexity function is realized by the uniform complexity function of a super-stationary set. It follows that the uniform complexity function p Ω ( k ) is either 2 k for any k or a polynomial function of k for large k .

Maximal pattern complexity for Toeplitz words

The notion of the maximal pattern complexity of words was introduced by Kamae and Zamboni. In this paper, we obtain an almost exact formula for the maximal pattern complexity $p^*_\alpha(k)$ of Toeplitz words $\alpha$ on an alphabet ${\mathbb A}$ defined by a sequence of coding words $(\eta^{(n)})^\infty\in({\mathbb A}\cup\{?\})^{\mathbb N}(n=1,2,\dotsc)$ including just one ‘?’ in their cycles $\eta^{(n)}$ . Using this formula, we characterize pattern Sturmian words (i.e. $p^*_\alpha(k)=2k$ (for all $k$ )) in this class. Moreover, we give a characterization of simple Toeplitz words in the sense of Kamae and Zamboni in terms of pattern complexity. In the case where $\eta^{(1)}=\eta^{(2)}=\dotsb$ , we obtain the value $\lim_{k\to\infty}p_\alpha^*(k)/k$ . We construct a Toeplitz word $\alpha\in{\mathbb A}^{\mathbb N}$ with $\#\A=2$ such that $p^*_\alpha(k)=2^k~(k=1,2,\dotsc)$ , while Toeplitz words in our sense always have discrete spectra.

Language structure of pattern Sturmian words

Pattern Sturmian words introduced by Kamae and Zamboni [Sequence entropy and the maximal pattern complexity of infinite words, Ergodic Theory Dynamical Systems 22 (2002) 1191–1199; Maximal pattern complexity for discrete systems, Ergodic Theory Dynamical Systems 22 (2002) 1201–1214] are an analogy of Sturmian words for the maximal pattern complexity instead of the block complexity. So far, two kinds of recurrent pattern Sturmian words are known, namely, rotation words and Toeplitz words. But neither a structural characterization nor a reasonable classification of the recurrent pattern Sturmian words is known. In this paper, we introduce a new notion, pattern Sturmian sets, which are used to study the language structure of pattern Sturmian words. We prove that there are exactly two primitive structures for pattern Sturmian words. Consequently, we suggest a classification of pattern Sturmian words according to structures of pattern Sturmian sets and prove that there are at most three classes in this classification. Rotation words and Toeplitz words fall into two different classes, but no examples of words from the third class are known.

Maximal pattern complexity of two-dimensional words

The maximal pattern complexity of one-dimensional words has been studied in several papers [T. Kamae, L. Zamboni, Sequence entropy and the maximal pattern complexity of infinite words, Ergodic Theory Dynam. Systems 22(4) (2002) 1191–1199; T. Kamae, L. Zamboni, Maximal pattern complexity for discrete systems, Ergodic Theory Dynam. Systems 22(4) (2002) 1201–1214; T. Kamae, H. Rao, Pattern Complexity over ℓ letters, E. Comb. J., to appear; T. Kamae, Y.M. Xue, Two dimensional word with 2 k maximal pattern complexity, Osaka J. Math. 41(2) (2004) 257–265]. We study the maximal pattern complexity p α * ( k ) of two-dimensional words α . A two-dimensional version of the notion of strong recurrence is introduced. It is shown that if α is strongly recurrent, then either α is doubly periodic or p α * ( k ) ⩾ 2 k ( k = 1 , 2 , … ) . Accordingly, we define a two-dimensional pattern Sturmian word as a strongly recurrent word α with p α * ( k ) = 2 k . Examples of pattern Sturmian words are given.

Maximal pattern complexity of words over ℓ letters

The maximal pattern complexity function p α ∗ ( k ) of an infinite word α = α 0 α 1 α 2 ⋯ over ℓ letters, is introduced and studied by [3] , [4] . In the present paper we introduce two new techniques, the ascending chain of alphabets and the singular decomposition , to study the maximal pattern complexity. It is shown that if p α ∗ ( k ) < ℓ k holds for some k ≥ 1 , then α is periodic by projection . Accordingly we define a pattern Sturmian word over ℓ letters to be a word which is not periodic by projection and has maximal pattern complexity function p α ∗ ( k ) = ℓ k . Two classes of pattern Sturmian words are given. This generalizes the definition and results of [3] where ℓ = 2 .

Two dimensional word with 2k maximal pattern complexity

Two dimensional word with 2k maximal pattern complexity

Maximal pattern complexity for discrete systems

For an infinite word \alpha=\alpha_0\alpha_1\alpha_2\dotsc over a finite alphabet, the authors introduced a new notion of complexity called maximal pattern complexity defined by p_\alpha^*(k):=\sup_\tau\sharp\{\alpha_{n+\tau(0)}\alpha_{n+\tau(1)}\dots\alpha_{n+\tau(k-1)};n=0,1,2,\dotsc\} where the supremum is taken over all sequences of integers 0=\tau(0)&lt;\tau(1)&lt;\dotsc&lt;\tau(k-1) of length k. The authors proved that \alpha is aperiodic if and only if p_\alpha^*(k)\ge 2k for every k=1,2,\dotsc. A word \alpha with p_\alpha^*(k)=2k for every k\geq 1 is called pattern Sturmian. In this paper, we give a simple criterion to be pattern Sturmian and exhibit a new class of recurrent pattern Sturmian words which do not arise from rotations. We also investigate the maximal pattern complexity of various discrete dynamical systems including irrational rotations on the circle, and self-similar systems generated by substitutions. We show that, for each irrational rotation on the circle, there exists a twofold partition of the circle, with respect to which the system generated has full maximal pattern complexity with probability one. Using the arithmetic properties of the underlying numeration system associated to a substitution dynamical system, we prove that the maximal pattern complexity of the fixed point of the Rauzy substitution 1\mapsto 12, 2\mapsto 13, 3\mapsto 1 has exponential growth. It is well known that the system generated by the Rauzy substitution is isomorphic in measure to an irrational rotation on the 2-torus.

Sequence entropy and the maximal pattern complexity of infinite words

For an infinite word \alpha=\alpha_0\alpha_1\alpha_2\dots, over a finite alphabet A, we define the maximal pattern complexity by p_\alpha^*(k)=\sup_\tau\sharp\{\alpha_{n+\tau(0)} \alpha_{n+\tau(1)}\dots\alpha_{n+\tau(k-1)}; n=0,1,2,\dots\} where the ‘sup’ is taken over all subsequences 0=\tau(0)<\tau(1)<\dots<\tau(k-1) of integers of length k. We prove that \alpha is eventually periodic if and only if p_\alpha^*(k)\le 2k-1 for some k. Infinite words \alpha, with p_\alpha^*(k)=2k for any k, are called pattern Sturmian words and are studied.