分解复杂性

Properties of a Ternary Infinite Word

We study the properties of the ternary infinite word p = 012102101021012101021012 ... , that is, the fixed point of the map h:0->01, 1->21, 2->0. We determine its factor complexity, critical exponent, and prove that it is 2-balanced. We compute its abelian complexity and determine the lengths of its bispecial factors. Finally, we give a characterization of p in terms of avoided factors.

Subword complexity and power avoidance

We begin a systematic study of the relations between subword complexity of infinite words and their power avoidance. Among other things, we show that -- the Thue-Morse word has the minimum possible subword complexity over all overlap-free binary words and all $(\frac 73)$-power-free binary words, but not over all $(\frac 73)^+$-power-free binary words; -- the twisted Thue-Morse word has the maximum possible subword complexity over all overlap-free binary words, but no word has the maximum subword complexity over all $(\frac 73)$-power-free binary words; -- if some word attains the minimum poss…

子字复杂度和因子集的分解

Subword complexity and decomposition of the set of factors

在本文中，我们探讨了一个新的层次的类的语言和无限的话，它与复杂性类。也就是说，我们说一种语言属于$L_k$类，如果它是$k$语言$S_1\cdots S_k$的一个子集，其中每个$S_i$中长度为$n$的单词的数量由一个常数限定。其因子集在$L_k$中的无限词类记为$W_k$。本文主要研究了无限字的子字复杂度与类$W_k$之间的关系，子字复杂度通常定义为长度为$n$的字的因子数。特别是，我们证明了类$W_{2}$符合类的无限词的线性复杂度。另一方面，尽管类$W_{k}$被包括在复杂性字类$O(n^{k-1})$中，但是这种包括对于$k> 2$是严格的。

Decompositions and complexity of linear automata

The Krohn-Rhodes complexity theory for pure (without linearity) automata is well-known. This theory uses an operation of wreath product as a decomposition tool. The main goal of the paper is to introduce the notion of complexity of linear automata. This notion is ultimately related with decompositions of linear automata. The study of these decompositions is the second objective of the paper. In order to define complexity for linear automata, we have to use three operations, namely, triangular product of linear automata, wreath product of pure automata and wreath product of a linear automaton w…

通过交替环分解简化$ω$-自动机的复杂度

The Complexity of Simplifying $ω$-Automata through the Alternating Cycle Decomposition

本文研究了计算交替环分解（ACD）及其DAG版本的复杂度，证明在Muller自动机接受条件的合适表示下可在多项式时间内完成，并据此推导出判定自动机类型性、奇偶指数及最小化接受条件颜色数的复杂度结果。