On Language Varieties Without Boolean Operations

The state complexity of a class of operations involving roots and boolean operations

Modifiers are a sets of functions acting on tuple of automata and allowing one to construct regular operations. We define and study the class of friendly modifiers that describes a class of regular operations involving compositions of boolean operations and roots. We also give an explicit tight bound for the state complexity of these operations.

State complexity of catenation combined with boolean operations

We exhaustively investigate possible combinations of a boolean operation together with a catenation. In many cases we prove and improve some conjectures by Brzozowski. For each family of operation, we endeavour to provide a common witness with a small size alphabet.

结合星的布尔运算和级联运算状态复杂性的泛证明

Universal Witnesses for State Complexity of Boolean Operations and Concatenation Combined with Star

研究了布尔运算和乘积（级联、连接）与星的结合的状态复杂度。我们推导出严格的上界对称的差异和差异的两种语言，其中之一或两者都是星号，并为产品的两个星号的语言。我们证明，以前发现的工会和交叉的语言与一个或两个星级的参数，产品的两种语言之一是星级，和两种语言的产品的明星的界限都可以满足最近推出的普遍证人和他们的变种。

State complexity of the star of a Boolean operation

Monsters and modifiers are two concepts recently developed in the state complexity theory. A monster is an automaton in which every function from states to states is represented by at least one letter. A modifier is a set of functions allowing one to transform a set of automata into one automaton. The paper describes a general strategy that can be used to compute the state complexity of many operations. We illustrate it on the problem of the star of a Boolean operation. After applying modifiers on monsters, the states of the resulting automata are assimilated to combinatorial objects: the tabl…

广义Eilenberg定理I：语言的局部变体

Generalized Eilenberg Theorem I: Local Varieties of Languages

我们调查代数和coalgebraic识别的语言之间的对偶性，以获得一个推广的本地版本的Eilenberg定理。这个定理指出，在导子下闭的字母表上的所有正则语言的布尔代数的格同构于所有由代数生成的幺半群的伪簇的格。通过将我们的方法应用于不同的范畴，我们得到了三个相关的结果：一个是由于Gehrke，Grigorieff和Pin，将布尔代数弱化为分配格，一个是将布尔代数弱化为并半格，最后一个是考虑二元域上的向量空间。

对称群与布尔运算的商复杂性

Symmetric Groups and Quotient Complexity of Boolean Operations

一个正则语言L的商复杂度是L的左商复杂度的个数，它与L的状态复杂度相同。设L和L'是商复杂度为m和n的二元正则语言，且接受L和L'的最小确定自动机的转移半群分别是m和n次对称群S_m和S_n。用o表示任何二元布尔运算，它不是常数，也不是只有一个参数的函数。对于m，n ≥ 2且（m，n）不在{（2，2），（3，4），（4，3），（4，4）}中，我们证明了LoL'的商复杂度为mn当且仅当（a）m不等于n或（b）m=n且S_m和S_n的基（有序生成元对）不共轭.对于（m，n）\in {（2，2），（3，4），（4，3），（4，4）}，我们给出例子来说明这不一定成立。在证明这些结果时，我们将一致极小性的概念推广到自动机的直积上。我们还建立了一个非平凡的布尔运算的复杂性和群论之间的联系。