论文标题
拓扑理论和自动机
Topological theories and automata
论文作者
论文摘要
该论文解释了布尔半度性和自动机及其概括中具有缺陷和价值的单元的拓扑理论之间的联系。有限状态自动机与普通语言密切相关。对于每一对普通语言和循环的普通语言，我们将拓扑理论与一维流形的拓扑理论相关联，与用该语言字母标记的零维缺陷相关联。该理论在布尔半程中占据价值。拓扑理论的普遍构建在这种情况下引起了与缺陷的一维共同体的布尔半连续性组合的单素类别。后一类可以被解释为与常规语言相关的标准结构的半连接刚性单体闭合，包括该语言和语法单体的最小确定性和非确定性有限状态自动机。循环语言扮演着正规器的角色，从而定义了这些结构的刚性封闭。当普通语言的单个点的状态空间描述了分布式晶格时，就有一种独特的相关循环语言，因此所产生的理论是布尔值TQFT。
The paper explains the connection between topological theories for one-manifolds with defects and values in the Boolean semiring and automata and their generalizations. Finite state automata are closely related to regular languages. To each pair of a regular language and a circular regular language we associate a topological theory for one-dimensional manifolds with zero-dimensional defects labelled by letters of the language. This theory takes values in the Boolean semiring. Universal construction of topological theories gives rise in this case to a monoidal category of Boolean semilinear combinations of one-dimensional cobordisms with defects modulo skein relations. The latter category can be interpreted as a semilinear rigid monoidal closure of standard structures associated to a regular language, including minimal deterministic and nondeterministic finite state automata for the language and the syntactic monoid. The circular language plays the role of a regularizer, allowing to define the rigid closure of these structures. When the state space of a single point for a regular language describes a distributive lattice, there is a unique associated circular language such that the resulting theory is a Boolean TQFT.
