Abstract Automata theory based on quantum logic has been established by Ying. Nondeterministic fuzzy finite automata theory has been proposed by Cao, and further generalized by Pan et al. In this paper, we propose the notion of nondeterministic finite automaton based on quantum logic whose underlying structure is a complete orthomodular lattice l, called nondeterministic l-valued finite automaton (Nl-FA, for short). We show that nondeterministic l-valued finite automata, deterministic l-valued finite automata, and nondeterministic l-valued finite automata with e-moves are equivalent if and only if l satisfies the distributive law. Next, two l-valued relations in terms of language equivalence are given. Some properties of the two l-valued relations under various operations such as reversal, union, product, and concatenation, are investigated. Finally, we give the degree of the closeness between two Nl-FAs with the same set of states to reflect the robustness, and obtain some interesting results. The theory developed in this paper enriches the automata theory based on quantum logic.

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基于量子逻辑的非确定性有限自动机：语言等价关系和鲁棒性

摘要 英建立了基于量子逻辑的自动机理论。非确定性模糊有限自动机理论已由曹提出，并由潘等人进一步推广。在本文中，我们提出了基于量子逻辑的非确定性有限自动机的概念，其底层结构是一个完整的正交模格 l，称为非确定性 l 值有限自动机（简称 Nl-FA）。我们表明，当且仅当 l 满足分配律时，非确定性左值有限自动机、确定性左值有限自动机和具有 e 移动的非确定性左值有限自动机是等效的。接下来，给出了语言等价方面的两个 l 值关系。研究了在反转、联合、乘积和串联等各种操作下的两个 l 值关系的一些性质。最后，我们给出了具有相同状态集的两个 Nl-FA 之间的接近程度来反映鲁棒性，并获得一些有趣的结果。本文发展的理论丰富了基于量子逻辑的自动机理论。