Stone-type dualities provide a powerful mathematical framework for studying properties of logical systems. They have recently been fruitfully explored in understanding minimisation of various types of automata. In Bezhanishvili et al. (2012), a dual equivalence between a category of coalgebras and a category of algebras was used to explain minimisation. The algebraic semantics is dual to a coalgebraic semantics in which logical equivalence coincides with trace equivalence. It follows that maximal quotients of coalgebras correspond to minimal subobjects of algebras. Examples include partially observable deterministic finite automata, linear weighted automata viewed as coalgebras over finite-dimensional vector spaces, and belief automata, which are coalgebras on compact Hausdorff spaces. In Bonchi et al. (2014), Brzozowski's double-reversal minimisation algorithm for deterministic finite automata was described categorically and its correctness explained via the duality between reachability and observability. This work includes generalisations of Brzozowski's algorithm to Moore and weighted automata over commutative semirings. In this paper we propose a general categorical framework within which such minimisation algorithms can be understood. The goal is to provide a unifying perspective based on duality. Our framework consists of a stack of three interconnected adjunctions: a base dual adjunction that can be lifted to a dual adjunction between coalgebras and algebras and also to a dual adjunction between automata. The approach provides an abstract understanding of reachability and observability. We illustrate the general framework on range of concrete examples, including deterministic Kripke frames, weighted automata, topological automata (belief automata), and alternating automata.

中文翻译：

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逻辑形式的最小化

石头型二元性为研究逻辑系统的特性提供了强大的数学框架。他们最近在理解各种类型的自动机的最小化方面进行了富有成效的探索。在 Bezhanishvili 等人。(2012), 一类代数和一类代数之间的对偶等价被用来解释最小化。代数语义与逻辑等价与迹等价重合的合代数语义是对偶的。因此，余代数的最大商对应于代数的最小子对象。示例包括部分可观察的确定性有限自动机、被视为有限维向量空间上的余代数的线性加权自动机，以及作为紧凑 Hausdorff 空间上的余代数的置信自动机。在邦奇等人。(2014), Brzozowski' 确定性有限自动机的双反转最小化算法被明确描述，其正确性通过可达性和可观察性之间的对偶性来解释。这项工作包括将 Brzozowski 算法推广到 Moore 和交换半环上的加权自动机。在本文中，我们提出了一个通用的分类框架，可以在其中理解此类最小化算法。目标是提供基于二元性的统一视角。我们的框架由一堆三个相互关联的附属组成：一个基本的对偶附属，它可以提升为余代数和代数之间的对偶附属，以及自动机之间的对偶附属。该方法提供了对可达性和可观察性的抽象理解。我们说明了一系列具体例子的一般框架，