两种认知模式：代数、余代数和语言
Two modes of recognition: algebra, coalgebra, and languages
摘要：
本文的目的是建立一个连接的两种方法之间的范畴语言理论：共代数和代数语言理论的单子。对于一对单子建模的分支和线性类型，我们定义了正规映射，推广正则语言在经典的非确定性自动机理论。这些映射是某些自动机的行为（即它们具有余代数性质），但它们源于Eilenberg-Moore代数及其同态（通过利用Eilenberg-Moore代数和饱和余代数之间的对偶）。给出一些额外的假设，我们表明，正规映射形成的Kleisli范畴的单子，这是分支和线性类型的组合的某个子范畴。此外，我们陈述了一个Kleene-like定理，该定理用在某些运算下闭的最小子范畴来刻画正则态射范畴。此外，每当分支类型单子被视为幂集单子，我们表明，正规映射被描述为某些函子，其余域是所有有限hom-set范畴的映射。我们在经典的非确定自动机，树自动机，模糊自动机和加权自动机上实例化我们的框架。
英文摘要：
The aim of the paper is to build a connection between two approaches towards categorical language theory: the coalgebraic and algebraic language theory for monads. For a pair of monads modelling the branching and the linear type we defined regular maps that generalize regular languages known in classical non-deterministic automata theory. These maps are behaviours of certain automata (i.e. they possess a coalgebraic nature), yet they arise from Eilenberg-Moore algebras and their homomorphisms (by exploiting duality between the category of Eilenberg-Moore algebras and saturated coalgebras). Given some additional assumptions, we show that regular maps form a certain subcategory of the Kleisli category for the monad which is the composition of the branching and linear type. Moreover, we state a Kleene-like theorem characterising the regular morphisms category in terms of the smallest subcategory closed under certain operations. Additionally, whenever the branching type monad is taken to be the powerset monad, we show that regular maps are described as maps recognized by certain functors whose codomains are categories with all finite hom-sets. We instantiate our framework on classical non-deterministic automata, tree automata, fuzzy automata and weighted automata.
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