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Sound and Complete Axiomatizations of Coalgebraic Language Equivalence

机译:等效代数语言的合理和完整公理化

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Coalgebras provide a uniform framework for studying dynamical systems, including several types of automata. In this article, we make use of the coalgebraic view on systems to investigate, in a uniform way, under which conditions calculi that are sound and complete with respect to behavioral equivalence can be extended to a coarser coalgebraic language equivalence, which arises from a generalized powerset construction that determinizes coalgebras. We show that soundness and completeness are established by proving that expressions modulo axioms of a calculus form the rational fixpoint of the given type functor. Our main result is that the rational fixpoint of the functor FT, where T is a monad describing the branching of the systems (e.g., non-determinism, weights, probability, etc.), has as a quotient the rational fixpoint of the deter-minized type functor F, a lifting of F to the category of T-algebras. We apply our framework to the concrete example of weighted automata, for which we present a new sound and complete calculus for weighted language equivalence. As a special case, we obtain nondeterministic automata in which we recover Rabinovich's sound and complete calculus for language equivalence.
机译:Coalgebras提供了一个统一的框架来研究动力系统,包括几种类型的自动机。在本文中,我们利用系统的Coalgebraic观点以统一的方式进行研究,在这种情况下,可以将行为等效方面合理且完整的演算扩展到较粗的Coalgebraic语言等效,这是由广义的确定煤场的动力装置构造。我们证明,通过证明微积分的模公理表达式形成给定类型函子的有理固定点,可以建立健全性和完整性。我们的主要结果是,函子FT的有理固定点(其中T是描述系统分支的单子)(例如,非确定性,权重,概率等)作为确定商的有理数。最小型函子F,将F提升为T代数的类别。我们将框架应用于加权自动机的具体示例,为此,我们提出了一种新的声音和完整的演算,以实现加权语言等效性。作为一种特殊情况,我们获得了不确定的自动机,在该自动机中,我们恢复了拉比诺维奇的声音并为语言对等完成了完整的演算。

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