Categorical Theory of Automata

基于有限状态的结构是计算机科学、工程和自然科学的基本工具。对于不同类型的行为，存在着几十种类型的有限状态设备，例如有限或无限词或树的规则语言，加权或随机语言，以及数据语言。他们的理论有许多相似之处，例如，有限代数识别器和一类一元二阶逻辑以及类似算法的机器独立表征。然而，每种类型的系统大多是单独研究的。在CATHY中，我们将开发一个新的统一理论，涵盖有限状态设备的整个行为谱。一致性建立在应用范畴论概念的基础上，范畴论是一种强大的数学理论，非常擅长将相似性形式化，并将不同的结构系统地联系起来。我们的目标是建立一个完全成熟的理论，说明机器行为的“规律性”与它们的类型无关。特别是，在经典理论中，机器和有限代数结构之间通过对偶性存在着深刻的联系。事实上，这是Eilenberg/ reiterman型对应的基础，它们将规则行为的属性与代数识别器的属性联系起来，并允许用无限方程来说明这些属性。此外，拓扑方法在正则语言理论中带来了新的见解和对现象的更深入的理解。他们通过观察到语言的可识别性可以通过相应谓词的连续性等效地描述而进入这一阶段。一个通用框架的开始，例如，一个一般的Eilenberg/ reiterman类型的通信现在已经到位。然而，拓扑方法甚至机器的一般概念仍然缺失。在CATHY中，我们将通过在一般框架内开发拓扑方法和扩展对偶性来推进基于对偶的代数语言理论方法的最新进展。我们还将引入可识别行为的通用机器的概念。重点将放在他们的算法上，特别是基于协代数自动机学习的最新进展的学习算法。为了证明通用方法的健壮性，我们投入了大量的精力来开发通用框架的新的重要应用程序。这包括更温和的例子，比如文本语言，但也有挑战性的例子，比如数据语言、随机语言和量子有限自动机接受的规则语言。综上所述，我们将通过提供适用于各种有限状态行为的通用数学基础、结构和算法的综合体系，为有限状态设备的统一基础做出重大贡献。

Finite state-based structures are fundamental tools in computer science, engineering and in the natural sciences. There exist dozens of types of finite-state devices for different kinds of behavior, e.g. regular languages of finite or infinite words or trees, weighted or stochastic languages, and data languages. Their theories share many similarities, e.g. machine-independent characterizations by finite algebraic recognizers and by a type of monadic second-order logic as well as similar algorithmics. However, each type of system has mostly been studied separately.In CATHY, we will develop a new unifying theory that covers the whole spectrum of behaviors of finite-state devices. Uniformity is founded on applying the concepts of category theory, a powerful mathematical theory that is very good at formalizing similarities and relating different structures systematically. The goal is a fully fledged theory of 'regularity' of the behavior of machines independent of their type. In particular, in the classical theory there is a deep connection between machines and finite algebraic structures for regular behaviors via duality. In fact, this is the basis for Eilenberg/Reiterman-type correspondences relating properties of regular behaviors with properties of their algebraic recognizers and allowing the specification of such properties by profinite equations. In addition, topological methods have led to new insights and a closer understanding of phenomena in the theory of regular languages. They enter the stage through the observation that recognizability of a language can be equivalently described by continuity of the corresponding predicate. Beginnings of a generic framework featuring e.g. a general Eilenberg/Reiterman-type correspondence are now in place. However, topological methods or even a generic notion of machine are still missing.In CATHY, we will advance the state of the art in the duality based approach to algebraic language theory by developing topological methods and extended dualities within a generic framework. We will also introduce a notion of generic machine for recognizable behavior. A focus will be on their algorithmics, in particular on learning algorithms based on recent advances on coalgebraic automata learning. To demonstrate the robustness of the generic approach we devote substantial efforts towards developing new non-trivial applications of the generic framework. This includes more tame instances such as text languages, but also challenging ones such as data languages, stochastic languages, and regular languages accepted by quantum finite automata. Summing up, we will substantially contribute to unified foundations for finite-state devices by providing a comprehensive body of generic mathematical foundations, constructions and algorithms applicable to a wide range of types of finite-state behaviors.