幺半计算机 III：可计算性与复杂性的余代数视角
Monoidal computer III: A coalgebraic view of computability and complexity
AI总结：
本文用余代数方法刻画幺半计算机，证明解释器与特化器的存在等价于通用状态空间的存在，从而为可计算性与复杂性研究提供图解语言。
摘要：
幺半计算机是一种内涵计算的范畴模型，其中许多不同的程序对应于相同的输入-输出行为。另一种计算模型的优势在于，范畴形式化应当为计算理论提供一种亟需的高层语言，它足够灵活，既能在实现细节无关紧要时将其抽象掉，又能在真正需要时将其纳入考虑。通过幺半范畴进行研究的一个显著特征是弦图的正式图形语言，它支持对程序与计算进行可视化推理。在本文中，我们给出了幺半计算机的余代数刻画。结果表明，使幺半范畴成为幺半计算机的解释器与特化器的可用性，等价于一个*通用状态空间*的存在性，该状态空间对任意输入类型和输出类型携带弱终状态机。能够在幺半计算机中编程状态机，使我们能够表示图灵机、捕捉其执行过程、计算其步数，以及例如它们所使用的存储单元。因此，幺半计算机的余代数视角为研究可计算性与复杂性提供了一种便捷的图解语言。
英文摘要：
Monoidal computer is a categorical model of intensional computation, where many different programs correspond to the same input-output behavior. The upshot of yet another model of computation is that a categorical formalism should provide a much needed high level language for theory of computation, flexible enough to allow abstracting away the low level implementation details when they are irrelevant, or taking them into account when they are genuinely needed. A salient feature of the approach through monoidal categories is the formal graphical language of string diagrams, which supports visual reasoning about programs and computations. In the present paper, we provide a coalgebraic characterization of monoidal computer. It turns out that the availability of interpreters and specializers, that make a monoidal category into a monoidal computer, is equivalent with the existence of a *universal state space*, that carries a weakly final state machine for any pair of input and output types. Being able to program state machines in monoidal computers allows us to represent Turing machines, to capture their execution, count their steps, as well as, e.g., the memory cells that they use. The coalgebraic view of monoidal computer thus provides a convenient diagrammatic language for studying computability and complexity.
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