递归程序模式的范畴论解
The Category Theoretic Solution of Recursive Program Schemes
摘要：
本文提供了一个一般的帐户递归程序计划的概念，研究未解释和解释的解决方案。它可以被看作是代数语义学经典领域的范畴论版本。所需的总体假设确实很小：只在具有“足够的最终余代数”的类别中工作，我们展示了如何制定，解决和研究递归程序计划。我们的一般理论是代数的，因此避免使用有序或度量结构。我们的工作概括了以前的方法，这些方法通过隔离研究无限树中的替换所需的关键概念来使用这种额外的结构，包括二阶替换。作为我们的解释解决方案的特殊情况下，我们得到了通常的指称语义使用完全偏序，并使用完整的度量空间。我们的理论还包括隐式定义的对象，通常不采取相关的递归程序计划。例如，经典的康托三分之二集合作为递归程序方案的解释解（在我们的意义上）而失败。
英文摘要：
This paper provides a general account of the notion of recursive program schemes, studying both uninterpreted and interpreted solutions. It can be regarded as the category-theoretic version of the classical area of algebraic semantics. The overall assumptions needed are small indeed: working only in categories with "enough final coalgebras" we show how to formulate, solve, and study recursive program schemes. Our general theory is algebraic and so avoids using ordered, or metric structures. Our work generalizes the previous approaches which do use this extra structure by isolating the key concepts needed to study substitution in infinite trees, including second-order substitution. As special cases of our interpreted solutions we obtain the usual denotational semantics using complete partial orders, and the one using complete metric spaces. Our theory also encompasses implicitly defined objects which are not usually taken to be related to recursive program schemes. For example, the classical Cantor two-thirds set falls out as an interpreted solution (in our sense) of a recursive program scheme.
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