抽象数据类型的双代数结构

Bialgebraic Structure of Abstract Data Types

对抽象数据类型的语法构造和动态行为的性质及两者的关系而言,单纯利用代数或共代数方法进行研究存在一定的不足.文中结合范畴论及分配律给出抽象数据类型的双代数结构,并通过λ-双代数自然地描述了语法构造与动态行为之间的转换关系;分别利用分配律对共代数函子及代数函子进行函子化提升,证明这种函子化提升可以将初始代数(或终结共代数)提升为初始(或终结)λ-双代数,并将其应用于递归及共递归函数的定义及计算中.实例表明,这种函子化提升可以扩展代数中的归纳原理和共代数中的共归纳原理,从而提高程序语言对抽象数据类型的描述与性质证明能力.

As algebraic or coalgebraic methods have some disadvantages in analyzing the relationships and properties between the syntactic constructions and the dynamic behaviors of Abstract data types,this paper presents a bialgebraic structure of Abstract data types based on the category theory and the distributive laws,uses λ-bialgebras to naturally describe the transformation between the syntactic constructions and the dynamic behaviors,and employs distributive laws to functorially lift coalgebraic and algebraic functors,thus lifting initial algebras(or final coalgebras) to initial(or final) λ-bialgebra.Moreover,the functorial lifting is applied to the definition and computation of recursive and corecursive functions.Case study indicates that,as the functorial lifting extends the inductive principles of algebras and the coinductive principles of coalgebras,it helps to improve the abilities of programming languages in describing or proving the properties of Abstract data types.