Abstract: Inductive data types mainly focus on the finite syntactic structures inductively in terms of algebras from the construction perspective, but have some disadvantages in describing dynamic behaviors. As their categorical dual nodons, coinductive data types aim to coinductivcly describe the observable behaviors of data types in terms of coalgebras from the observation perspective. We firstly gave the definitions of inductive data types in programming languages from the categorical and algebraic viewpoints. After that, we continued to present the definition of coinductive data types with coalgebras and analyze the corresponding corecursion operations according to the finality of coinductive data types. Finally, we pointed out how to use J,-bialgebras and distributive laws to combine inductive and coinductive data types and discuss the relations between syntactic constructions and dynamic behaviors of data types.

Key words: Inductive data type, Coinductive data type, Category theory, Algebras, Coalgebras, Bialgcbras