Bootstrapping Inductive and Coinductive Types in HasCASL (bibtex)

by Lutz Schröder

Abstract:

We discuss the treatment of initial datatypes and final process types in the wide-spectrum language HasCASL. In particular, we present specifications that illustrate how datatypes and process types arise as bootstrapped concepts using HasCASL's type class mechanism, and we describe constructions of types of finite and infinite trees that establish the conservativity of datatype and process type declarations adhering to certain reasonable formats. The latter amounts to modifying known constructions from HOL to avoid unique choice; in categorical terminology, this means that we establish that quasitoposes with an internal natural numbers object support initial algebras and final coalgebras for a range of polynomial functors, thereby partially generalising corresponding results from topos theory. Moreover, we present similar constructions in categories of internal complete partial orders in quasitoposes.

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Reference:

Lutz Schröder: Bootstrapping Inductive and Coinductive Types in HasCASL, In Logical Methods in Computer Science, 4(4:17), pp. 1–27, 2008.

Bibtex Entry:

@Article{Schroder09,
  author = {Lutz Schr{\"o}der},
  title = {Bootstrapping Inductive and Coinductive Types in HasCASL},
  year = {2008},
  journal = {Logical Methods in Computer Science},
  volume = {4},
  pages = {1-27},
  number = {4:17},
  keywords = {datatypes, process types, software speci&#64257;cation, quasitoposes, partial {$\lambda$}-calculus.},
  url = {http://www.lmcs-online.org/ojs/viewarticle.php?id=379&layout=abstract},
  abstract = {We discuss the treatment of initial datatypes and final process types in the wide-spectrum language HasCASL. In particular, we present specifications that illustrate how datatypes and process types arise as bootstrapped concepts using HasCASL's type class mechanism, and we describe constructions of types of finite and infinite trees that establish the conservativity of datatype and process type declarations adhering to certain reasonable formats. The latter amounts to modifying known constructions from HOL to avoid unique choice; in categorical terminology, this means that we establish that quasitoposes with an internal natural numbers object support initial algebras and final coalgebras for a range of polynomial functors, thereby partially generalising corresponding results from topos theory. Moreover, we present similar constructions in categories of internal complete partial orders in quasitoposes.},
}