Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type

Partiality, Revisited
The Partiality Monad as a Quotient Inductive-Inductive Type
Thorsten Altenkirch, Nils Anders Danielsson and Nicolai Kraus
Foundations of Software Science and Computation Structures, 20th International Conference, FOSSACS 2017. This is the authors' version of the paper. The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-662-54458-7_31. This version contains some insignificant improvements to the text that are not present in the published version. [pdf, highlighted Agda code, zip file with code]

Abstract

Capretta's delay monad can be used to model partial computations, but it has the "wrong" notion of built-in equality, strong bisimilarity. An alternative is to quotient the delay monad by the "right" notion of equality, weak bisimilarity. However, recent work by Chapman et al. suggests that it is impossible to define a monad structure on the resulting construction in common forms of type theory without assuming (instances of) the axiom of countable choice.

Using an idea from homotopy type theory—a higher inductive-inductive type—we construct a partiality monad without relying on countable choice. We prove that, in the presence of countable choice, our partiality monad is equivalent to the delay monad quotiented by weak bisimilarity. Furthermore we outline several applications.