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

Altenkirch, Thorsten; Danielson, Nils Anders; Kraus, Nicolai

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.

Citation

Altenkirch, T., Danielson, N. A., & Kraus, N. (2017, April). Partiality, Revisited: The Partiality Monad as a Quotient Inductive-Inductive Type. Presented at 20th International Conference, FOSSACS 2017, Held as Part of the European Joint Conferences on Theory and Practice of Software, Uppsala, Sweden

• Presentation Conference Type: Edited Proceedings
• Conference Name: 20th International Conference, FOSSACS 2017, Held as Part of the European Joint Conferences on Theory and Practice of Software
• Start Date: Apr 22, 2017
• End Date: Apr 29, 2017
• Acceptance Date: Dec 22, 2016
• Online Publication Date: Mar 15, 2017
• Publication Date: Mar 16, 2017
• Deposit Date: Mar 24, 2017
• Publicly Available Date: Mar 24, 2017
• Journal: FOSSACS
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 10203 LNCS
• Pages: 534-549
• Series Title: Lecture Notes in Computer Science
• Series Number: 10203
• Series ISSN: 0302-9743
• Book Title: Foundations of Software Science and Computation Structures: FoSSaCS 2017
• Chapter Number: 31
• ISBN: 9783662544570
• DOI: https://doi.org/10.1007/978-3-662-54458-7_31
• Public URL: https://nottingham-repository.worktribe.com/output/832490
• Publisher URL: https://link.springer.com/chapter/10.1007%2F978-3-662-54458-7_31
• Additional Information: First Online: 16 March 2017; Conference Acronym: FoSSaCS; Conference Name: International Conference on Foundations of Software Science and Computation Structures; Conference City: Uppsala; Conference Country: Sweden; Conference Year: 2017; Conference Start Date: 24 April 2017; Conference End Date: 27 April 2017; Conference Number: 20; Conference ID: fossacs2017; Conference URL: http://www.etaps.org/index.php/2017/fossacs; Free to read: This content has been made available to all.
• Contract Date: Mar 24, 2017