Quotient inductive-inductive types

Altenkirch, Thorsten; Capriotti, Paolo; Dijkstra, Gabe; Kraus, Nicolai; Nordvall Forsberg, Fredrik

Abstract

Higher inductive types (HITs) in Homotopy Type Theory (HoTT) allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types and allow to define types which are not sets in the sense of HoTT (i.e. do not satisfy uniqueness of equality proofs) such as spheres, suspensions and the torus. However, there are also interesting uses of HITs to define sets, such as the Cauchy reals, the partiality monad, and the internal, total syntax of type theory. In each of these examples we define several types that depend on each other mutually, i.e. they are inductive-inductive definitions. We call those HITs quotient inductive-inductive types (QIITs). Although there has been recent progress on the general theory of HITs, there isn't yet a theoretical foundation of the combination of equality constructors and induction-induction, despite having many interesting applications. In the present paper we present a first step towards a semantic definition of QIITs. In particular, we give an initial-algebra semantics and show that this is equivalent to the section induction principle, which justifies the intuitively expected elimination rules.

Citation

Altenkirch, T., Capriotti, P., Dijkstra, G., Kraus, N., & Nordvall Forsberg, F. (2018, April). Quotient inductive-inductive types. Presented at 21st International Conference, FOSSACS 2018, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2018, Thessaloniki, Greece

• Presentation Conference Type: Edited Proceedings
• Conference Name: 21st International Conference, FOSSACS 2018, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2018
• Start Date: Apr 14, 2018
• End Date: Apr 20, 2018
• Acceptance Date: Dec 22, 2017
• Online Publication Date: Apr 14, 2018
• Publication Date: 2018
• Deposit Date: Apr 18, 2018
• Publicly Available Date: Apr 18, 2018
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Pages: 293-310
• Series Title: Lecture notes in computer science
• Series Number: 10803
• Series ISSN: 1611-3349
• Book Title: Foundations of Software Science and Computation Structures
• ISBN: 9783319893655
• DOI: https://doi.org/10.1007/978-3-319-89366-2_16
• Keywords: Logic in Computer Science
• Public URL: https://nottingham-repository.worktribe.com/output/855899
• Publisher URL: https://link.springer.com/chapter/10.1007/978-3-319-89366-2_16
• Contract Date: Apr 18, 2018