Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-24T19:40:35.946Z Has data issue: false hasContentIssue false
  • English
  • Français

Multisets in type theory

Part of: General logic

Published online by Cambridge University Press:  27 March 2019

HÅKON ROBBESTAD GYLTERUD
HÅKON ROBBESTAD GYLTERUD*
Affiliation:
Institutt for informatikk, Universitet i Bergen Postboks 7803, N - 5020 Bergen, Norway. e-mail: hakon.gylterud@uib.no
Get access Rights & Permissions [Opens in a new window]

Abstract

A multiset consists of elements, but the notion of a multiset is distinguished from that of a set by carrying information of how many times each element occurs in a given multiset. In this work we will investigate the notion of iterative multisets, where multisets are iteratively built up from other multisets, in the context Martin–Löf Type Theory, in the presence of Voevodsky’s Univalence Axiom.

In his 1978 paper, “the type theoretic interpretation of constructive set theory” Aczel introduced a model of constructive set theory in type theory, using a W-type quantifying over a universe, and an inductively defined equivalence relation on it. Our investigation takes this W-type and instead considers the identity type on it, which can be computed from the univalence axiom. Our thesis is that this gives a model of multisets. In order to demonstrate this, we adapt axioms of constructive set theory to multisets, and show that they hold for our model.

MSC classification

Primary: 03B15: Higher-order logic and type theory
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 1 , July 2020 , pp. 1 - 18
Copyright
© Cambridge Philosophical Society 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Aczel, P.. The type theoretic interpretation of constructive set theory. Logic Colloquirm 77, Ed. by MacIntyre, A., Pacholski, L. and Paris, J.. (North–Holland, Amsterdam-New York, 1978) pp 55–66Google Scholar
Aczel, P. and Michael, R.. Notes on Constructive Set Theory. Tech. Rep. Institut Mittag–Leffler (2001).Google Scholar
Awodey, S., Pelayo, Á. and Warren, M. A.. Voevodsky’s univalence axiom in homotopy type theory (2013) arXiv:1302.4731 [math.HO].CrossRefGoogle Scholar
Blizard, W. D.. Multiset theory Notre Dame Journal of Formal Logic 30.1 (1988) pp. 36–66.CrossRefGoogle Scholar
Danielsson, N. A.. Positive h-levels are closed under W. https://homotopytypetheory.org/2012/09/21/positive-h-levels-are-closed-under-w/, (2012).Google Scholar
Escardó, M. H.. A self-contained, brief and complete formulation of Voevodsky’s univalence axiom. (2018) arXiv:1803.02294.Google Scholar
Gylterud, H. R.. Formalisation of iterative multisets and sets in Agda, (2016) URL: http://staff.math.su.se/gylterud/agda/.Google Scholar
Martin–Löf, P.. Intuitionistic type theory. Notes by Giovanni Sambin. Vol. 1. Studies in Proof Theory (Bibliopolis, Naples, (1984)) pp. iv+91.Google Scholar
Nordström, B., Petersson, K. and Smith, J. M.. Programming in Martin–Löf’s Type Theory. (1990).Google Scholar
Rado, R.. The cardinal module and some theorems on families of sets. Annali di Matematica Pura ed Applicata 102.1 (1975), pp. 135–154.CrossRefGoogle Scholar
The Univalent Foundations Program (2013). Homotopy type theory: univalent foundations of mathematics. Institute for Advanced Study (2013). homotopytypetheory.org/book.Google Scholar
Wiener, N.. A simplification of the logic of relations. Proc. of Camb. Phil. Soc. 17, (1917), pp. 387–390.Google Scholar