nLab parametric dependent type theory

Redirected from "modal internal parametricity".

Context

Type theory

natural deduction metalanguage, practical foundations

judgement
hypothetical judgement, sequent
- antecedents $\vdash$ consequent, succedents

type theory (dependent, intensional, observational type theory, homotopy type theory)

calculus of constructions

syntax object language

theory, axiom
proposition/type (propositions as types)
definition/proof/program (proofs as programs)
theorem

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language

homotopy levels

semantics

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Idea
Related concepts
References

Idea

Parametric dependent type theory is dependent type theory with parametricity (i.e. parametric polymorphism).

Parametric dependent type theories are distinguished by the kind of parametricity that they have:

External parametricity only applies to closed types and terms, and is a meta-theoretic statement about the type theory rather than something that is provable inside the theory itself.
Internal parametricity is a statement provable inside the theory itself and applies to all types and terms, but contradicts classical axioms such as the axiom of choice and excluded middle.
Modal internal parametricity is a statement also provable inside the theory itself, but only applies to types and terms guarded by a modal operator. Unlike the usual internal parametricity, modal internal parametricity is consistent with classical axioms such as the axiom of choice and excluded middle.

Related concepts

References

Jean-Philippe Bernardy, Patrik Jansson?, and Ross Paterson?. 2010. Parametricity and dependent types. In Proceeding of the 15th ACM SIGPLAN international conference on Functional programming, ICFP 2010, Baltimore, Maryland, USA, September 27-29, 2010, Paul Hudak and Stephanie Weirich (Eds.). ACM, 345–356. [doi:10.1145/1863543.1863592]
Jean-Philippe Bernardy and Guilhem Moulin?. 2012. A Computational Interpretation of Parametricity. In Proceedings of the 27th Annual IEEE Symposium on Logic in Computer Science, LICS 2012, Dubrovnik, Croatia, June 25-28, 2012. IEEE Computer Society, 135–144. [doi:10.1109/LICS.2012.25]
Robert Atkey, Neil Ghani and Patricia Johann, A Relationally Parametric Model of Dependent Type Theory, In Proceedings of the 41st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL 2014). 2014. (web)
Jean-Philippe Bernardy, Thierry Coquand, and Guilhem Moulin?. 2015. A Presheaf Model of Parametric Type Theory. In The 31st Conference on the Mathematical Foundations of Programming Semantics, MFPS 2015, Nijmegen, The Netherlands, June 22-25, 2015 (Electronic Notes in Theoretical Computer Science, Vol. 319), Dan R. Ghica (Ed.). Elsevier, 67–82. [doi:10.1016/j.entcs.2015.12.006]
Auke Bart Booij, Martín Hötzel Escardó, Peter LeFanu Lumsdaine, and Michael Shulman. 2016. Parametricity, Automorphisms of the Universe, and Excluded Middle. In 22nd International Conference on Types for Proofs and Programs, TYPES 2016, May 23-26, 2016, Novi Sad, Serbia (LIPIcs, Vol. 97), Silvia Ghilezan, Herman Geuvers, and Jelena Ivetic (Eds.). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 7:1–7:14. [doi:10.4230/LIPICS.TYPES.2016.7]
Guilhem Moulin?. Internalizing parametricity. PhD thesis, Chalmers University, 2016. [pdf]
Evan Cavallo. 2021. Higher Inductive Types and Internal Parametricity for Cubical Type Theory. Ph. D. Dissertation. Carnegie Mellon University, USA. [doi:10.1184/r1/14555691]
Evan Cavallo and Robert Harper. 2021. Internal Parametricity for Cubical Type Theory. Log. Methods Comput. Sci. 17, 4 (2021). [doi:10.46298/lmcs-17(4:5)2021]
Thorsten Altenkirch, Yorgo Chamoun?, Ambrus Kaposi, Michael Shulman, Internal parametricity, without an interval, Proceedings of the ACM on Programming Languages (POPL) 8 (2024) 2340-2369 [arXiv:2307.06448, doi:10.1145/3632920]
Antoine Van Muylder?, Andreas Nuyts, and Dominique Devriese?. 2024. Internal and Observational Parametricity for Cubical Agda. In Proceedings of the ACM on Programming LanguagesVolume 8Issue POPLArticle No.: 8pp 209–240. [doi:10.1145/3632850]
Astra Kolomatskaia, Michael Shulman, Displayed Type Theory and Semi-Simplicial Types [arXiv:2311.18781]
C.B. Aberlé, Parametricity via Cohesion [arXiv:2404.03825]
Daniel Gratzer, Jonathan Weinberger, Ulrik Buchholtz, Directed univalence in simplicial homotopy type theory (arXiv:2407.09146)

A proof assistant implementing an observational parametric type theory with both internal and external parametricity of variable arity:

Narya

Last revised on August 9, 2024 at 14:08:40. See the history of this page for a list of all contributions to it.