nLab observational type theory

Contents

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
See also
References

Idea

What is called observational type theory (OTT) is a flavor of type theory in between extensional type theory and intensional type theory.

This may be regarded as a first-stage approximation to homotopy type theory (HoTT, and for more motivation see at higher observational type theory):

The propositions of OTT correspond to the h-propositions of HoTT, and the types in OTT correspond to h-sets in HoTT. The notion of equality on OTT is based on symmetric prosets, which is a special case of higher internal groupoids. Since equality is defined type-by-type, OTT enjoys function extensionality, and a form of propositional extensionality at least for a specified universe of propositions (not necessarily including all h-propositions).

There are a few technical differences (e.g. proofs of propositions are definitionally equal in OTT, whereas proofs of hprops are only propositionally equal in HoTT) but on the whole HoTT looks a lot like a higher homotopy version of OTT.

References

Observational type theory was introduced in:

Thorsten Altenkirch, Conor McBride, Towards observational type theory, draft (2006) [pdf, pdf, pdf]
Thorsten Altenkirch, Conor McBride, Wouter Swierstra, Observational Equality, Now!, PLPV ‘07: Proceedings of the 2007 workshop on Programming languages meets program verification (2007) 57-68 [ISBN:978-1-59593-677-6, doi:10.1145/1292597.1292608, pdf]

A blog post about an Agda implementation, including propositional extensionality (which is not mentioned in ABS07):

Conor McBride, Hier Soir, an OTT Hierarchy

The above comparison between OTT and HoTT is taken from

Dan Licata, comment (2012)

Metatheoretical properties of OTT are proved in

Loïc Pujet, Nicolas Tabareau, Observational Equality: Now for Good, POPL’22, pdf
Loïc Pujet, Nicolas Tabareau, Impredicative Observational Equality, POPL’23, pdf
Loïc Pujet, Nicolas Tabareau, Observational Equality Meets CIC, ESOP’24, pdf

Last revised on August 19, 2024 at 12:48:09. See the history of this page for a list of all contributions to it.