Constructing quotient inductive-inductive types

Kaposi, Ambrus; Kovács, András; Altenkirch, Thorsten

doi:10.1145/3290315

Cited by 36 publications

(52 citation statements)

References 24 publications

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“…That unordered countably-branching trees are a QW-type is significant since no previous work on various subclasses of QITs (or indeed QIITs [18,9]) supports infinitary QITs [5,25,27,11,18,9]. See Example 5 for another, more substantial infinitary QW-type.…”

Section: Quotient-inductive Typesmentioning

confidence: 99%

Constructing Infinitary Quotient-Inductive Types

Fiore

Pitts

Steenkamp

2020

Lecture Notes in Computer Science

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This paper introduces an expressive class of quotient-inductive types, called QW-types. We show that in dependent type theory with uniqueness of identity proofs, even the infinitary case of QW-types can be encoded using the combination of inductive-inductive definitions, Hofmann-style quotient types, and Abel's size types. The latter, which provide a convenient constructive abstraction of what classically would be accomplished with transfinite ordinals, are used to prove termination of the recursive definitions of the elimination and computation properties of our encoding of QW-types. The development is formalized using the Agda theorem-prover.

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Section: Quotient-inductive Typesmentioning

confidence: 99%

Constructing Infinitary Quotient-Inductive Types

Fiore

Pitts

Steenkamp

2020

Lecture Notes in Computer Science

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“…A category with families (CwF), where all four underlying sets (of objects, morphisms, types and terms) are in Set i . Following notation in [4], we denote these respectively as Con : Set i , Sub : Con → Con → Set i , Ty : Con → Set i and Tm : (Γ : Con) → Ty Γ → Set i . We use id and -• -to denote identity and composition for substitution.…”

Section: Qiit Signaturesmentioning

confidence: 99%

“…include its own signature and provide its own metatheory. This was not possible previously in [4], where only finitary QIITs were described. In this paper we use self-representation to bootstrap the model theory of signatures, without having to assume any pre-existing internal syntax.…”

Section: Introductionmentioning

confidence: 95%

Large and Infinitary Quotient Inductive-Inductive Types

Kovács

Kaposi

2020

Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

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Quotient inductive-inductive types (QIITs) are generalized inductive types which allow sorts to be indexed over previously declared sorts, and allow usage of equality constructors. QIITs are especially useful for algebraic descriptions of type theories and constructive definitions of real, ordinal and surreal numbers. We develop new metatheory for large QI-ITs, large elimination, recursive equations and infinitary constructors. As in prior work, we describe QIITs using a type theory where each context represents a QIIT signature. However, in our case the theory of signatures can also describe its own signature, modulo universe sizes. We bootstrap the model theory of signatures using self-description and a Church-coded notion of signature, without using complicated raw syntax or assuming an existing internal QIIT of signatures. We give semantics to described QI-ITs by modeling each signature as a finitely complete CwF (category with families) of algebras. Compared to the case of finitary QIITs, we additionally need to show invariance under algebra isomorphisms in the semantics. We do this by modeling signature types as isofibrations. Finally, we show by a term model construction that every QIIT is constructible from the syntax of the theory of signatures.

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“…Specifically, the proof seems to require the construction of a filler for a 4-dimensional cube which is rather laborious. In [KKA19] it is shown that for QIITs (i.e., set-truncated HIITs) elimination and initiality are equivalent, but the extension to higher dimensional HIITs seems non-trivial. In particular it may require developing the higher order categorical structure of the category of algebras.…”

Section: Open Questionsmentioning

confidence: 99%

The Integers as a Higher Inductive Type

Altenkirch

Scoccola

2020

Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

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We consider the problem of defining the integers in Homotopy Type Theory (HoTT). We can define the type of integers as signed natural numbers (i.e., using a coproduct), but its induction principle is very inconvenient to work with, since it leads to an explosion of cases. An alternative is to use set-quotients, but here we need to use set-truncation to avoid non-trivial higher equalities. This results in a recursion principle that only allows us to define function into sets (types satisfying UIP). In this paper we consider higher inductive types using either a small universe or bi-invertible maps. These types represent integers without explicit set-truncation that are equivalent to the usual coproduct representation. This is an interesting example since it shows how some coherence problems can be handled in HoTT. We discuss some open questions triggered by this work. The proofs have been formally verified using cubical Agda.

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Constructing quotient inductive-inductive types

Cited by 36 publications

References 24 publications

Constructing Infinitary Quotient-Inductive Types

Constructing Infinitary Quotient-Inductive Types

Large and Infinitary Quotient Inductive-Inductive Types

The Integers as a Higher Inductive Type

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