Partiality, Revisited

Altenkirch, Thorsten; Danielsson, Nils Anders; Kraus, Nicolai

doi:10.1007/978-3-662-54458-7_31

Cited by 24 publications

(10 citation statements)

References 17 publications

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“…2 The usual correspondence between type families and fibrations extends to algebras, and so we formulate the elimination rule for X as X being section inductive in the category of algebras in the following sense:…”

Section: The Section Induction Principlementioning

confidence: 99%

“…Instead of defining the real numbers as a quotient of sequences of rationals, a HIT is used to define them as the Cauchy completion of the rational numbers, with the quotienting happening simultaneously with the completion definition. Similarly, a definition of the partiality monad, which represents potentially diverging operations over a given type, was given using a HIT [2,13,35], again avoiding the axiom of choice when showing e.g. that the construction is a monad [12].…”

Section: Introductionmentioning

confidence: 99%

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Quotient Inductive-Inductive Types

Altenkirch

Capriotti

Dijkstra³

et al. 2018

Lecture Notes in Computer Science

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Abstract. Higher inductive types (HITs) in Homotopy TypeTheory allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types, and allow to define types with non-trivial higher equality types, such as spheres, suspensions and the torus. However, there are also interesting uses of HITs to define types satisfying uniqueness of equality proofs, such as the Cauchy reals, the partiality monad, and the well-typed 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 a general theory of HITs, there is not yet a theoretical foundation for the combination of equality constructors and induction-induction, despite 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. We further derive a section induction principle, stating that every algebra morphism into the algebra in question has a section, which is close to the intuitively expected elimination rules.

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Section: The Section Induction Principlementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Quotient Inductive-Inductive Types

Altenkirch

Capriotti

Dijkstra³

et al. 2018

Lecture Notes in Computer Science

Self Cite

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“…This allows specification of infinitely branching trees. Examples of infinitary QIITs in previous works include real, surreal numbers [7], ordinal numbers [8] and a partiality monad [9]. Of special note here is that the theory of QIIT signatures is itself large and infinitary, thus it can "eat itself", i.e.…”

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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“…We discuss some positive results (structures that have already been constructed) as well as some open problems (structures that have not already been constructed) in Sect. 3.…”

Section: Ulrik Buchholtzmentioning

confidence: 99%

“…10.5], the Cauchy-complete real numbers [52, Sect. 11.3], as well as the partiality monad [3]. I won't say more about these, since this Chapter is supposed to be about higher structure.…”

Section: Bmentioning

confidence: 99%

Higher Structures in Homotopy Type Theory

Buchholtz

2019

Synthese Library

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The intended model of the homotopy type theories used in Univalent Foundations is the ∞-category of homotopy types, also known as ∞-groupoids. The problem of higher structures is that of constructing the homotopy types needed for mathematics, especially those that aren't sets. The current repertoire of constructions, including the usual type formers and higher inductive types, suffice for many but not all of these. We discuss the problematic cases, typically those involving an infinite hierarchy of coherence data such as semi-simplicial types, as well as the problem of developing the meta-theory of homotopy type theories in Univalent Foundations. We also discuss some proposed solutions.

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Partiality, Revisited

Cited by 24 publications

References 17 publications

Quotient Inductive-Inductive Types

Quotient Inductive-Inductive Types

Large and Infinitary Quotient Inductive-Inductive Types

Higher Structures in Homotopy Type Theory

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