A Categorical Semantics for Inductive-Inductive Definitions

Altenkirch, Thorsten; Morris, Peter D.; Forsberg, Fredrik Nordvall; Setzer, Anton

doi:10.1007/978-3-642-22944-2_6

Cited by 12 publications

(11 citation statements)

References 16 publications

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“…Elimination rules can also be formulated [5]. Here, we just give the elimination rules for the data type of sorted lists (Example 2) as an example, and show how one can use them to define a function which inserts a number into a sorted list.…”

Section: Elimination Rules By Examplementioning

confidence: 98%

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A Finite Axiomatisation of Inductive-Inductive Definitions

Forsberg

Setzer²

2012

Logic, Construction, Computation

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Induction-induction is a principle for mutually defining data types A : Set and B : A → Set. Both A and B are defined inductively, and the constructors for A can refer to B and vice versa. In addition, the constructor for B can refer to the constructor for A. Induction-induction occurs in a natural way when formalising dependent type theory in type theory. We give some examples of inductive-inductive definitions, such as the set of surreal numbers. We then give a new finite axiomatisation of the principle of induction-induction, and prove its consistency by constructing a model.

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Section: Elimination Rules By Examplementioning

confidence: 98%

“…It differs from our earlier axiomatisation [24] in that it is finite, and is hopefully easier to understand. The current article is also somewhat different in scope from our CALCO paper [5], which focuses on a categorical semantics and shows that the elimination rules (not treated here) are equivalent to the initiality of certain algebras.…”

Section: Introductionmentioning

confidence: 95%

A Finite Axiomatisation of Inductive-Inductive Definitions

Forsberg

Setzer²

2012

Logic, Construction, Computation

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“…The β-rules can be added to the system by pattern matching on the constructorsthese rules are the same for the recursor and the eliminator. The categorical semantics of inductive-inductive types has been explored in [6,26]. From a computational point of view inductiveinductive types are unproblematic and pattern matching can be reduced to using elimination constants which are derived from the type signature and constructor types.…”

Section: Inductive-inductive Typesmentioning

confidence: 99%

“…(1) can be addressed by using inductive-inductive definitions [6] (see section 2.1) -indeed doing Type Theory in Type Theory was one of the main motivations behind introducing inductive-inductive types. It seemed that this would also suffice to address (2), since we can define the conversion relation mutually with the rest of the syntax.…”

Section: Introductionmentioning

confidence: 99%

Type theory in type theory using quotient inductive types

AltenkirchThorsten

KaposiAmbrus

2016

SIGPLAN Not.

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A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription. AbstractWe present an internal formalisation of a type heory with dependent types in Type Theory using a special case of higher inductive types from Homotopy Type Theory which we call quotient inductive types (QITs). Our formalisation of type theory avoids referring to preterms or a typability relation but defines directly well typed objects by an inductive definition. We use the elimination principle to define the set-theoretic and logical predicate interpretation. The work has been formalized using the Agda system extended with QITs using postulates.

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“…Last, but not least, the starting idea of this paper is of course inspired by the dialgebras of Hagino [13]. These have also been applied to give semantics to induction-induction [4] schemes.…”

Section: Introductionmentioning

confidence: 99%

Dependent Inductive and Coinductive Types are Fibrational Dialgebras

Basold¹

2015

Electron. Proc. Theor. Comput. Sci.

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In this paper, I establish the categorical structure necessary to interpret dependent inductive and coinductive types. It is well-known that dependent type theories \`a la Martin-L\"of can be interpreted using fibrations. Modern theorem provers, however, are based on more sophisticated type systems that allow the definition of powerful inductive dependent types (known as inductive families) and, somewhat limited, coinductive dependent types. I define a class of functors on fibrations and show how data type definitions correspond to initial and final dialgebras for these functors. This description is also a proposal of how coinductive types should be treated in type theories, as they appear here simply as dual of inductive types. Finally, I show how dependent data types correspond to algebras and coalgebras, and give the correspondence to dependent polynomial functors.Comment: In Proceedings FICS 2015, arXiv:1509.0282

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A Categorical Semantics for Inductive-Inductive Definitions

Cited by 12 publications

References 16 publications

A Finite Axiomatisation of Inductive-Inductive Definitions

A Finite Axiomatisation of Inductive-Inductive Definitions

Type theory in type theory using quotient inductive types

Dependent Inductive and Coinductive Types are Fibrational Dialgebras

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