Type Theory based on Dependent Inductive and Coinductive Types

We develop a dependent type theory that is based purely on inductive and coinductive types, and the corresponding recursion and corecursion principles. This results in a type theory with a small set of rules, while still being fairly expressive. For example, all well-known basic types and type forme...

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Published in:Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science pp. 327 - 336
Main Authors: Basold, Henning, Geuvers, Herman
Format: Conference Proceeding
Language:English
Published: New York, NY, USA ACM 05.07.2016
Series:ACM Conferences
Subjects:
Theory of computation > Logic
Theory of computation > Models of computation > Computability
Inductive Types
Coinductive Types
Fibrations
Dependent Types
ISBN:9781450343916, 1450343910
Online Access:Get full text
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Abstract We develop a dependent type theory that is based purely on inductive and coinductive types, and the corresponding recursion and corecursion principles. This results in a type theory with a small set of rules, while still being fairly expressive. For example, all well-known basic types and type formers that are needed for using this type theory as a logic are definable: propositional connectives, like falsity, conjunction, disjunction, and function space, dependent function space, existential quantification, equality, natural numbers, vectors etc. The reduction relation on terms consists solely of a rule for recursion and a rule for corecursion. The reduction relations for well-known types arise from that. To further support the introduction of this new type theory, we also prove fundamental properties of its term calculus. Most importantly, we prove subject reduction and strong normalisation of the reduction relation, which gives computational meaning to the terms. The presented type theory is based on ideas from categorical logic that have been investigated before by the first author, and it extends Hagino's categorical data types to a dependently typed setting. By basing the type theory on concepts from category theory we maintain the duality between inductive and coinductive types, and it allows us to describe, for example, the function space as a coinductive type.
AbstractList We develop a dependent type theory that is based purely on inductive and coinductive types, and the corresponding recursion and corecursion principles. This results in a type theory with a small set of rules, while still being fairly expressive. For example, all well-known basic types and type formers that are needed for using this type theory as a logic are definable: propositional connectives, like falsity, conjunction, disjunction, and function space, dependent function space, existential quantification, equality, natural numbers, vectors etc. The reduction relation on terms consists solely of a rule for recursion and a rule for corecursion. The reduction relations for well-known types arise from that. To further support the introduction of this new type theory, we also prove fundamental properties of its term calculus. Most importantly, we prove subject reduction and strong normalisation of the reduction relation, which gives computational meaning to the terms. The presented type theory is based on ideas from categorical logic that have been investigated before by the first author, and it extends Hagino's categorical data types to a dependently typed setting. By basing the type theory on concepts from category theory we maintain the duality between inductive and coinductive types, and it allows us to describe, for example, the function space as a coinductive type.
Author Geuvers, Herman
Basold, Henning
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  givenname: Herman
  surname: Geuvers
  fullname: Geuvers, Herman
  email: herman@cs.ru.nl
  organization: Radboud University, Technical University Eindhoven
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Keywords Inductive Types
Coinductive Types
Fibrations
Dependent Types
Language English
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Snippet We develop a dependent type theory that is based purely on inductive and coinductive types, and the corresponding recursion and corecursion principles. This...
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StartPage 327
SubjectTerms Theory of computation -- Logic
Theory of computation -- Models of computation -- Computability
Title Type Theory based on Dependent Inductive and Coinductive Types
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