natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
Inductive families generalize inductive types. Instead of defining a single type inductively, one simultaneously defines a whole family of types. An alternative term is “indexed inductive definition”.
A simple example of an inductive family is the type of vectors Vect n indexed by the dimension n. This is defined by two constructors: one for the empty vector of dimension 0, and another for the operation which constructs a vector of dimension n+1 by adding a component to a vector of dimension n. The family of finite types Fin n (indexed by n) can also be defined as an inductive family: the constructor 0 is in any Fin n, and there is a successor operation which constructs an element in Fin (n+1) from an element in Fin n.
By the identification of propositions as types, inductive families correspond to inductively defined predicates. For example, the identity type on a type A can be defined inductively by the reflexivity rule stipulating that a is identical to a for any a : A, that is, identity is the least reflexive relation. The identity family of types in intuitionistic type theory results from the identification of this relation with a family of types.
The identity types of an indexed W-type are another indexed W-type. This has been formalized by Huginin.
The inductive covers in formal topology and real analysis are an example of a higher inductive family.
Standard inductive types, W-types can be interpreted in any topos with natural numbers object (Moerdijk-Palmgren). Gambino and Hyland construct initial algebras for dependent polynomial functors. Indexed containers are the same as dependent polynomial functors. Indexed containers are claimed to form a foundation for inductive families.
van den Berg and Moerdijk show that (standard) W-types can be interpreted in certain model categories.
In homotopy type theory with universes, one can reduce indexed W-types to W-types. This has been formalized here, here and here. Sattler outlines a generalization of the reduction to homotopy type theory without the need of universes.
Historical references on the definition of inductive types.
A first type theoretic formulation of general inductive definitions:
The induction principle for identity types (also known as “path induction” or the “J-rule”) is first stated in:
and in the modern form of inference rules in:
The special case of inductive types now known as $\mathcal{W}$-types is first formulated in:
Per Martin-Löf, pp. 171 of: Constructive Mathematics and Computer Programming, in: Proceedings of the Sixth International Congress of Logic, Methodology and Philosophy of Science (1979), Studies in Logic and the Foundations of Mathematics 104 (1982) 153-175 $[$doi:10.1016/S0049-237X(09)70189-2, ISBN:978-0-444-85423-0$]$
Per Martin-Löf (notes by Giovanni Sambin), Intuitionistic type theory, Lecture notes Padua 1984, Bibliopolis, Napoli (1984) $[$pdf, pdf$]$
Early proposals for a general formal definition of inductive types:
Robert L. Constable, N. Paul Francis Mendler, Recursive definitions in type theory, in Logic of Programs 1985, Lecture Notes in Computer Science 193 Springer (1985) $[$doi:10.1007/3-540-15648-8_5$]$
Paul Francis Mendler, Inductive Definition in Type Theory, Cornell (1987) $[$hdl:1813/6710$]$
Roland Backhouse, On the Meaning and Construction of the Rules of Martin-Löf’s Theory of Types, in: Workshop on General Logic, ECS-LFCS-88-52 (1988) $[$pdf, pdf$]$
The modern notion of inductive types and inductive families in intensional type theory is independently due to
Peter Dybjer, Inductive sets and families in Martin-Löf’s type theory and their set-theoretic semantics, Logical frameworks (1991) 280-306 $[$doi:10.1017/CBO9780511569807.012, pdf$]$
Peter Dybjer, Inductive families, Formal Aspects of Computing 6 (1994) 440–465 $[$doi:10.1007/BF01211308, doi:10.1007/BF01211308, pdf$]$
and due to
which became the basis of the calculus of inductive constructions used in the Coq-proof assistant:
Frank Pfenning, Christine Paulin-Mohring, Inductively defined types in the Calculus of Constructions, in: Mathematical Foundations of Programming Semantics MFPS 1989, Lecture Notes in Computer Science 442, Springer (1990) $[$doi:10.1007/BFb0040259$]$
Christine Paulin-Mohring, Inductive definitions in the system Coq – Rules and Properties, in: Typed Lambda Calculi and Applications TLCA 1993, Lecture Notes in Computer Science 664 Springer (1993) $[$doi:10.1007/BFb0037116$]$
reviewed in
Zhaohui Luo, §9.2.2 in: Computation and Reasoning – A Type Theory for Computer Science, Clarendon Press (1994) $[$ISBN:9780198538356, pdf$]$
Christine Paulin-Mohring, §2.2. in: Introduction to the Calculus of Inductive Constructions, contribution to: Vienna Summer of Logic (2014) $[$hal:01094195, pdf, pdf slides$]$
with streamlined exposition in:
The generalization to inductive-recursive types is due to
Peter Dybjer, A general formulation of simultaneous inductive-recursive definitions in type theory, The Journal of Symbolic Logic 65 2 (2000) 525-549 $[$doi:10.2307/2586554, pdf$]$
Peter Dybjer, Anton Setzer, Indexed induction-recursion, in Proof Theory in Computer Science PTCS 2001. Lecture Notes in Computer Science2183 Springer (2001) $[$doi:10.1007/3-540-45504-3_7, pdf$]$
See also:
Thorsten Altenkirch, Neil Ghani, Peter Hancock, Conor McBride, and Peter Morris, Indexed containers (pdf)
Nicola Gambino and Martin Hyland, Wellfounded Trees and Dependent Polynomial Functors PDF
Benno van den Berg, Ieke Moerdijk, W-types in Homotopy Type Theory (arXiv:1307.2765)
Christian Sattler, On relating indexed W-types with ordinary ones slides
Last revised on January 8, 2024 at 17:36:22. See the history of this page for a list of all contributions to it.