In
type theory
In mathematics, logic, and computer science, a type theory is the formal presentation of a specific type system, and in general type theory is the academic study of type systems. Some type theories serve as alternatives to set theory as a fou ...
, a system has inductive types if it has facilities for creating a new type from constants and functions that create terms of that type. The feature serves a role similar to
data structures in a programming language and allows a type theory to add concepts like
number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers c ...
s,
relations, and
trees
In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...
. As the name suggests, inductive types can be self-referential, but usually only in a way that permits
structural recursion Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some other mathematical fields. It is a generalization of mathematical induction over natural nu ...
.
The standard example is encoding the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
s using
Peano's encoding.
Inductive nat : Type :=
, 0 : nat
, S : nat -> nat.
Here, a natural number is created either from the constant "0" or by applying the function "S" to another natural number. "S" is the
successor function which represents adding 1 to a number. Thus, "0" is zero, "S 0" is one, "S (S 0)" is two, "S (S (S 0))" is three, and so on.
Since their introduction, inductive types have been extended to encode more and more structures, while still being
predicative and supporting structural recursion.
Elimination
Inductive types usually come with a function to prove properties about them. Thus, "nat" may come with:
nat_elim : (forall P : nat -> Prop, (P 0) -> (forall n, P n -> P (S n)) -> (forall n, P n)).
This is the expected function for structural recursion for the type "nat".
Implementations
W- and M-types
W-types are
well-founded types in
intuitionistic type theory
Intuitionistic type theory (also known as constructive type theory, or Martin-Löf type theory) is a type theory and an alternative foundation of mathematics.
Intuitionistic type theory was created by Per Martin-Löf, a Swedish mathematician an ...
(ITT). They generalize natural numbers, lists, binary trees, and other "tree-shaped" data types. Let be a
universe of types. Given a type : and a
dependent family : → , one can form a W-type
. The type may be thought of as "labels" for the (potentially infinitely many) constructors of the inductive type being defined, whereas indicates the (potentially infinite)
arity of each constructor. W-types (resp. M-types) may also be understood as well-founded (resp. non-well-founded) trees with nodes labeled by elements : and where the node labeled by has ()-many subtrees. Each W-type is isomorphic to the initial algebra of a so-called
polynomial functor
In algebra, a polynomial functor is an endofunctor on the category \mathcal of finite-dimensional vector spaces that depends polynomially on vector spaces. For example, the symmetric powers V \mapsto \operatorname^n(V) and the exterior powers V \ ...
.
Let 0, 1, 2, etc. be finite types with inhabitants 1
1 : 1, 1
2, 2
2:2, etc. One may define the natural numbers as the W-type
with : 2 → is defined by (1
2) = 0 (representing the constructor for zero, which takes no arguments), and (2
2) = 1 (representing the successor function, which takes one argument).
One may define lists over a type : as
where
and 1
1 is the sole inhabitant of 1. The value of
corresponds to the constructor for the empty list, whereas the value of
corresponds to the constructor that appends to the beginning of another list.
The constructor for elements of a generic W-type
has type
We can also write this rule in the style of a
natural deduction In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use ax ...
proof,
The elimination rule for W-types works similarly to
structural induction Structural induction is a proof method that is used in mathematical logic (e.g., in the proof of Łoś' theorem), computer science, graph theory, and some other mathematical fields. It is a generalization of mathematical induction over natural nu ...
on trees. If, whenever a property (under the
propositions-as-types interpretation)
holds for all subtrees of a given tree it also holds for that tree, then it holds for all trees.
In extensional type theories, W-types (resp. M-types) can be defined up to
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
as
initial algebra
In mathematics, an initial algebra is an initial object in the category of -algebras for a given endofunctor . This initiality provides a general framework for induction and recursion.
Examples
Functor
Consider the endofunctor sending ...
s (resp. final coalgebras) for
polynomial functor
In algebra, a polynomial functor is an endofunctor on the category \mathcal of finite-dimensional vector spaces that depends polynomially on vector spaces. For example, the symmetric powers V \mapsto \operatorname^n(V) and the exterior powers V \ ...
s. In this case, the property of initiality (res. finality) corresponds directly to the appropriate induction principle. In intensional type theories with the
univalence axiom
In mathematical logic and computer science, homotopy type theory (HoTT ) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory a ...
, this correspondence holds up to homotopy (propositional equality).
M-types are
dual to W-types, they represent
coinductive (potentially infinite) data such as
streams. M-types can be derived from W-types.
Mutually inductive definitions
This technique allows ''some'' definitions of multiple types that depend on each other. For example, defining two
parity predicates on
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
s using two mutually inductive types in
Coq
Coq is an interactive theorem prover first released in 1989. It allows for expressing mathematical assertions, mechanically checks proofs of these assertions, helps find formal proofs, and extracts a certified program from the constructive proof ...
:
Inductive even : nat -> Prop :=
, zero_is_even : even O
, S_of_odd_is_even : (forall n:nat, odd n -> even (S n))
with odd : nat -> Prop :=
, S_of_even_is_odd : (forall n:nat, even n -> odd (S n)).
Induction-recursion
Induction-recursion started as a study into the limits of ITT. Once found, the limits were turned into rules that allowed defining new inductive types. These types could depend upon a function and the function on the type, as long as both were defined simultaneously.
Universe types can be defined using induction-recursion.
Induction-induction
Induction-induction
In intuitionistic type theory (ITT), some discipline within mathematical logic, induction-induction is for simultaneously declaring some inductive type and some inductive predicate over this type.
An inductive definition is given by rules for ge ...
allows definition of a type and a family of types at the same time. So, a type and a family of types
.
Higher inductive types
This is a current research area in
Homotopy Type Theory
In mathematical logic and computer science, homotopy type theory (HoTT ) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory a ...
(HoTT). HoTT differs from ITT by its identity type (equality). Higher inductive types not only define a new type with constants and functions that create elements of the type, but also new instances of the identity type that relate them.
A simple example is the type, which is defined with two constructors, a basepoint;
:
and a loop;
:
The existence of a new constructor for the identity type makes a higher inductive type.
See also
*
Coinduction
In computer science, coinduction is a technique for defining and proving properties of systems of concurrent interacting objects.
Coinduction is the mathematical dual to structural induction. Coinductively defined types are known as codata and ...
permits (effectively) infinite structures in type theory.
References
* {{cite book, author=Univalent Foundations Program, title=Homotopy Type Theory: Univalent Foundations of Mathematics, year=2013, publisher=Institute for Advanced Study, url=http://homotopytypetheory.org/book/
External links
Induction-Recursion SlidesInduction-Induction SlidesHigher Inductive Types: a tour of the menagerie
Type theory